\documentstyle[12pt]{article}
%\documentclass[12pt]{amsart}
\usepackage{amssymb}
\textheight=23.7cm
\textwidth=16cm
\topmargin=0pt
\oddsidemargin=0pt
\evensidemargin=0pt
\frenchspacing
\newtheorem{theorem}{Theorem}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{conjecture}{Conjecture}
\newtheorem{remark}{Remark}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newcommand*{\sm}{\setminus}
\newcommand*{\rra}{\rightrightarrows}
\newcommand*{\rla}{\rightleftarrows}
\newcommand*{\ua}{\uparrow}
\newcommand*{\da}{\downarrow}
\newcommand*{\CC}{{\mathfrak C}}
\newcommand*{\K}{{\mathcal K}}
\newcommand*{\LL}{{\mathcal L}}
\newcommand*{\Z}{{\mathbb Z}}
\newcommand*{\R}{{\mathbb R}}
\newcommand*{\RP}{{\mathbb {RP}}}
%\newcommand*{\bnm}[2]{\binom{#1}{#2}}
\newcommand*{\bnm}[2]{{#1}\choose{#2}}
\begin{document}

\title[Invariants of self-intersecting curves]{First order
invariants and cohomology of spaces of embeddings of
self-intersecting curves into $\R^n$}
\author{V.A.~Vassiliev}
\thanks{Supported in part by the Russian President's grant
��-1972.2003.01, the program ``Modern problems of theoretical
mathematics'' of the Mathematical section of RAS, and by the grant
ACI-NIM-2004-243 (Noeuds et tresses)} \date{}

\maketitle

\begin{abstract}
We study cohomology classes of the space of generic immersions
$\R^1 \to \R^n,$ $n \ge 3,$ with a fixed set of transversal
self-intersections; in particular, isotopy invariants of such
immersions for $n=3$. For $n>3$ lower cohomology groups of this space
are calculated, for $n=3$ the groups of {\em first order invariants}
of such immersions are defined and calculated. The representability of
these invariants by rational combinatorial formulas, generalizing the
classical formula for the linking number of two curves in
$\R^3$, is investigated. The existence of such combinatorial formulas
with half-integer coefficients is proved, and the topological obstruction
to the integrality of these combinatorial formulas is constructed.
As a corollary, it is proved, that one of 4th order knot invariants
cannot be represented by an integral Polyak-Viro formula.

The structure of the investigated cohomology groups depends on the
existence of a planar curve with the given self-intersection type.
On the other hand, from the self-intersection type one constructs
automatically a chain complex calculating these cohomology
groups, which gives an easy homological criterion for the
existence of such a planar curve.
\end{abstract}

%\maketitle


\section{Introduction}

\begin{definition} \rm
For any natural $m$, a {\em $m$-chord diagram} in $\R^1$ is an arbitrary
collection of $2m$ distinct points in $\R^1$, matched somehow into pairs.
\end{definition}

These pairs are depicted by smooth arcs (``chords''), connecting the points
of any pair; thus the chord diagrams are depicted by pictures like
ones in Fig.~\ref{exchd}.
\begin{figure}
\unitlength 2.00mm
\linethickness{0.4pt}
\begin{picture}(14.00,3.00)
\put(00.00,1.00){\line(1,0){14.00}}
\put(4.00,1.00){\oval(4.00,4.00)[t]}
\put(10.00,1.00){\oval(4.00,4.00)[t]}
\end{picture}
\mbox{ \qquad  \qquad }
\unitlength 2.00mm
\linethickness{0.4pt}
\begin{picture}(14.00,4.00)
\put(0.00,1.00){\line(1,0){14.00}}
\put(5.00,1.00){\oval(6.00,5.00)[t]}
\put(9.00,1.00){\oval(6.00,3.00)[t]}
\put(7.00,1.00){\oval(6.00,4.00)[b]}
\end{picture}
\mbox{ \qquad  \qquad }
\unitlength 2.00mm
\linethickness{0.4pt}
\begin{picture}(10.00,3.00)
\put(00.00,1.00){\line(1,0){10.00}}
\put(4.00,1.00){\oval(4.00,4.00)[t]}
\put(6.00,1.00){\oval(4.00,4.00)[b]}
\end{picture}
\caption{Examples of chord diagrams}
\label{exchd}
\end{figure}
Let us fix some $m$-chord diagram $\Theta$, and also a
linear embedding $l: \R^1 \to \R^n$.

\begin{definition} \rm
The space $\K(\Theta,n)$ is the space of all $C^\infty$-smooth
maps $f:\R^1 \to \R^n$ such that

a) $f(x)=f(y)$ for any of $m$ pairs $(x,y) \subset \R^1$, constituting
the chord diagram $\Theta$;

b) $f$ coincides with the standard embedding
$l$ outside some compact set in $\R^1$.

The {\em discriminant}  $\Sigma(\Theta,n)$ in this space is the set of all
maps $f \in \K(\Theta,n)$, having at least one additional
self-intersection (in addition to $m$ ones following from the fact that
$f \in \K(\Theta,n)$) or a point at which $f'=0$, or a self-tangency at
some of these $m$ self-intersection points. The image of the
embedding $l$ is called the {\em axis} of the space $\R^n$.
\end{definition}

It is obvious that $\K(\Theta,n)$ is an affine space, the discriminant
is closed in it, and for $n>2$ the complement of the discriminant is dense in
$\K(\Theta,n).$

We study the cohomology classes of the complementary space
\begin{equation}
\K(\Theta,n) \sm \Sigma(\Theta, n),
\label{gene}
\end{equation}
in particular the numerical isotopy invariants of {\em generic} maps
$f \in \K(\Theta,3)$ (tautologically, such invariants are the 0-dimensional
cohomology classes of the space (\ref{gene}) with $n=3$).

In particular, we calculate two lower non-trivial cohomology groups
of this space for $n>4$ and one lower group for $n=4$. To describe them,
let us give two definitions more.

\begin{definition}[see \cite{CDL}] \rm
The {\em intersection graph}, related with the chord diagram $\Theta$,
is the graph, whose vertices correspond to the chords of the diagram
$\Theta$, and two vertices are connected by an edge if and only if
the endpoints of the corresponding chords alternate in
$\R^1$ (i.e. two endpoints of any of these chords are separated by
an endpoint of the other chord). The number of connected components of the
intersection graph is denoted by $\#(\Theta)$. The chord diagram
$\Theta$ is called {\em irreducible} if $\#(\Theta)=1.$
\end{definition}

For instance, the intersection graphs of three diagrams of Fig.~\ref{exchd}
are correspondingly a pair of isolated points, a triangle, and a segment.
With any connected component of the intersection graph in $\Theta$
a chord subdiagram of $\Theta$ is related: it is the union of all chords of
$\Theta$, corresponding to the vertices of this component.

\begin{definition} \rm
The chord diagram $\Theta$ is {\em planar}, if there exists an immersion
$\R^1 \to \R^2$ of class $\K(\Theta,2) \sm \Sigma(\Theta,2)$. For instance,
two first diagrams of Fig.~\ref{exchd} are planar, and the third one is not.
The number of connected components of the intersection graph, such that
the corresponding chord subdiagrams are planar (respectively, non-planar)
is denoted by $\boxminus(\Theta)$ (respectively, $\boxplus(\Theta)$).
\end{definition}

\begin{theorem}
\label{gomolm}
For any $n>3$ and any $m$-chord diagram $\Theta$, the group
$H^i(\K(\Theta,n) \sm \Sigma(\Theta, n))$

a) is trivial for all positive $i$ such that $i<n-3$ or $ n-2 < i <2n-6$;

b) for $i=n-3$ is free Abelian of rank
${\bnm{m}{2}}+ \#(\Theta)$ if $n$ is odd and of rank
${\bnm{m}{2}}+ \boxminus(\Theta)$ if $n$ is even;

c) for $i=n-2$ has rank equal to $\#(\Theta)$ if $n$ is odd, is
isomorphic to $\Z^{\boxminus(\Theta)}\oplus \Z_2^{\boxplus(\Theta)}$
if $n$ is even and greater than 4, and contains a subgroup isomorphic to
$\Z^{\boxminus(\Theta)}\oplus \Z_2^{\boxplus(\Theta)}$
if $n=4$.
\end{theorem}

This theorem is a direct corollary of theorems
\ref{red}, \ref{homoddi} and \ref{homeveni} formulated and proved below.
\hfill $\square$
\medskip

On the other hand, we construct explicitly a chain complex, calculating
these homology groups, from the chord diagram $\Theta$. This gives us a
linear algebraic algorithm recognizing the planarity of chord diagrams;
concerning the latter problem see also \cite{LM}, \cite{CE}, \cite{vasya}.

The ring
\begin{equation}
H^*(\K(\Theta,n) \sm \Sigma(\Theta, n))
\label{ring}
\end{equation}
admits a natural increasing filtration similar to the filtration of the
cohomology group of the space of knots, introduced in
\cite{knsp} and in particular defining the notion of a finite
order invariant of knots in $\R^3$.  We calculate here the first
non-trivial term of this filtration, i.e. the subgroup
\begin{equation}
H^*_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))
\label{perpor}
\end{equation}
of  {\em first order cohomology classes} in the group (\ref{ring}).
Further, we construct combinatorial formulas for these cohomology classes
(in particular, for invariants if $n=3$) and indicate the topological
obstruction to the integrality of these formulas.
This obstruction is simultaneously the first obstruction to the integrality
of {\em Polyak--Viro combinatorial formulas \cite{PV}} for knot invariants
of order $m+1$; using it we show that the Polyak--Viro formulas for one
of basic knot invariants of order 4 necessarily include coefficients with
even positive denominators.

The definition of the first order cohomology groups will be given in
\S \ref{tavtol}, and now the most important example will be considered.
If $n=3$, then 0-dimensional cohomology classes of the space
(\ref{gene}) are the invariants of generic (i.e. non-discriminant) maps
of class $\K(\Theta,3)$. Such first order invariants can be distinguished
by the following condition: the jump of the value of such an invariant
at the simplest local surgery
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(19.40,4.80)
\put(0.70,2.40){\line(1,0){4.60}}
\put(3.00,2.80){\line(0,1){2.00}}
\put(3.00,2.00){\line(0,-1){2.00}}
\put(10.00,2.40){\makebox(0,0)[cc]{$\Rightarrow$}}
\put(17.00,4.70){\line(0,-1){4.60}}
\put(16.60,2.40){\line(-1,0){2.00}}
\put(17.40,2.40){\line(1,0){2.00}}
\end{picture}
(occurring at an instant additional self-intersection of the curve
$f(\R^1)$), i.e. the difference of values of the invariant on such curves
after and before the surgery, depends only on the behavior of our curve
close to this additional  self-intersection point (and on the disposition of
the points of the preimage of this self-intersection in $\R^1$
among the points of the chord diagram $\Theta$). This condition can be
expressed also in the terms of the {\em planar diagram} of the curve
$f(\R^1)$, i.e. of its projection into the plane $\R^2$, supplied with the
information on which its branch goes above the other at the crossing
points of this projection, not occurring from the self-intersections
of the curve $f(\R^1)$ itself. Namely, any first order invariant should
satisfy the condition
\begin{equation}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(124.00,32.00)
\put(10.00,19.00){\line(1,1){9.00}}
\put(10.00,28.00){\line(1,-1){4.00}}
\put(19.00,19.00){\line(-1,1){4.00}}
\put(10.00,4.00){\line(1,1){9.00}}
\put(10.00,13.00){\line(1,-1){4.00}}
\put(19.00,4.00){\line(-1,1){4.00}}
\put(43.00,28.00){\line(1,-1){9.00}}
\put(52.00,28.00){\line(-1,-1){4.00}}
\put(43.00,19.00){\line(1,1){4.00}}
\put(43.00,4.00){\line(1,1){9.00}}
\put(43.00,13.00){\line(1,-1){4.00}}
\put(52.00,4.00){\line(-1,1){4.00}}
\put(76.00,19.00){\line(1,1){9.00}}
\put(76.00,28.00){\line(1,-1){4.00}}
\put(85.00,19.00){\line(-1,1){4.00}}
\put(76.00,13.00){\line(1,-1){9.00}}
\put(85.00,13.00){\line(-1,-1){4.00}}
\put(76.00,4.00){\line(1,1){4.00}}
\put(109.00,28.00){\line(1,-1){9.00}}
\put(109.00,19.00){\line(1,1){4.00}}
\put(118.00,28.00){\line(-1,-1){4.00}}
\put(109.00,13.00){\line(1,-1){9.00}}
\put(109.00,4.00){\line(1,1){4.00}}
\put(118.00,13.00){\line(-1,-1){4.00}}
\put(2.00,16.00){\makebox(0,0)[cc]{{\Large $F($}}}
\put(25.00,16.00){\makebox(0,0)[cc]{\Large $)$}}
\put(29.00,16.00){\makebox(0,0)[cc]{\Large $-$}}
\put(35.00,16.00){\makebox(0,0)[cc]{{\Large $F($}}}
\put(58.00,16.00){\makebox(0,0)[cc]{\Large $)$}}
\put(62.00,16.00){\makebox(0,0)[cc]{\Large =}}
\put(68.00,16.00){\makebox(0,0)[cc]{{\Large $F($}}}
\put(91.00,16.00){\makebox(0,0)[cc]{\Large $)$}}
\put(95.00,16.00){\makebox(0,0)[cc]{\Large $-$}}
\put(101.00,16.00){\makebox(0,0)[cc]{{\Large $F($}}}
\put(124.00,16.00){\makebox(0,0)[cc]{\Large $)$}}
\put(7.00,0.00){\line(0,1){32.00}}
\put(22.00,32.00){\line(0,-1){32.00}}
\put(40.00,0.00){\line(0,1){32.00}}
\put(73.00,0.00){\line(0,1){32.00}}
\put(106.00,0.00){\line(0,1){32.00}}
\put(55.00,32.00){\line(0,-1){32.00}}
\put(88.00,32.00){\line(0,-1){32.00}}
\put(121.00,32.00){\line(0,-1){32.00}}
\end{picture}
\end{center}
\label{cret}
\end{equation}
(cf. \cite{bw}, \cite{BN}); here $F$ is our invariant, and the expressions in
parentheses indicate arbitrary four maps of class
$\K(\Theta,n) \sm \Sigma(\Theta, n),$
whose images coincide outside some two small discs, and in these discs
they behave as is shown in the picture. The equality (\ref{cret}) should hold
for any such quadruple of maps of this class, independently of their (common)
behavior outside these discs.

\begin{theorem}
\label{red}
For any $n>3$ and any $m$-chord diagram $\Theta$, the group
$H^i(\K(\Theta,n) \sm \Sigma(\Theta, n))$
is trivial for positive $i<n-3$ and canonically coincides with its subgroup
$H^{i}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$
for $n-3\le i < 2n-6$.
\end{theorem}

As we shall see from Theorems \ref{homoddi}, \ref{homeveni}, this subgroup
$H^{i}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$
is non-trivial only if $i$ is equal $n-3$ or $n-2$.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(50.00,55.00)
\put(0.00,5.00){\vector(1,0){50.00}}
\put(40.00,5.00){\vector(0,1){50.00}}
\put(40.00,5.00){\line(-1,2){25.00}}
\bezier{24}(35.00,5.00)(35.00,8.00)(38.00,9.00)
\put(23.00,8.50){\makebox(0,0)[cc]{$\tan \alpha = n-2$}}
\put(32.00,21.00){\line(1,0){8.00}}
\put(36.00,13.00){\line(1,0){4.00}}
\put(48.00,7.50){\makebox(0,0)[cc]{$p$}}
\put(43.00,53.00){\makebox(0,0)[cc]{$q$}}
\put(36.00,13.00){\line(0,1){40.00}}
\put(32.00,53.00){\line(0,-1){32.00}}
\put(28.00,29.00){\line(0,1){24.00}}
\put(24.00,53.00){\line(0,-1){16.00}}
\put(20.00,45.00){\line(0,1){8.00}}
\put(36.00,17.00){\line(1,0){4.00}}
\put(42.50,3.00){\makebox(0,0)[cc]{$0$}}
\put(38.00,3.00){\makebox(0,0)[cc]{$-1$}}
\put(36.00,25.00){\line(-1,0){4.00}}
\put(28.00,29.00){\line(1,0){8.00}}
\put(36.00,33.00){\line(-1,0){8.00}}
\put(24.00,37.00){\line(1,0){12.00}}
\put(36.00,41.00){\line(-1,0){12.00}}
\put(20.00,45.00){\line(1,0){16.00}}
\put(36.00,49.00){\line(-1,0){16.00}}
\put(36.00,53.00){\line(-1,0){20.00}}
\end{picture}
\caption{Support of the spectral sequence}
\label{sse}
\end{center}
\end{figure}

\medskip
{\em Proof of theorem \ref{red}.} Our filtration in the ring (\ref{ring})
is induced by some spectral sequence calculating elements of this ring
and defined in \S \ref{ssse} below. Non-trivial terms $E^{p,q}_r$
of this spectral sequence with $r \ge 1$ lie in the wedge
$\{p<0, q+(n-2)p \ge 0\}$ (see Fig. \ref{sse}, corresponding to the case
$n=4$). For $n > 3$ this spectral sequence converges to entire group
$H^*(\K(\Theta,n) \sm \Sigma(\Theta,n))$
(while for $n=3$ it converges only to some its important subgroup, the
group of finite order homology classes). Theorem \ref{red} follows
immediately from the existence and explicit shape of this spectral
sequence.
\hfill $\square$
\medskip

The group $H^{i}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$ is
described in the following two theorems.

\begin{theorem}
\label{homoddi}
Let $n$ be odd, $n>1$. Then for any $m$-chord diagram $\Theta$,
the group $H^i_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$

a) is trivial for $i$ not equal to $n-3$ or $n-2$;

b) for $i=n-2$ does not contain 2-torsion, and its rank is equal to the
number of connected components of the intersection graph of the diagram
$\Theta$;

c) for $i=n-3$ is free Abelian, with rank equal to the sum of the
binomial coefficient ${\bnm{m}{2}}$ and the number of connected
components of $\Theta$.
\end{theorem}

\begin{theorem}
\label{homeveni}
Let $n$ be even, $n >2$. Then for any $m$-chord diagram $\Theta$, the group
$H^i_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$

a) is trivial for $i$ not equal to $n-3$ or $n-2$;

b) for $i=n-2$
is the direct sum of summands $\Z$, corresponding to all connected
components of the intersection graph, such that the corresponding chord
subdiagrams are planar, and summands $\Z_2$, corresponding to all components,
related with non-planar chord subdiagrams;

c) for $i=n-3$ is free Abelian, whose rank is equal to the sum of
the binomial coefficient
${\bnm{m}{2}}$ and the number of connected components of the intersection
graph corresponding to planar chord subdiagrams.
\end{theorem}

These two theorems will be proved in \S \ref{homcalc}, these
proofs are based on some homological
calculations from \S \ref{coors} and a theorem by
V.O.Manturov, establishing a geometrical criterion of the planarity
of a chord diagram.

\begin{example} \rm
\label{trexi}
Suppose that the endpoints of any chord of the chord diagram $\Theta$
are not separated in $\R^1$ by endpoints of other its chords,
see Fig. \ref{trivexi}.
\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(78.00,9.00)
\put(13.00,5.00){\oval(10.00,8.00)[t]}
\put(31.00,5.00){\oval(10.00,8.00)[t]}
\put(66.00,5.00){\oval(10.00,8.00)[t]}
\put(2.00,5.00){\line(1,0){40.00}}
\put(55.00,5.00){\line(1,0){26.00}}
\put(48.00,5.00){\makebox(0,0)[cc]{$\dots$}}
\put(5.00,2.00){\makebox(0,0)[cc]{\small $1$}}
\put(13.00,2.00){\makebox(0,0)[cc]{\small $2$}}
\put(22.00,2.00){\makebox(0,0)[cc]{\small $3$}}
\put(31.00,2.00){\makebox(0,0)[cc]{\small $4$}}
\put(66.00,2.00){\makebox(0,0)[cc]{\small $2m$}}
\put(79.00,2.00){\makebox(0,0)[cc]{\small $2m+1$}}
\end{picture}
\caption{The simplest $m$-chord diagram}
\label{trivexi}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(125.00,14.00)
\bezier{116}(0.00,0.00)(19.00,0.00)(19.00,10.00)
\bezier{40}(19.00,10.00)(19.00,14.00)(14.00,14.00)
\bezier{44}(14.00,14.00)(9.00,14.00)(9.00,10.00)
\bezier{100}(9.00,10.00)(9.00,0.00)(29.00,0.00)
\bezier{116}(29.00,0.00)(49.00,0.00)(49.00,10.00)
\bezier{40}(49.00,10.00)(49.00,14.00)(44.00,14.00)
\bezier{44}(44.00,14.00)(39.00,14.00)(39.00,10.00)
\bezier{100}(39.00,10.00)(39.00,0.00)(58.00,0.00)
\bezier{52}(67.00,0.00)(72.00,0.00)(76.00,1.00)
\bezier{52}(76.00,1.00)(79.00,2.00)(81.00,2.00)
\bezier{52}(81.00,2.00)(83.00,2.00)(86.00,1.00)
\bezier{52}(86.00,1.00)(90.00,0.00)(96.00,0.00)
\put(81.00,8.50){\oval(10.00,11.00)[]}
\bezier{52}(96.00,0.00)(102.00,0.00)(106.00,1.00)
\bezier{52}(106.00,1.00)(109.00,2.00)(111.00,2.00)
\bezier{52}(111.00,2.00)(113.00,2.00)(116.00,1.00)
\bezier{52}(116.00,1.00)(120.00,0.00)(125.00,0.00)
\put(111.00,8.50){\oval(10.00,11.00)[]}
\put(62.00,6.00){\makebox(0,0)[cc]{$\Longrightarrow$}}
\end{picture}
\caption{Smoothing the angles}
\label{ugoli}
\end{center}
\end{figure}

Then the rank of the group $H^{0}_{(1)}(\K(\Theta,3) \sm \Sigma(\Theta,3))$
is equal to ${\bnm{m+1}{2}}$. Basic invariants of maps
$f \in \K(\Theta,3) \sm \Sigma(\Theta,3)$
generating this group have the following realization. For any
generic immersion $f: \R^1 \to \R^3$ with this self-intersection type
(see Fig. \ref{ugoli} left), we can smooth down its image at any
self-intersection point in such a way that it splits into
$m$ compact curves arising from intervals with numbers
2, 4, \dots, $2m$, and one non-singular long knot, consisting of
(slightly deformed) images of intervals 1, 3, \dots, $2m-1$, $2m+1$.
For any two of these $m+1$ curves (supplied with obvious orientations)
their linking number is well-defined and invariant under deformations
of the immersion $f$ in the class $\K(\Theta,3) \sm \Sigma(\Theta,3)$.
\end{example}

\begin{example} \rm
\label{exnonti}
Consider now the simplest non-trivial chord diagram $\Theta$ of
Fig. \ref{crossi} and the corresponding space
$\K(\Theta,3) \sm \Sigma(\Theta,n)$.
\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(110.00,13.00)
\put(43.00,6.00){\vector(1,0){14.00}}
\put(0.00,6.00){\line(1,0){35.00}}
\put(14.00,6.00){\oval(14.00,12.00)[t]}
\put(21.00,6.00){\oval(14.00,12.00)[b]}
\put(3.00,4.00){\makebox(0,0)[cc]{\small $1$}}
\put(10.00,4.00){\makebox(0,0)[cc]{\small $2$}}
\put(17.00,4.00){\makebox(0,0)[cc]{\small $3$}}
\put(25.00,4.00){\makebox(0,0)[cc]{\small $4$}}
\put(32.00,4.00){\makebox(0,0)[cc]{\small $5$}}
\put(67.00,6.00){\line(1,0){16.50}}
\bezier{92}(83.50,6.00)(91.50,6.00)(91.50,9.00)
\bezier{80}(91.50,9.00)(91.50,11.50)(84.50,11.50)
\bezier{100}(84.50,11.50)(76.50,11.00)(76.50,6.00)
\bezier{84}(76.50,6.00)(77.00,1.00)(81.50,5.00)
\bezier{88}(81.50,5.00)(85.00,8.50)(90.75,8.50)
\bezier{32}(92.25,8.50)(94.00,8.50)(95.50,7.00)
\bezier{40}(95.50,7.00)(96.50,6.00)(100.00,6.00)
\put(100.00,6.00){\line(1,0){10.00}}
\put(76.60,6.00){\circle*{1.00}}
\put(82.80,6.00){\circle*{1.00}}
\put(71.00,4.00){\makebox(0,0)[cc]{\small $1$}}
\put(85.00,13.00){\makebox(0,0)[cc]{\small $3$}}
\put(79.00,1.00){\makebox(0,0)[cc]{\small $4$}}
\put(104.00,4.00){\makebox(0,0)[cc]{\small $5$}}
\put(80.00,7.40){\makebox(0,0)[cc]{\small $2$}}
\end{picture}
\caption{Non-trivial chord diagram and immersion}
\label{crossi}
\end{center}
\end{figure}

By Theorem \ref{homoddi} the group of first order invariants (i.e.
0-dimensional cohomology classes) of maps of this class is isomorphic to
$\Z^2$. Again, its generators can be easily reduced to linking numbers.
Indeed, one of these generators can be defined as follows.
We take two cycles in the graph $f(\R^1)$, the first of them consisting of
images of intervals $1, 4, 5$, and the second one consisting of images
of intervals $2, 3$; then we suitably diverse these cycles in
$\R^3$ and take the linking number of obtained two curves.
At the first common point of these cycles
(in which they meet by ``angles'') such a divorce is canonical, cf.
Fig.~\ref{ugoli}, but at the second one we need to choose the way in which
we separate two smooth local curves. This makes the choice of this generator
non-invariant under the mirror symmetry in $\R^3$. In a similar way, for
the second generator of our group we can take
the linking number of somehow separated
cycles $1+2+5$ and $3+4$.
\end{example}

Now let us discuss one-dimensional homology and cohomology classes of spaces
$\K(\Theta,3) \sm \Sigma(\Theta, 3).$ For any non-empty chord diagram
$\Theta,$ one generator of the group
$H_1(\K(\Theta,3) \sm \Sigma(\Theta, 3))$
is obvious: it is defined by rotations of the space
$\R^3$ around its axis $l(\R^1).$ Moreover, with any connected component
of the intersection graph it is easy to associate a similar
one-dimensional cycle, consisting of maps, coinciding at those
self-intersection points of the curve $f(\R^1)$, that do not belong
to this component, and rotating the part of $f(\R^1)$
related with this component. For the exact construction of these
cycles see \S \ref{realodd}, in which we shall see also that the group
$H^1_{(1)}(\K(\Theta,3) \sm \Sigma(\Theta, 3))$ of first order
cohomology classes is dual to the group generated by these rotations.
In a similar way, for any {\em odd} $n$ a set of
free generators of the group
$H^{n-2}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$
can be realized by embedded $(n-2)$-dimensional spheres. In this
realization it is essential that such a sphere admits a tangent
non-zero vector field. For even $n$ the similar realization generally
does not exist: one does not need such a vector field only in constructing
spheres, corresponding to those components of the intersection graph,
which define planar chord subdiagrams.
\medskip

Further, we investigate the representability of elements of the group
(\ref{perpor}) by combinatorial formulas similar to the classical formula
for the linking number of two curves (counting for the points of one
curve lying over the points of the other) or to the Polyak--Viro formulas
\cite{PV} for knot invariants.

\begin{definition}[cf.~\cite{PV}] \rm
\label{gadi}
Let $\Theta$ be an arbitrary $m$-chord diagram. A
{\em non-oriented} (respectively, {\em oriented})
{\em Gau\ss \ diagram of order one with base} $\Theta$ is the object
consisting of the diagram $\Theta$ and a non-ordered
(respectively, ordered) pair of components of the set
$\R^1 \sm \Theta$.
\end{definition}

In this work, we consider only Gau\ss \ diagrams of order one
and for simplicity call them just
{\em Gau\ss \ diagrams}.

The notation of such a diagram consists of the collection of $m$ smooth arcs,
depicting $\Theta$, and a broken line, connecting some point of one of these
two components of $\R^1 \sm \Theta$
with some point of the other one; if the Gau\ss \ diagram
is oriented, i.e. these components are ordered, then the broken line
should be oriented from the point of the first component to the point of
the second. Obviously, for any $m$-chord diagram $\Theta$ there are
exactly ${\bnm{2m+1}{2}}$ non-oriented and $2{\bnm{2m+1}{2}}$
oriented Gau\ss \ diagram with base $\Theta$.

Any invariant of first order, i.e. an element of the group
$H^0_{(1)}(\K(\Theta,3) \sm \Sigma(\Theta, 3)),$
can be defined (up to a constant function) by a linear combination
of non-oriented Gau\ss \  diagrams. Indeed, this invariant is
characterized by the jumps of its values at passing over the
non-singular pieces of the discriminant, i.e. at local surgeries
of type
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(29.40,4.80)
\put(0.70,2.40){\line(1,0){4.60}}
\put(3.00,2.80){\line(0,1){2.00}}
\put(3.00,2.00){\line(0,-1){2.00}}
\put(9.00,2.40){\makebox(0,0)[cc]{$\Rightarrow$}}
\put(15.00,4.70){\line(0,-1){4.60}}
\put(14.60,2.40){\line(-1,0){2.00}}
\put(15.40,2.40){\line(1,0){2.00}}
\put(21.00,2.40){\makebox(0,0)[cc]{$\Rightarrow$}}
\put(27.00,4.70){\line(0,-1){4.60}}
\put(26.60,2.40){\line(-1,0){2.00}}
\put(27.40,2.40){\line(1,0){2.00}}
\put(15.00,2.40){\circle*{1.00}}
\end{picture} .
There is a canonical way to call one of nearby curves, separated
by such a surgery, positive, and the other negative. The jump of the
invariant at a non-singular point of the discriminant is defined as the
difference of its values at the positive and the negative curves.
Equality (\ref{cret}) implies that such a jump depends only on two
components of the set $\R^1 \sm \Theta$, in which in the instant of the
surgery the additional intersection occurs, but does not depend on the
behavior of the map far from these points. The desired linear combination
contains the non-oriented Gau\ss \ diagram, defined by this pair of
components, with the coefficient equal to the value of the jump at
this surgery. More generally, the elements of groups
$H^{n-3}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta, n))$
with arbitrary $n \ge 3$ also can be defined by linear combinations of
non-oriented Gau\ss \ diagrams.
\medskip

A different way to define invariants exploits oriented Gau\ss \ diagrams;
it simulates the classical formula for the linking number of two closed
curves in $\R^3$.

Consider an arbitrary map $f: \R^1 \to \R^3$ of class $\K(\Theta,3)$,
generic with respect to the projection of $\R^3$ onto the
``blackboard plane'' $\R^2;$ the genericity condition means that the
composition of $f$ and this projection is an immersion with only double
transverse selfintersection points. Consider also an oriented Gau\ss \
diagram, consisting of $\Theta$ and an ordered pair of integer numbers $(i
\ne j)$ from the segment $[1,2m+1]$. The value of this diagram on the map $f$
is defined as the {\em weighted} number of pairs of points
$(x,y) \in \R^1$ such that $x$ belongs to the $i$-th (from the left)
component of the set $\R^1 \sm \Theta,$ $y$ belongs to the $j$-th
component, projections of points $f(x)$ and $f(y)$ into $\R^2$
coincide, and the point $f(x)$ lies {\em below} $f(y)$, i.e. the projection
of the point $f(x)$ onto the oriented line, orthogonal to
$\R^2$, is smaller than the projection of $f(y)$; the weightedness means
that such a pair $(x,y)$ is counted with coefficient $1$ or $-1$ depending
on the orientation of the ordered pair of vectors $\R^2,$ obtained as
projections of vectors $f'(x), f'(y).$\footnote{In the original work by
Polyak and Viro \cite{PV} the opposite rule is proposed: the endpoint of the
arrow corresponds to the lower point. This rule disturbs very much in
the calculation of differentials in our complexes, therefore we change it by
the opposite one: in our case the end looks up.}

By linearity we define the value on the map $f$ of an arbitrary linear
combination of oriented Gau\ss \ diagrams. This value obviously does not
change when $f$ moves continuously in the class of maps generic with
respect to the projection into $\R^2$. However, to be an invariant,
i.e. a constant function on any connected component of the space
$\K(\Theta,3) \sm \Sigma(\Theta,3),$ such a linear combination should
satisfy a certain homological condition.

We investigate the question whether any first order invariant in this space
can be realized by a linear combination of oriented Gau\ss \ diagrams
with integer or rational coefficients; a similar question can be
posed also for all other first order elements of groups
$H^*(\K(\Theta,n) \sm \Sigma(\Theta,n)),$ $n \ge 3.$
For instance, if $n=3$ and
the chord diagram $\Theta$ is the simplest one, see
Example \ref{trexi} in page \pageref{trexi}, then such a realization
always exists. Indeed, the invariant, defined as the linking number of
two circles, can be represented by the unique oriented Gau\ss \ diagram,
connecting some points of intervals, from which these circles are obtained;
in a similar way, the linking number of such a circle and the long curve
can be represented by the sum of $m+1$ such diagrams. For an arbitrary chord
diagram $\Theta$ and arbitrary $n \ge 3$ the answer is as follows.

\begin{theorem}
\label{predsti}
1. For any $n \ge 3$ and $\Theta,$ any element of the group
$H^{n-3}_{(1)}(\K(\Theta,n)\sm \Sigma(\Theta,n))$ can be realized by a
linear combination of oriented Gau\ss \ diagrams with integer and
half-integer coefficients.

2. If $n$ is even or the chord diagram $\Theta$ is planar, then any such
element can be realized by a linear combination of oriented Gau\ss \
diagrams with only integer coefficients.
\end{theorem}

\begin{theorem}
\label{predsti2}
If $n$ is odd and $\Theta$ is non-planar, then there exists an element of the
group $H^{n-3}_{(1)}(\K(\Theta,n)\sm \Sigma(\Theta,n))$, not
representable by integer linear combinations of oriented Gau\ss \ diagrams.
\end{theorem}

A proof of these two theorems (and exact definition of the representability
of elements, participating in it) will be given in \S \ref{obstcof}.

Let us describe an easy criterion recognizing the representability
of an invariant by integer linear combinations of oriented Gau\ss \
diagrams in the case of an irreducible chord diagram $\Theta$.

It was shown above, that any integer first order invariant can be
defined (up to a constant function, i.e. up to the choice of its value at
the distinguished point of the space  $\K(\Theta,3) \sm \Sigma(\Theta,3)$)
by an integer linear combination $F$ of non-oriented Gau\ss \ diagrams:
\begin{equation}
\label{licomi}
F= \sum_{a<b} \lambda_{a,b} (a\cdot b),
\end{equation}
where $a \cdot b $ is the notation of the non-oriented chord diagram,
consisting of the $a$-th  and the $b$-th (from the left) connected
component of the set $\R^1 \sm \Theta$, and $\lambda_{a,b} \in \Z$.
Consider an arbitrary immersion $f: \R^1 \to \R^3$ of the class
$\K(\Theta,3)\sm \Sigma(\Theta,3).$ Let $f_1: \R^1 \to \R^2$
be the composition of the immersion $f$ and the projection
$\R^3 \to \R^2$. If the immersion $f$ is generic in the class $\K(\Theta,3)$,
and the direction of the projection does not coincide with the axis
$l(\R^1)$ in $\R^3$, then $f_1$ also is a generic immersion, in particular has
no triple points and self-tangencies. However, $f_1$ can have more
double points than $f$. Define the number $\varpropto(\Theta,F,f)$ by the
formula
\begin{equation}
\varpropto(\Theta,F,f) =
\sum_{
\begin{array}{c}f_1(x)=f_1(y) \\
f(x) \ne f(y)
\end{array}}
\lambda_{\{x\},\{y\}}
\label{maprep}
\end{equation}
where the sum is taken over all double points of the curve $f_1(\R^1)$,
which are not images of double points of the curve $f(\R^1)$, and for any
such double point $f_1(x)=f_1(y)$, $\lambda_{\{x\},\{y\}}$
is the coefficient from (\ref{licomi}), where $\{x\}, \{y\}$
are the numbers of components of the set
$\R^1 \sm \Theta$, that contain the points $x,y$.

\begin{theorem}
1. The parity of the number $\varpropto(\Theta,F,f)$ does not depend on the
choice of the generic immersion
$f$ $($and depends only on $\Theta$ and $F)$.

2. If this number is odd for some generic immersion $f$, then the invariant,
defining the linear combination $F$, cannot be realized by an integer
linear combination of oriented Gau\ss \ diagrams.

3. If the diagram $\Theta$ is irreducible, then the converse is also true:
if this number is even, then the invariant can be represented by an
integer linear combination of oriented Gau\ss \ diagrams.
\label{necconi}
\end{theorem}

{\em Proof} will be given in \S \ref{obstcof}.
\medskip

For instance, in the case of the simplest non-planar diagram
$\Theta$, described in Example \ref{exnonti}, for both generators
of the group
$H^0_{(1)}(\K(\Theta,3) \sm \Sigma(\Theta,3))$
indicated there, this number,
defined by the immersion of Fig.~\ref{crossi}, is odd.

\begin{remark} \rm
The question on the integrality of combinatorial formulas is motivated
by the theory of knot invariants. Any $k$-th order invariant of knots in
$\R^3$ (see \cite{knsp}) defines an invariant of first order of generic
$k-1$ times self-intersecting immersed curves, an invariant of second order
of $k-2$ times self-intersecting curves, etc. The most convenient method of
explicit {\em integration of the weight system},  i.e. the construction of an
invariant with the given principal part (and additionally of a combinatorial
formula for this invariant) is the inverse induction over the
number of self-intersections, when one constructs consecutively the
combinatorial formulas of these invariants of self-intersecting curves
in this order, see \cite{tt}, \cite{bryden}, \cite{combcomp}.
In particular, the easiest construction of Polyak--Viro formulas for
knot invariants \cite{PV}  consists in a sequence of such combinatorial
formulas of a special form. By a theorem of M.N.Goussarov \cite{GPV},
any finite order knot invariant can be represented by a Polyak--Viro
formula with rational coefficients. If a knot invariant of order
$m+1$ admits an {\em integer} Polyak--Viro formula, then the corresponding
first order invariant of $m$ times self-intersecting curves is
representable by an integer linear combination of oriented Gau\ss \ diagrams.
Therefore, the obstruction to the existence of such an integer representation
of invariants of self-intersecting curves, described in Theorem
\ref{necconi}, is also the first obstruction to the integrality of
Polyak--Viro formulas of knot invariants.

For instance, in the case of the unique third order knot invariant
the corresponding first order invariant of twice self-intersecting curves
is equal to the sum of two generators from Example \ref{exnonti},
therefore our obstruction is trivial for it. Moreover, this knot
invariant admits an integral Polyak--Viro formula, see \cite{tyurina},
so that all the further obstructions to this also necessarily vanish.

On the other hand, in \S \ref{obstcof} we shall prove that our obstruction is
non-trivial for one of basis integer knot invariants of order 4, and hence
this knot invariant surely has no integer Polyak--Viro formulas.
Moreover, even if we wish to construct a
$\Z_2$-valued combinatorial formula of this invariant reduced mod 2,
then we need some terms different from the oriented Gau\ss \ diagrams,
e.g. the terms proposed in \cite{combcomp}.
\end{remark}

All generators of groups
$H^{n-2}_{(1)}(\K(\Theta,n)\sm\Sigma(\Theta,n))$
also can be realized by combinatorial formulas. For instance, let us
consider the classical case $n=3.$ Any chord of the diagram $\Theta$
defines a generator of this group
$H^{1}_{(1)}(\K(\Theta,3)\sm\Sigma(\Theta,3))$.
The value of this generator on a 1-cycle, i.e. on a smooth one-parameter
family of maps $f_\tau \in \K(\Theta,3)\sm\Sigma(\Theta,3)$,
$\tau \in S^1$,
is defined as the weighted number of values of the parameter
$\tau$ such that the image $f_\tau(x)$ of some point $x$,
lying in $\R^1$ to the left of the left-hand endpoint $u$
of our chord $(u,v)$, lies above the point
$f_\tau(u) \equiv f_\tau(v)$
in $\R^3.$ We shall prove (see Proposition \ref{rea1} on page \pageref{rea1})
that these elements, corresponding to chords from one and the same
component of the intersection graph, are homologous to one another,
and all such elements, corresponding to any collection of chords,
taken one from each component of this graph, is a basis of the group
$H^{1}_{(1)}(\K(\Theta,3)\sm\Sigma(\Theta,3))$.
\medskip

I thank V.O.Manturov for the proof of Theorem \ref{vassya}, and the
referee for many suggestions on improving the explanation.


\section{Homological calculation}
\label{homcalc}

Similarly to \cite{b94}, \cite{fasis}, the calculation of cohomology
classes of the space $\K(\Theta,n) \sm \Sigma(\Theta, n)$ can be reduced to
the calculation of {\em Borel--Moore cohomology of finite codimension}
of the space $\Sigma(\Theta,n)$ itself by a kind of the
Alexander duality\footnote{In the case of ordinary finite-dimensional
discriminants, i.e. sets of polynomials of fixed degree with multiple
roots, this reduction was first used by V.Arnold \cite{ar70}.}
\begin{equation}
\tilde H^i(\K(\Theta,n) \sm \Sigma(\Theta, n))
\simeq \bar H_{\omega-i-1}(\Sigma(\Theta,n)).
\label{evr}
\end{equation}
Here $\omega$ symbolizes the dimension of the space
$\K(\Theta,n),$ $\bar H_*$ denotes the Borel--Moore homology groups,
i.e. homology groups of the one-point compactification reduced modulo the
added point, and the meaning of the entire group in the right-hand part
can be formulated by means of finite-dimensional approximations of the space
$\K(\Theta,n)$, cf.  \cite{b94}, \cite{fasis}, and comments to formula
(\ref{finap}) below. In particular, if $n=3$ then the invariants of curves
(i.e. zero-dimensional cohomology classes) correspond to the elements of the
group $ \bar H_{\omega-1}(\Sigma(\Theta,n))$. The spectral sequence,
shown in Fig.~\ref{sse}, occurs in the calculation of the right-hand group
in (\ref{evr}) with the help of the {\em simplicial resolution}
of the discriminant; its construction generalizes the similar construction
from the knot theory (corresponding to the case $m=0$),
see \cite{knsp}, \cite{b94}, \cite{fasis} and \S \ref{ssse} below.

The first order homology groups, calculated in this work,
coincide with the homology groups of the
{\em tautological resolution} of the discriminant, which we shall
define now; it corresponds to the first column of this spectral
sequence.


\subsection{Tautological resolution of the discriminant}
\label{tavtol}

It follows immediately from the definition that
the discriminant $\Sigma(\Theta, n) \subset \K(\Theta,n)$
is the closure of the set $\breve \Sigma(\Theta,n)$, consisting
of maps $f \in \K(\Theta,n)$ such
that there exists a pair of non-singular (i.e. not belonging to
$\Theta$) different points $u,v \in \R^1$ for which $f(u)=f(v)$.  The
tautological resolution $\sigma_1(\Theta,n)$ of $\Sigma(\Theta,n)$
(which we define in the next paragraph)
can be obtained from this definition by removing the existence
quantifier: we take the set of all pairs $((u,v);f)$ satisfying the same
conditions, and take its closure in some natural space, which we describe now.

Consider the vector space $\Phi(\Theta)$ of all smooth functions
$\R^1 \to \R^1,$ taking equal values at the endpoints of any chord of
$\Theta$, and denote by $I(\Theta)$ the Grassmannian manifold, whose points
are all
possible vector subspaces of codimension 1 in $\Phi(\Theta)$. The space
$B(\R^1 \sm \Theta,2)$ of unordered couples of
distinct points in $\R^1 \sm \Theta$ is naturally embedded into
$I(\Theta)$: to any 2-configuration $(u,v) \subset \R^1 \sm \Theta$
there corresponds the subspace
consisting of functions taking equal values at the points of this
configuration. The completed
configuration space $B_2(\Theta)$ is defined as the closure of the
submanifold $B(\R^1 \sm \Theta,2)$ in $I(\Theta)$. Finally, the
tautological resolution $\sigma_1(\Theta,n)$ of the discriminant
$\Sigma(\Theta,n)$ is defined as a certain subset of the direct product
\begin{equation}
\label{taut}
B_2(\Theta) \times \K(\Theta,n),
\end{equation}
namely as the closure in it of the set $\breve \sigma_1(\Theta,n)$
consisting of pairs
$((u,v); f)$ such that $\{u \ne v\} \subset \R^1 \sm \Theta$ and
$f(u) = f(v)$. The restriction of the projection
$B_2(\Theta) \times \K(\Theta,n) \to B_2(\Theta)$
on $\sigma_1(\Theta,n)$
is an affine fibre bundle, whose fibre is a subspace of
codimension $n$ in $\K(\Theta,n)$.
This space $\sigma_1(\Theta,n)$ is naturally stratified.
Its maximal strata are the connected components of the above-described
non-closed subset
$\breve \sigma_1(\Theta,n) \subset B(\R^1 \sm \Theta,2) \times \K(\Theta,n)$;
they are in a one-to-one correspondence with
unordered pairs of (maybe coinciding) components
of the set $\R^1 \sm \Theta$.
Remaining strata are counted below, see diagram (\ref{complex})
on page \pageref{complex} and comments to it.

The projection $\sigma_1(\Theta,n) \to \Sigma(\Theta,n),$ induced by the
obvious projection
$B_2(\Theta) \times \K(\Theta,n) \to \K(\Theta,n),$
together with the Alexander isomorphism (\ref{evr})
defines a homomorphism of the
``Borel--Moore homology group of finite codimension'' into the
usual reduced cohomology group of the complementary space:
\begin{equation}
\label{finap}
\bar H_{\omega -*}(\sigma_1(\Theta,n)) \to \bar
H_{\omega-*}(\Sigma(\Theta,n)) \to
\tilde H^{*-1}(\K(\Theta,n) \sm \Sigma(\Theta,n)) .
\end{equation}
The exact sense of the left-hand group of this formula
and of entire this homomorphism
is as follows, cf.
\cite{b94}, \cite{fasis}. We approximate the infinite-dimensional space
$\K(\Theta,n)$
by an increasing sequence of finite-dimensional affine subspaces
$T^{N_1} \subset T^{N_2} \subset \cdots$,
$N_j \to \infty,$ in general position with respect to the discriminant
$\Sigma(\Theta,n) \subset \K(\Theta,n).$
For any $j=1,2, \dots$ we define $\sigma_1^{(j)}$ as
$\sigma_1(\Theta,n) \cap (B_2(\Theta) \times T^{N_j}).$
It is easy to see that for sufficiently large $k<j$ there is a
natural homeomorphism
\begin{equation}
\sigma_1^{(j)} \simeq \sigma_1^{(k)} \times \R^{N_j - N_k},
\end{equation}
in particular an isomorphism of Borel--Moore homology groups
\begin{equation}
\label{stabil}
\bar H_{N_j-*}(\sigma_1^{(j)}) \simeq \bar H_{N_k-*}(\sigma_1^{(k)}).
\end{equation}

\begin{definition} \rm
An element of the left-hand group in
(\ref{finap}) is a sequence of elements of all groups
$\bar H_{N_j-*}(\sigma_1^{(j)})$
with sufficiently large $j$, corresponding to each other by
isomorphisms (\ref{stabil}).
\end{definition}

It is easy to prove that this definition does not depend on the choice of
the approximating sequence $\{T^{N_j}\},$ i.e. there is a
natural isomorphism between the groups of all such elements defined by
any two such sequences of subspaces.

Since all subspaces $T^{N_j}$ are generic, the projections
$\sigma_1^{(j)} \to \Sigma(\Theta,n) \cap T^{N_j}$ are proper maps and
induce homomorphisms of Borel--Moore homology groups of these spaces.
For any $j$, the composition
\begin{equation}
\label{alnestab}
\bar H_{N_j-*}(\sigma_1^{(j)}) \to
\bar H_{N_j-*}(\Sigma(\Theta,n) \cap T^{N_j})
\to \tilde H^{*-1}(\K(\Theta,n) \cap T^{N_j} \sm \Sigma(\Theta,n))
\end{equation}
of such a homomorphism and the Alexander isomorphism in $T^{N_j}$
sends any element of the left-hand group
$\bar H_{N_j-*}(\sigma_1^{(j)})$ into some
reduced cohomology class of the space
$\K(\Theta,n) \cap T^{N_j} \sm \Sigma(\Theta,n)$.
Moreover, the homology classes of spaces
$\sigma_1^{(k)}, \sigma_1^{(j)},$
corresponding to each other by means of the isomorphism (\ref{stabil}),
define cohomology classes of one and the same dimension, which are mapped
one into the other by the identical embedding
$T^{N_k} \sm \Sigma(\Theta,n) \subset T^{N_j} \sm \Sigma(\Theta,n),$
and hence stabilize to a well-defined cohomology class of the entire
space $\K(\Theta,n) \sm \Sigma(\Theta,n).$

\begin{definition} \rm
The first order cohomology group of the space
$\K(\Theta,n) \sm \Sigma(\Theta,n)$ is the image of the
composite map (\ref{finap}), i.e. the union of all
cohomology classes obtained as limits at $j \to \infty$ of the
right-hand parts of (\ref{alnestab}).
\end{definition}

In the special case of $n=3$ and
0-dimensional cohomology classes, i.e. just invariants, this definition
is consistent with the one given in the introduction.

\begin{proposition}
Any first order invariant of maps $f \in \K(\Theta,3)$
$($i.e. an invariant satisfying the equation
(\ref{cret})$)$ can be defined $($up to adding a constant$)$
by the linking number with the image of some top dimensional cycle
of $\sigma_1(\Theta,3)$.
\end{proposition}

Indeed, such an invariant can be defined by an appropriate linear
combination of non-oriented Gau\ss \ diagrams;
for the desired cycle we need to take the
linear combination (with the same coefficients) of the maximal strata in
$\sigma_1(\Theta,3)$ corresponding to these diagrams and supplied with
appropriate orientations, see \S \ref{coors}.
\hfill $\square$ \medskip

In the terms of the space $\K(\Theta,3)$ an oriented Gau\ss \ diagram
should be considered as a singular chain of full dimension in this
space (the exact sense of the notion of a singular chain here
also is based on finite-dimensional approximations), while the value
of the diagram on a generic map $f \in \K(\Theta,3)$ is the multiplicity
with which a neighborhood of the point $f$ is covered by this chain.
A linear combination of Gau\ss \ diagrams defines an invariant
if and only if the corresponding linear combination of singular chains
is a relative cycle in
$\K(\Theta,3) \mbox{ mod } \Sigma(\Theta,3),$
and its boundary is a cycle corresponding to this invariant by
the isomorphism (\ref{evr}).

\subsection{The simplicial resolution of the discriminant}
\label{ssse}

In this section we construct the spectral sequence, calculating cohomology
groups of the space $\K(\Theta,n) \sm \Sigma(\Theta,n)$ and promised in the
proof of Theorem \ref{red}. To do it, we construct the
{\em simplicial resolution} $\sigma(\Theta,n)$ of the discriminant
$\Sigma(\Theta,n)$. This is a topological space with natural projection
\begin{equation}
\label{proek}
\sigma(\Theta,n) \to \Sigma(\Theta,n)
\end{equation}
and increasing infinite filtration
$\sigma_1(\Theta,n) \subset \sigma_2(\Theta,n) \subset \dots,$
whose first term is described in the previous subsection. For all other terms
$\sigma_i(\Theta,n)$ of this filtration their Borel--Moore homology groups
of finite codimension are also well-defined. Similarly to the previous
subsection, one can define maps of these homology groups into the usual
cohomology groups of the space $\K(\Theta,n) \sm \Sigma(\Theta,n)$
using the finite-dimensional approximations of the space
$\K(\Theta,n)$. This allows us to define the desired spectral sequence
and filtration in these cohomology groups. Similarly to the case of
$\sigma_1(\Theta,n),$ any term
$\sigma_i(\Theta,n) \sm \sigma_{i-1}(\Theta,n)$ of our filtration is
the space of an affine fiber bundle, whose fiber is equal to
$\R^{\omega-in}$ and the base is finite-dimensional and has an easy
decomposition into open cells. This allows us to reduce the homology groups
of finite codimension of these spaces to usual Borel--Moore homology groups
of finite-dimensional spaces.

The following construction of the simplicial resolution essentially
repeats the similar construction from \cite{kotor}, \cite{combcomp}, see also
\cite{knsp}, \cite{fasis}. (In the present work this construction is
used only in the proof of Theorem \ref{red}, therefore a reader,
which can believe on the statement of this theorem or is interested
mainly in the case $n=3$, can omit its description.)

Recall that the space $B_2(\Theta)$ is embedded into $I(\Theta)$,
hence with any its point a subspace of codimension 1 in $\Phi(\Theta)$
is tautologically associated. For any natural $i$, consider the set of all
vector subspaces of codimension $i$ in $\Phi(\Theta)$, which can be obtained
as intersections of several subspaces of codimension 1, associated with
points of the space $B_2(\Theta)$.

This set is a stratified subset of dimension $2i$ in the Grassmannian
manifold $I_i(\Theta)$, i.e. in the set of all subspaces of codimension
$i$ in $\Phi(\Theta)$. Denote by $B_{2i}$ the closure of this set in
$I_i(\Theta)$.

On the disjoint union
$B_2(\Theta) \sqcup B_4(\Theta) \sqcup \dots \sqcup B_{2i}(\Theta)$
there is a partial order defined by incidences of corresponding
subspaces of codimensions $1, 2, \dots, i$ in $\Phi(\Theta).$
Define the space $\Delta_i(\Theta)$  as the
{\em continuous order complex} related with this partially ordered set.
Namely, let us consider the join
$B_2(\Theta) * B_4(\Theta) * \dots * B_{2i}(\Theta)$
of these spaces, i.e. the canonically topologized union of all possible
simplices, whose vertices belong to different such spaces.
$\Delta_i(\Theta)$ is  its subset consisting of only those simplices, whose
vertices are related with one another by this partial order
(i.e. correspond to chains of embedded subspaces).
If $k<i$, then the space $\Delta_k(\Theta)$ is canonically embedded into
$\Delta_i(\Theta)$, and we obtain the natural filtration
$\Delta_1(\Theta) \subset \Delta_2(\Theta) \subset \dots \subset
\Delta_i(\Theta).$
The space $\Delta(\Theta)$ is defined as the union of all
$\Delta_i(\Theta)$ with the topology of a direct limit.

Over any set
$\Delta_i(\Theta) \sm \Delta_{i-1}(\Theta)$
an affine fiber bundle is defined, whose fibers are certain subspaces
of codimension $i n$ in $\K(\Theta,n).$ Indeed, to any point $X$
of this set a point $L(X)$ of the set $B_{2i}(\Theta)$ corresponds,
namely, the maximal (in the sense of our partial order) vertex of the
minimal simplex of our order complex, containing the point $X$.
This point $L(X)$ defines tautologically a subspace of codimension
$i$ in $\Phi(\Theta)$ (which we also will denote by $L(X)$). The desired
subspace $\LL(X,n) \subset \K(\Theta,n)$ consists of all maps
$f: \R^1 \to \R^n$ of the class $\K(\Theta,n),$ all $n$ coordinate
components of which belong to this subspace $L(X).$

Define the space
$\sigma_i(\Theta,n) \setminus \sigma_{i-1}(\Theta,n)$
as the subset in the direct product
$(\Delta_i(\Theta) \setminus \Delta_{i-1}(\Theta)) \times \K(\Theta,n),$
consisting of all possible pairs
$(X,f)$ such that $f \in \LL(X).$ Define the space
$\sigma_i(\Theta,n)$ inductively as
$(\sigma_i(\Theta,n) \setminus \sigma_{i-1}(\Theta,n)) \cup
\sigma_{i-1}(\Theta,n)$ and the space
$\sigma(\Theta,n)$ as the union of all
spaces $\sigma_i(\Theta,n),$ $i=1,2, \dots$,
supplied with the topology of the direct limit.

The projection (\ref{proek}) sends any pair $(X,f) \in \sigma(\Theta,n)$
into the point $f$. If the point $f \in \Sigma(\Theta,n)$ belongs only to
a finite number of hyperplanes of the form $\LL(X)$ (which is true
for all $f$ except for a subset of infinite codimension in $\K(\Theta,n)$
for any $n\ge 2$), then its preimage under this map is a contractible finite
simplicial complex; all maximal simplices of this complex contain a common
point, corresponding to the smallest plane $\LL(X)$ containing $f$.

For any affine subspace $T^{N_j} \subset \K(\Theta,n)$ a subspace in
$\sigma_i(\Theta,n)$ is defined, consisting of pairs $(X,f)$ such that
$f \in T^{N_j}$. If the sequence of subspaces $T^{N_j}$ is of general
position with respect to the discriminant, then for any natural
$i$ there exists $j_0$ such that for any  $j > j_0$
this subspace
$\sigma_i(\Theta,n) \cap (\Delta_i(\Theta) \times T^{N_j})$
is homeomorphic to the direct product of the space
$\sigma_i(\Theta,n) \cap (\Delta_i(\Theta) \times T^{N_{j_0}})$
and $\R^{N_j - N_{j_0}}$, and this homeomorphism respects the
natural filtrations in these spaces.

This allows us to carry the argumentation, related with finite-dimensional
approximations, in particular to define Borel--Moore homology classes
of finite codimension of spaces
$\sigma_i(\Theta,n)$ and $\sigma(\Theta,n)$, and also the maps
of these classes into the usual reduced cohomology groups of the space
$\K(\Theta,n) \sm \Sigma(\Theta,n)$. All this can be done in exactly the same
way as it was done in the previous subsection for the subspace
$\sigma_1(\Theta,n).$

All above-mentioned properties of the occurring spectral sequence,
generated by our filtration in $\sigma(\Theta,n),$ in particular the fact
that its support looks as is shown in Fig.~\ref{sse},
follows immediately from the Thom transversality theorem,
cf.~\cite{b94}, \cite{fasis}.



\subsection{Stratification and cell structure of the space
$\sigma_1(\Theta,n)$}

The space $\sigma_1(\Theta,n)$ (as well as its base $B_2(\Theta)$)
consists of strata of three neighboring (co)dimensions and seven
different types $\Pi, \dots, \Upsilon,$ counted in the next diagram
(\ref{complex}) and in comments after it.
All these strata are open topological cells; they generate
a cell complex calculating the Borel--Moore homology groups of
the space $\sigma_1(\Theta,n).$ The expressions
$\Pi(\Theta)$, $\Xi(\Theta)$ etc in (\ref{complex}) are the
notations of corresponding groups of cell chains, i.e. of free
Abelian groups, whose generators are fundamental cycles of
strata of corresponding types.

\begin{equation}
\begin{array}{c}
\unitlength 1mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(90.00,55.00)
\put(6.00,50.00){\makebox(0,0)[cc]{$\Pi(\Theta) \simeq \Z^{\bnm{2m+1}{2}}$}}
\put(51.00,50.00){\makebox(0,0)[cc]{$\Xi(\Theta) \simeq \Z^{m(2m+1)}$}}
\put(94.00,50.00){\makebox(0,0)[cc]{$\Omega(\Theta) \simeq \Z^{\bnm{m}{2}}$}}
\put(50.00,30.00){\makebox(0,0)[cc]{$\|(\Theta) \simeq \Z^{2m}$}}
\put(90.00,30.00){\makebox(0,0)[cc]{$\Psi(\Theta) \simeq \Z^{2m}$}}
\put(8.00,10.00){\makebox(0,0)[cc]{$\Pi'(\Theta) \simeq \Z^{2m+1}$}}
\put(50.00,10.00){\makebox(0,0)[cc]{$\Upsilon(\Theta) \simeq \Z^{2m+1}$}}
\put(21.00,50.00){\vector(1,0){12.00}}
\put(68.00,50.00){\vector(1,0){12.00}}
\put(22.00,45.00){\vector(2,-1){18.00}}
\put(63.00,45.00){\vector(2,-1){17.00}}
\put(64.00,30.00){\vector(1,0){13.00}}
\put(23.00,10.00){\vector(1,0){13.00}}
\put(22.00,15.00){\vector(2,1){18.00}}
\put(19.00,18.00){\vector(1,1){24.00}}
\put(63.00,15.00){\vector(2,1){17.00}}
\end{picture}
}
\end{array}
\label{complex}
\end{equation}

The strata of maximal dimension consist of pairs of the form
(a couple of points of the punctured line $\R^1 \sm \Theta$;
a map $f \in \K(\Theta,n)$, gluing these two points together). Any such
stratum is defined by the choice of some two of $2m+1$ intervals into which
$\R^1$ is divided by the points of the chord diagram $\Theta$. It is
convenient to distinguish those of these strata, for which these two
intervals coincide.  The corresponding group is denoted by $\Pi'(\Theta)$,
and the group related with non-coinciding intervals by $\Pi(\Theta)$.  In
particular, the canonical generators of the group $\Pi(\Theta)$ are numbered
exactly by the non-oriented Gau\ss \ diagrams with base $\Theta.$

Strata of vice-maximal dimension are of three types. The strata
of type $\Xi$ are indexed by the choices of one
of $m$ chords and one of $2m+1$ components of the set
$\R^1 \sm \Theta$. Their points are
the pairs of the form
(a point of the chosen interval; a map
$f \in \K(\Theta,n)$ gluing this point together with both endpoints of
the chosen chord). Strata of type $\|$ are indexed by the
choices of a chord and one of characters $\rra$ or $\rla$. They consist
respectively of such maps, whose derivatives at the endpoints of the
chosen chord are co-directed or counter-directed. Strata of type
$\Upsilon$ are indexed by the choices of one of $2m+1$ components of the
set $\R^1 \sm \Theta$ and consist of pairs of the form
(a point of the chosen interval; a map $f$, whose derivative vanishes
at this point).

Strata of codimension 2 are of two types more. Strata of type
$\Omega$ are indexed by the pairs of chords and
consist of maps, gluing together all endpoints of some two chords.
Strata of type $\Psi$ are indexed by the endpoints of chords
and consist of maps with vanishing derivative at the corresponding
point; this condition is depicted by an asterisk at this point.

It is easy to see that all these strata are the spaces of
affine fibre bundles of infinite dimension but finite (equal to $n$)
codimension in $\K(\Theta,n)$ over open cells of dimensions
2, 1 or 0. For instance, the generator
$\alpha_\rra$ or $\alpha_\rla$ of the group $\|(\Theta)$
is fibered over one of components of the punctured projective line
$\RP^1 \sm \{0,\infty\}$: the fibre over the point
$\rho$ consists of maps $f \in \K(\Theta,n)$ such that the derivative of
$f$ at the right-hand endpoint of the chord $\alpha$ is equal to
the derivative at the left-hand endpoint, dilated with
coefficient $\rho$.

We consider the 3-term chain complex
\begin{equation}
0 \to \CC_2(\Theta,n) \to \CC_1(\Theta,n) \to \CC_0(\Theta,n) \to 0
\label{short}
\end{equation}
with groups
$\CC_2(\Theta,n)=\Pi(\Theta)\oplus \Pi'(\Theta),$
$\CC_1(\Theta,n)=\Xi(\Theta)\oplus \|(\Theta)\oplus \Upsilon(\Theta)$ and
$\CC_0(\Theta,n)=\Omega(\Theta)\oplus \Psi(\Theta)$,
whose generators are fundamental cycles of somehow oriented
corresponding strata, while the boundary operators, depicted by
the arrows in (\ref{complex}), reflect the incidence coefficients
of strata of neighboring dimensions. These incidences correspond to
degenerations of conditions defining the bigger stratum. In the case
when this stratum is a maximal one, these degenerations are shown
in Fig. \ref{rmx}. Although there is a one-to-one correspondence between
sets of
strata corresponding to cases of different $n$, these incidence
coefficients (and homology groups of the complex) depend strongly on
the parity of $n$ (which is important for counting the orientations).
Below we shall see that all complexes
$\CC_*(\Theta,n)$ corresponding to numbers $n$ of one and the same
parity are canonically isomorphic to each other.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(119.67,86.00)
\bezier{96}(2.00,84.00)(12.00,74.00)(12.00,64.00)
\put(9.00,64.00){\oval(6.00,8.00)[b]}
\bezier{104}(2.00,34.00)(9.00,24.00)(9.00,10.00)
\bezier{104}(9.00,10.00)(9.00,24.00)(16.00,34.00)
\put(9.00,52.00){\vector(0,-1){10.00}}
\put(9.00,1.00){\makebox(0,0)[cc]{\bf \u R1}}
\put(84.00,85.00){\line(-5,-4){16.00}}
\put(92.00,85.00){\line(5,-4){27.00}}
\put(110.67,71.00){\line(4,3){9.00}}
\put(92.00,57.00){\line(4,3){17.00}}
\put(89.00,1.00){\makebox(0,0)[cc]{\bf \u R3}}
\put(79.00,53.00){\vector(1,-4){3.00}}
\put(97.00,53.00){\vector(-1,-3){4.00}}
\bezier{96}(6.00,64.00)(7.00,76.00)(16.00,84.00)
\put(74.00,86.00){\line(0,-1){31.00}}
\put(102.00,86.00){\line(0,-1){31.00}}
\put(84.00,85.00){\line(-5,-4){27.00}}
\put(84.00,57.00){\line(-6,5){16.00}}
\put(66.00,72.00){\line(-5,4){9.00}}
\put(88.00,37.00){\line(0,-1){24.00}}
\put(83.00,22.00){\line(5,-3){25.00}}
\put(93.00,22.00){\line(-5,-3){25.00}}
\put(9.00,75.00){\circle*{1.00}}
\put(74.00,77.00){\circle*{1.00}}
\put(102.00,77.00){\circle*{1.00}}
\put(102.00,64.33){\circle*{1.00}}
\put(74.00,65.33){\circle*{1.00}}
\put(88.00,19.00){\circle*{1.00}}
\bezier{172}(31.00,84.00)(47.00,70.00)(31.00,56.00)
\bezier{172}(41.00,84.00)(25.00,70.00)(41.00,56.00)
\bezier{136}(30.00,34.00)(42.00,20.00)(30.00,6.00)
\bezier{144}(42.00,34.00)(30.00,20.00)(42.00,6.00)
\put(36.00,20.00){\circle*{1.00}}
\put(36.00,78.50){\circle*{1.00}}
\put(36.00,61.50){\circle*{1.00}}
\put(36.00,52.00){\vector(0,-1){11.00}}
\put(36.00,1.00){\makebox(0,0)[cc]{\bf \u R2}}
\end{picture}
\end{center}
\caption{Adjacencies of discriminant strata}
\label{rmx}
\end{figure}

Further we shall denote the group $H_*(\CC_*(\Theta,n))$ simply by
$H_*(\Theta,n)$.

\begin{proposition}
\label{justa}
For any $n \ge 3$ and $\Theta$, the first order cohomology group
of the space $\K(\Theta,n) \sm \Sigma(\Theta,n)$ is related with
the homology group
of the complex $($\ref{short}$)$ by the canonical isomorphism
\begin{equation}
\label{just}
H_k(\Theta,n) \simeq
H^{n-k-1}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta,n)).
\end{equation}
\end{proposition}

{\em Proof.} By the construction, for any $i$ there is
a natural isomorphism
$H_k(\Theta,n) \simeq \bar H_{\omega-n+k}(\sigma_1(\Theta,n)).$
By definition, the group in the right-hand part of
(\ref{just}) is the image of the latter group under the
composite homomorphism (\ref{finap}). Therefore it remains to prove
that this homomorphism is
injective. For $n>4$ this follows immediately from
the shape of the spectral sequence (\ref{sse}), in which the
considered groups form the column $E_1^{-1,*}$ by the rule
$E_1^{-1,q} \simeq H_{n-q-1}(\Theta,n)$.
For $n=4$ we need to prove additionally that the arrow
$E_1^{-2,5} \to E_1^{-1,5}$ of the first page of this spectral sequence
is trivial; this follows from the fact that, as we shall see in
Theorem \ref{homeven}, the complex (\ref{short}) is acyclic in the term
$\CC_0$. For $n=3$ we need to check the triviality of a) all arrows,
acting into the cell $E_1^{-1,3}$, and b) of the arrow
$E_1^{-2,2} \to E_1^{-1,2}$. The first condition again follows from the
equality $H_0(\Theta,3)=0$ (see Theorem \ref{homodd}), while the second one
follows from the explicit calculation of the group $E_1^{-1,2}$, which we
accomplish in the proof of Theorem \ref{homodd},
and from Theorem \ref{realiz} describing the realization of all generators
of this group by one-dimensional cohomology classes of the space
$\K(\Theta,3) \sm \Sigma(\Theta,3)$.
\hfill $\square$ \medskip

Actually, the complex (\ref{short}) is the cellular complex calculating
the Borel--Moore homology groups of the space $B_2(\Theta)$ with coefficients
in the local system watching for the orientation of the affine fibre
bundle $\sigma_1(\Theta,n) \to B_2(\Theta).$ This orientation
behaves differently for numbers $n$ of different parities.
In particular, in the case of odd $n$ the numbers of two intervals,
defining a generator of the group $\Pi(\Theta)$, are
commuting indices, and in the case of even $n$ they are
anti-commuting ones. The corresponding notation for the
generator related, say, with the first and the fourth intervals
in the first case will be $1 \cdot 4,$ and in the second $1 \wedge 4$;
as usual we have $1 \cdot 4 = 4\cdot 1$ and $1\wedge 4 =- 4 \wedge 1$.
For instance, two generators of the group
$H^1_{(1)}(\K(\Theta,3) \sm \Sigma(\Theta,3))$, discussed in
Example \ref{exnonti}, can be written as
$(1+4+5)\cdot (2+3)$ and $(1+2+5)\cdot (3+4)$.

The Euler characteristic of the complex (\ref{short}) is obviously
equal to ${\bnm{m}{2}}$ for any $n.$
\medskip


We shall use the following two lemmas which follow easily from definitions.

\begin{lemma}
\label{alt2}
For any two different connected components of the intersection graph
of a chord diagram $\Theta$
either there is a non-singular point in $\R^1$ such that all endpoints
of chords of one component are to the right and of the other
to the left of it, or there is a pair of non-singular points such that
all endpoints of chords of one component lie in the interval between
them, and all endpoints of the other component lie outside this interval.
\hfill $\square$
\end{lemma}

We get a partial order on the set of these connected
components: one component
is {\em subordinate} to the other, if all endpoints of all chords of the
first component are placed in $\R^1$ between the endpoints of some
chord of the second component.

\begin{lemma}
\label{rballs}
For any $n \ge 3$ and any chord diagram $\Theta$ there exists an immersion
$\varphi: \R^1 \to \R^n$ of class
$\K(\Theta,n) \sm \Sigma(\Theta,n)$
and a collection of balls with non-intersecting boundaries in $\R^n$,
being in a one-to-one correspondence with connected components
of the intersection graph of $\Theta$, such that

1$)$ centers of all these balls lie on the {\em axis} of the space
$\R^n$, i.e. on the image of the standard linear embedding
$l:\R^1 \to \R^n$ participating in the definition of the space
$\K(\Theta,n)$,

2$)$ the curve $\varphi(\R^1)$ intersects the boundary of any of
these balls at exactly two antipodal points lying on this axis, and
coincides with this axis close to these intersection points;

3$)$ such a ball contains those and only those
self-intersection points of $\varphi(\R^1),$ which are depicted by the
chords of the connected component of the chord diagram
corresponding to this ball, and of components subordinate to this one;

4$)$ if some connected component of the intersection graph of
$\Theta$ defines a planar subdiagram of $\Theta$,
then inside the corresponding ball,
from which all balls corresponding to its subordinate
subdiagrams are removed, the curve
$\varphi(\R^1)$ lies in some two-dimensional plane.
\hfill $\square$
\end{lemma}

Now we formulate the main homological results of this work.

\begin{theorem}
\label{homodd}
If $n$ is odd then for any $m$-chord diagram $\Theta$
the group $H_0(\Theta,n)$ is trivial, the rank of the
group $H_2(\Theta,n)$ is equal to the sum of the binomial
coefficient ${\bnm{m}{2}}$ and the number of
connected component of the intersection graph of $\Theta$, and the
group $H_1(\Theta,n)$ does not contain 2-torsion, and its rank is equal
to the same number of connected components.
\end{theorem}

\begin{theorem}
\label{homeven}
Let $n$ be even. Then the group $H_0(\Theta,n)$ of the complex
$($\ref{complex}$)$ is trivial, and the group $H_1(\Theta,n)$ is the direct
sum of summands $\Z$, corresponding to all connected components of the
intersection graph of $\Theta$,
defining planar chord subdiagrams in $\Theta$,
and summands $\Z_2$, corresponding to all
non-planar components. For a generator of any such summand $\Z$ or $\Z_2$,
corresponding to any component of this graph, one can take the cycle
$\alpha_{\rra}+\alpha_{\rla} \in \|(\Theta)$, where $\alpha$ is an arbitrary chord
of this component. The group $H_2(\Theta,n)$ is free Abelian, its rank is
equal to the sum of the number ${\bnm{m}{2}}$ and the rank of the group
$H_1(\Theta,n)$ $($i.e. the number $\boxminus(\Theta)$ of connected component
of the intersection graph that correspond to planar subdiagrams in $\Theta)$.
\end{theorem}

Theorems \ref{homoddi} and \ref{homeveni} follow immediately from these two
theorems and Proposition \ref{justa}.


\subsection{Differentials in the complex (\ref{complex})}

In this subsection we start the proof of Theorems \ref{homodd} and
\ref{homeven}, namely, we describe the operators given by arrows
in the diagram (\ref{complex}).

The lower arrow $\Pi'(\Theta) \to \Upsilon(\Theta)$ of
this diagram corresponds to degenerations of type
{\bf \u R1} of Fig. \ref{rmx}, the arrows
$\Pi(\Theta) \to \|(\Theta)$ and $\Pi'(\Theta) \to \|(\Theta)$
to degenerations of type {\bf \u R2}, and
the arrows
$\Pi(\Theta) \to \Xi(\Theta)$ and $\Pi'(\Theta) \to \Xi(\Theta)$
to degenerations of type
{\bf \u R3}. In the upper parts of the second and the third columns of
Fig. \ref{rmx} two fat points, corresponding to self-intersections
of the curve $f(\R^1)$,
have different meanings: one of these intersections
(and the points of $\R^1$ participating in it) is fixed for all
points of the space $\K(\Theta,n)$ and is expressed by some chord
of the diagram $\Theta$, while the second one depends on the
discriminant point $f$, and the points participating in it
(or one of these points)
tend somehow to the endpoints of the previous chord; this
self-intersection is encoded by a broken line.

It is easy to see that for any $n$ the lower horizontal
arrow in (\ref{complex})
is an isomorphism, and hence calculating the homology of the complex
(\ref{short}) we can replace the complex (\ref{complex}) by
the complex (\ref{compred}), obtained from it by removing the lower
line and all arrows issuing from it.

\begin{equation}
\begin{array}{c}
\mbox{
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(71.00,24.00)
\put(3.00,21.00){\makebox(0,0)[cc]{$\Pi(\Theta)$}}
\put(37.00,21.00){\makebox(0,0)[cc]{$\Xi(\Theta)$}}
\put(71.00,21.00){\makebox(0,0)[cc]{$\Omega(\Theta)$}}
\put(37.00,3.00){\makebox(0,0)[cc]{$\|(\Theta)$}}
\put(71.00,3.00){\makebox(0,0)[cc]{$\Psi(\Theta)$}}
\put(13.00,21.00){\vector(1,0){13.00}}
\put(47.00,21.00){\vector(1,0){13.00}}
\put(13.00,16.00){\vector(2,-1){14.00}}
\put(47.00,16.00){\vector(2,-1){14.00}}
\put(47.00,3.00){\vector(1,0){13.00}}
\put(19.00,24.00){\makebox(0,0)[cc]{$\beta$}}
\put(22.00,14.00){\makebox(0,0)[cc]{$\zeta$}}
\put(54.00,24.00){\makebox(0,0)[cc]{$\epsilon$}}
\put(55.00,14.00){\makebox(0,0)[cc]{$\phi$}}
\put(54.00,6.00){\makebox(0,0)[cc]{$\psi$}}
\end{picture}
}
\end{array}
\label{compred}
\end{equation}

\begin{proposition}
\label{diffs}
For some natural choice of orientations of strata corresponding to
generators of the complex $($\ref{compred}$)$, the parts of boundary
operators, defining five arrows in
$($\ref{compred}$)$, are defined by the following pictures.

\begin{enumerate}

\item Operators
$\beta:\Pi(\Theta) \to \Xi(\Theta)$ and
$\zeta: \Pi(\Theta) \to \|(\Theta)$ by Figures
\ref{easiest0} and \ref{selftan0} respectively,
on these pictures the small ovals surround the endpoints of
chords tending to each other at the corresponding degeneration;

\item The operator $\psi: \|(\Theta) \to \Psi(\Theta)$ by Fig. \ref{difc};

\item The operator $\phi: \Xi(\Theta) \to \Psi(\Theta)$ by the left-hand
part of Fig. \ref{bf};

\item The operator $\epsilon: \Xi(\Theta) \to \Omega(\Theta)$ by
Fig.~\ref{bevar}. Strictly speaking, there are many $($namely, 24$)$
possible mutual dispositions of two chords and one broken line
issuing from one of them and colliding at this degeneration with the
other chord. However there are only four essentially different variants,
whose representatives are shown in Fig. \ref{bevar}: they differ
from one another by a)
whether the left-hand endpoint of the chord carrying the endpoint of the
broken line is further to the left than the left-hand endpoint of the other
chord, and b) by the side from which an endpoint of the broken
line tends to an endpoint of the chord. For all representatives of
any of these four classes the incidence coefficient will be the same as for
its representative shown in Fig.~\ref{bevar}.
\end{enumerate}
\end{proposition}

The choice of orientations of strata and the proof of this proposition
will be given in \S \ref{coors}.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(128.67,10.33)
\put(0.00,5.20){\makebox(0,0)[cc]{$\beta$}}
\put(8.33,5.00){\oval(8.00,6.00)[t]}
\put(17.33,10.00){\line(0,-1){5.00}}
\put(19.33,5.00){\line(-1,0){17.00}}
\put(21.67,5.00){\makebox(0,0)[cc]{=}}
\put(26.00,5.20){\makebox(0,0)[cc]{$- \beta$}}
\put(53.00,5.20){\makebox(0,0)[cc]{$- \beta$}}
\put(79.00,5.20){\makebox(0,0)[cc]{$\beta$}}
\put(36.34,5.00){\oval(8.00,6.00)[t]}
\put(62.33,5.00){\oval(8.00,6.00)[t]}
\put(86.33,5.00){\oval(8.00,6.00)[t]}
\put(117.67,5.00){\oval(8.00,6.00)[t]}
\put(44.34,10.00){\line(0,-1){5.00}}
\put(71.33,10.00){\line(0,-1){5.00}}
\put(96.33,10.00){\line(0,-1){5.00}}
\put(126.67,10.00){\line(0,-1){5.00}}
\put(46.34,5.00){\line(-1,0){17.00}}
\put(73.33,5.00){\line(-1,0){17.00}}
\put(128.67,5.00){\line(-1,0){17.00}}
\put(48.67,5.00){\makebox(0,0)[cc]{=}}
\put(75.67,5.00){\makebox(0,0)[cc]{=}}
\put(100.00,5.00){\makebox(0,0)[cc]{=}}
\put(3.33,5.00){\line(0,1){5.00}}
\put(3.33,10.00){\line(1,0){14.00}}
\put(67.33,5.00){\line(0,1){5.00}}
\put(67.33,10.00){\line(1,0){4.00}}
\put(89.83,5.00){\oval(3.00,2.00)[]}
\put(66.83,5.00){\oval(3.00,2.00)[]}
\put(32.84,5.00){\oval(3.00,2.00)[]}
\put(3.83,5.00){\oval(3.00,2.00)[]}
\put(117.66,8.00){\line(0,1){2.00}}
\put(117.66,10.00){\line(1,0){9.00}}
\put(44.34,10.00){\line(-1,0){11.00}}
\put(33.34,10.00){\line(0,-1){5.00}}
\put(96.33,10.00){\line(-1,0){7.00}}
\put(89.33,10.00){\line(0,-1){5.00}}
\put(107.00,5.67){\makebox(0,0)[cc]{$(-1)^n$}}
\put(81.00,5.00){\line(1,0){16.33}}
\end{picture}


\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(122.33,10.00)
\put(26.67,5.20){\makebox(0,0)[cc]{$- \beta$}}
\put(41.00,5.00){\oval(8.00,6.00)[t]}
\put(44.00,10.00){\line(0,-1){5.00}}
\put(47.00,5.00){\line(-1,0){17.00}}
\put(102.33,5.00){\makebox(0,0)[cc]{=}}
\put(44.50,5.00){\oval(3.00,2.00)[]}
\put(1.00,5.20){\makebox(0,0)[cc]{$\beta$}}
\put(13.00,5.00){\oval(8.00,6.00)[t]}
\put(18.00,10.00){\line(0,-1){5.00}}
\put(20.00,5.00){\line(-1,0){17.00}}
\put(22.33,5.00){\makebox(0,0)[cc]{=}}
\put(4.00,5.00){\line(0,1){5.00}}
\put(4.00,10.00){\line(1,0){14.00}}
\put(17.50,5.00){\oval(3.00,2.00)[]}
\put(122.33,5.00){\line(-1,0){17.00}}
\put(116.33,5.00){\oval(8.00,6.00)[t]}
\put(107.33,5.00){\line(0,1){5.00}}
\put(107.33,10.00){\line(1,0){9.00}}
\put(116.33,10.00){\line(0,-1){2.00}}
\put(53.67,5.20){\makebox(0,0)[cc]{$- \beta$}}
\put(68.00,5.00){\oval(8.00,6.00)[t]}
\put(74.00,5.00){\line(-1,0){17.00}}
\put(63.83,5.00){\oval(3.00,2.00)[]}
\put(80.33,5.20){\makebox(0,0)[cc]{$\beta$}}
\put(92.33,5.00){\oval(8.00,6.00)[t]}
\put(99.33,5.00){\line(-1,0){17.00}}
\put(77.00,5.00){\makebox(0,0)[cc]{=}}
\put(88.83,5.00){\oval(3.00,2.00)[]}
\put(44.00,10.00){\line(-1,0){12.00}}
\put(32.00,10.00){\line(0,-1){5.00}}
\put(49.33,5.00){\makebox(0,0)[cc]{=}}
\put(83.33,5.00){\line(0,1){5.00}}
\put(83.33,10.00){\line(1,0){6.00}}
\put(89.33,10.00){\line(0,-1){5.00}}
\put(58.00,5.00){\line(0,1){5.00}}
\put(58.00,10.00){\line(1,0){5.00}}
\put(63.00,10.00){\line(0,-1){5.00}}
\end{picture}


\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(130.00,9.00)
\put(0.00,5.50){\makebox(0,0)[cc]{$(-1)^n \beta$}}
\put(24.00,5.00){\line(-1,0){17.00}}
\put(27.00,5.00){\makebox(0,0)[cc]{=}}
\put(37.00,5.50){\makebox(0,0)[cc]{$(-1)^{n-1} \beta$}}
\put(68.00,5.50){\makebox(0,0)[cc]{$- \beta$}}
\put(93.33,5.50){\makebox(0,0)[cc]{$\beta$}}
\put(88.33,5.00){\line(-1,0){17.00}}
\put(63.67,5.00){\makebox(0,0)[cc]{=}}
\put(90.33,5.00){\makebox(0,0)[cc]{=}}
\put(114.33,5.00){\makebox(0,0)[cc]{=}}
\put(109.83,5.00){\oval(3.00,2.00)[]}
\put(86.16,5.00){\oval(3.00,2.00)[]}
\put(48.33,5.00){\oval(3.00,2.00)[]}
\put(8.50,5.00){\oval(3.00,2.00)[]}
\put(16.00,5.00){\oval(14.00,8.00)[t]}
\put(54.33,5.00){\oval(12.00,8.00)[t]}
\put(80.00,5.00){\oval(14.00,8.00)[t]}
\put(103.00,5.00){\oval(12.00,8.00)[t]}
\put(123.66,5.00){\oval(10.00,8.00)[t]}
\put(123.66,5.00){\line(0,1){4.00}}
\put(8.00,5.00){\line(0,-1){4.00}}
\put(8.00,1.00){\line(1,0){8.00}}
\put(16.00,1.00){\line(0,1){4.00}}
\put(49.33,5.00){\line(0,-1){4.00}}
\put(49.33,1.00){\line(1,0){5.00}}
\put(54.33,1.00){\line(0,1){4.00}}
\put(80.00,5.00){\line(0,-1){4.00}}
\put(80.00,1.00){\line(1,0){6.00}}
\put(86.00,1.00){\line(0,1){4.00}}
\put(103.00,5.00){\line(0,-1){4.00}}
\put(103.00,1.00){\line(1,0){8.00}}
\put(111.00,1.00){\line(0,1){4.00}}
\put(112.00,5.00){\line(-1,0){16.33}}
\put(130.00,5.00){\line(-1,0){12.67}}
\put(46.00,5.00){\line(1,0){15.33}}
\end{picture}
\end{center}
\caption{Boundary operator at the easiest degeneration}
\label{easiest0}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(84.00,13.00)
\put(42.00,9.00){\line(1,0){18.00}}
\put(25.00,9.00){\makebox(0,0)[cc]{=}}
\put(0.50,9.50){\makebox(0,0)[cc]{$\zeta$}}
\put(34.00,9.50){\makebox(0,0)[cc]{$(-1)^n \zeta$}}
\put(3.00,9.00){\line(1,0){18.00}}
\put(64.00,9.00){\makebox(0,0)[cc]{=}}
\put(76.00,9.00){\oval(12.00,8.00)[t]}
\put(84.00,9.00){\line(-1,0){16.00}}
\put(6.00,9.00){\oval(4.00,2.00)[]}
\put(18.00,9.00){\oval(4.00,2.00)[]}
\put(45.00,9.00){\oval(4.00,2.00)[]}
\put(57.00,9.00){\oval(4.00,2.00)[]}
\put(11.00,9.00){\oval(12.00,8.00)[t]}
\put(7.00,9.00){\line(0,-1){4.00}}
\put(7.00,5.00){\line(1,0){12.00}}
\put(19.00,5.00){\line(0,1){4.00}}
\put(52.00,9.00){\oval(12.00,8.00)[t]}
\put(56.00,9.00){\line(0,-1){4.00}}
\put(56.00,5.00){\line(-1,0){12.00}}
\put(44.00,5.00){\line(0,1){4.00}}
\put(73.00,5.00){\vector(1,0){6.00}}
\put(73.00,3.00){\vector(1,0){6.00}}
\put(82.00,2.00){\makebox(0,0)[cc]{1}}
\put(82.00,6.00){\makebox(0,0)[cc]{2}}
\end{picture}

\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(85.00,11.00)
\put(3.00,7.00){\line(1,0){18.00}}
\put(24.00,7.00){\makebox(0,0)[cc]{=}}
\put(0.50,7.50){\makebox(0,0)[cc]{$\zeta$}}
\put(33.00,7.50){\makebox(0,0)[cc]{$(-1)^n \zeta$}}
\put(41.00,7.00){\line(1,0){18.00}}
\put(64.00,7.00){\makebox(0,0)[cc]{=}}
\put(77.00,7.00){\oval(12.00,8.00)[t]}
\put(85.00,7.00){\line(-1,0){16.00}}
\put(50.00,7.00){\oval(14.00,8.00)[t]}
\put(45.00,7.00){\line(0,-1){4.00}}
\put(45.00,3.00){\line(1,0){10.00}}
\put(55.00,3.00){\line(0,1){4.00}}
\put(12.00,7.00){\oval(10.00,8.00)[t]}
\put(19.00,7.00){\line(0,-1){4.00}}
\put(19.00,3.00){\line(-1,0){14.00}}
\put(5.00,3.00){\line(0,1){4.00}}
\put(74.00,4.00){\vector(1,0){6.00}}
\put(80.00,2.00){\vector(-1,0){6.00}}
\put(82.00,5.00){\makebox(0,0)[cc]{1}}
\put(82.00,1.00){\makebox(0,0)[cc]{2}}
\put(44.00,7.00){\oval(4.00,2.00)[]}
\put(56.00,7.00){\oval(4.00,2.00)[]}
\put(6.00,7.00){\oval(4.00,2.00)[]}
\put(18.00,7.00){\oval(4.00,2.00)[]}
\end{picture}
\end{center}
\caption{Birth of a self-tangency}
\label{selftan0}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(100.00,28.00)
\put(1.00,6.00){\makebox(0,0)[cc]{$\psi$ }}
\put(4.00,6.00){\line(1,0){22.00}}
\put(14.00,6.00){\oval(14.00,10.00)[t]}
\put(10.00,3.00){\vector(1,0){8.00}}
\put(18.00,1.00){\vector(-1,0){8.00}}
\put(32.00,6.00){\makebox(0,0)[cc]{$=$}}
\put(38.00,6.00){\line(1,0){22.00}}
\put(76.00,6.00){\line(1,0){22.00}}
\put(48.00,6.00){\oval(14.00,10.00)[t]}
\put(86.00,6.00){\oval(14.00,10.00)[t]}
\put(55.00,6.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(79.00,6.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(68.00,6.00){\makebox(0,0)[cc]{$-$}}
\put(1.00,23.00){\makebox(0,0)[cc]{$\psi$ }}
\put(4.00,23.00){\line(1,0){22.00}}
\put(14.00,23.00){\oval(14.00,10.00)[t]}
\put(10.00,20.00){\vector(1,0){8.00}}
\put(32.00,23.00){\makebox(0,0)[cc]{$= \ -$}}
\put(40.00,23.00){\line(1,0){22.00}}
\put(78.00,23.00){\line(1,0){22.00}}
\put(50.00,23.00){\oval(14.00,10.00)[t]}
\put(88.00,23.00){\oval(14.00,10.00)[t]}
\put(57.00,23.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(81.00,23.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(70.00,23.00){\makebox(0,0)[cc]{$+ \ (-1)^n$}}
\put(10.00,18.00){\vector(1,0){8.00}}
\end{picture}
\caption{Boundary of a self-tangency}
\label{difc}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(125.00,19.00)
\put(1.00,13.50){\makebox(0,0)[cc]{$\phi$}}
\put(13.00,13.00){\oval(10.00,8.00)[t]}
\put(13.00,17.00){\line(0,1){2.00}}
\put(13.00,19.00){\line(-1,0){7.00}}
\put(6.00,19.00){\line(0,-1){6.00}}
\put(7.00,13.00){\oval(4.00,2.00)[]}
\put(4.00,13.00){\line(1,0){16.00}}
\put(23.00,13.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,13.50){\makebox(0,0)[cc]{$-$}}
\put(41.00,13.00){\oval(12.00,8.00)[t]}
\put(41.00,17.00){\line(-1,-1){4.00}}
\put(36.00,13.00){\oval(4.00,2.00)[]}
\put(33.00,13.00){\line(1,0){16.00}}
\put(52.00,13.00){\makebox(0,0)[cc]{$=$}}
\put(63.00,13.00){\oval(12.00,8.00)[t]}
\put(57.00,13.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(55.00,13.00){\line(1,0){16.00}}
\put(1.00,1.50){\makebox(0,0)[cc]{$\phi$}}
\put(40.00,1.00){\oval(10.00,8.00)[t]}
\put(40.00,5.00){\line(0,1){2.00}}
\put(40.00,7.00){\line(1,0){7.00}}
\put(47.00,7.00){\line(0,-1){6.00}}
\put(46.00,1.00){\oval(4.00,2.00)[]}
\put(49.00,1.00){\line(-1,0){16.00}}
\put(23.00,1.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,1.50){\makebox(0,0)[cc]{$-$}}
\put(12.00,1.00){\oval(12.00,8.00)[t]}
\put(12.00,5.00){\line(1,-1){4.00}}
\put(17.00,1.00){\oval(4.00,2.00)[]}
\put(20.00,1.00){\line(-1,0){16.00}}
\put(52.00,1.00){\makebox(0,0)[cc]{$=$}}
\put(63.00,1.00){\oval(12.00,8.00)[t]}
\put(69.00,1.00){\makebox(0,0)[cc]{{\large $\star$}}}
\put(55.00,1.00){\line(1,0){16.00}}
\bezier{116}(93.00,0.00)(103.00,10.00)(118.00,10.00)
\bezier{36}(118.00,10.00)(125.00,10.00)(125.00,7.00)
\bezier{32}(125.00,7.00)(125.00,4.00)(119.00,4.00)
\bezier{112}(119.00,4.00)(103.00,4.00)(93.00,14.00)
\put(93.50,13.50){\vector(-1,1){1.00}}
\put(30.00,13.50){\makebox(0,0)[cc]{$\phi$}}
\put(30.00,1.50){\makebox(0,0)[cc]{$\phi$}}
\end{picture}
\caption{Vanishing loop}
\label{bf}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(93.00,31.00)
\put(1.00,25.00){\makebox(0,0)[cc]{$\epsilon$ }}
\put(6.00,25.00){\line(0,1){6.00}}
\put(6.00,31.00){\line(1,0){12.00}}
\put(18.00,31.00){\line(0,-1){2.00}}
\put(18.00,25.00){\oval(8.00,8.00)[t]}
\put(24.00,25.00){\line(-1,0){20.00}}
\put(13.00,25.00){\oval(10.00,8.00)[b]}
\put(7.00,25.00){\oval(4.00,2.00)[]}
\put(34.00,25.00){\makebox(0,0)[cc]{$-\epsilon$ }}
\put(51.00,31.00){\line(0,-1){2.00}}
\put(51.00,25.00){\oval(8.00,8.00)[t]}
\put(57.00,25.00){\line(-1,0){20.00}}
\put(40.00,25.00){\oval(4.00,2.00)[]}
\put(45.00,25.00){\oval(12.00,8.00)[b]}
\put(41.00,25.00){\line(0,1){6.00}}
\put(41.00,31.00){\line(1,0){10.00}}
\put(28.00,25.00){\makebox(0,0)[cc]{$=$}}
\put(61.00,25.00){\makebox(0,0)[cc]{$=$}}
\put(74.00,25.00){\oval(12.00,8.00)[b]}
\put(78.00,25.00){\oval(12.00,8.00)[t]}
\put(78.00,29.00){\line(-1,-2){4.00}}
\put(66.00,25.00){\line(1,0){20.00}}
\put(10.00,5.00){\oval(8.00,8.00)[t]}
\put(10.00,9.00){\line(0,1){2.00}}
\put(10.00,11.00){\line(1,0){12.00}}
\put(22.00,11.00){\line(0,-1){6.00}}
\put(15.00,5.00){\oval(10.00,8.00)[b]}
\put(21.00,5.00){\oval(4.00,2.00)[]}
\put(4.00,5.00){\line(1,0){20.00}}
\put(43.00,5.00){\oval(8.00,8.00)[t]}
\put(43.00,9.00){\line(0,1){2.00}}
\put(54.00,5.00){\oval(4.00,2.00)[]}
\put(37.00,5.00){\line(1,0){20.00}}
\put(49.00,5.00){\oval(12.00,8.00)[b]}
\put(43.00,11.00){\line(1,0){10.00}}
\put(53.00,11.00){\line(0,-1){6.00}}
\put(1.00,5.00){\makebox(0,0)[cc]{$\epsilon$}}
\put(34.00,5.00){\makebox(0,0)[cc]{$-\epsilon$ }}
\put(28.00,5.00){\makebox(0,0)[cc]{$=$}}
\put(61.00,5.00){\makebox(0,0)[cc]{=}}
\put(68.00,5.00){\makebox(0,0)[cc]{$(-1)^n$}}
\put(85.00,5.00){\oval(12.00,8.00)[b]}
\put(81.00,5.00){\oval(12.00,8.00)[t]}
\put(81.00,9.00){\line(3,-5){4.80}}
\put(93.00,5.00){\line(-1,0){20.00}}
\end{picture}
\caption{Collision of chords}
\label{bevar}
\end{center}
\end{figure}

\begin{corollary}
1. For any $\Theta,$ all complexes $\CC_*(\Theta,n),$ corresponding
to the numbers $n \ge 3$ of one and the same parity, are isomorphic to
each other. In particular, the notation $\CC_*(\Theta,\mbox{\rm even}),$
$\CC_*(\Theta,\mbox{\rm odd}),$ $H_*(\Theta,\mbox{\rm even}),$
and $H_*(\Theta,\mbox{\rm odd})$
is correct.

2. For any $\Theta,$ all complexes $\CC_*(\Theta,n) \otimes \Z_2,$
corresponding to all possible $n \ge 3,$ are isomorphic to each other.
\hfill $\square$
\end{corollary}

Here is another description of the operator
$\beta: \Pi(\Theta) \to \Xi(\Theta)$
of the upper line in (\ref{complex}).

For any $a=1, \dots, 2m+1,$ i.e. the number of some interval
bounded by the endpoints of chords, denote by $a^{\ua}$ (respectively,
$a^{\da}$) the chord whose endpoint bounds the $a$-th interval from
the right (respectively, from the left) in $\R^1$
(if such a chord exists, i.e. if this interval does not tend to
the infinity). Of course, $a^{\ua} \equiv (a+1)^{\da}$. Then the
horizontal arrow $\beta: \Pi(\Theta) \to \Xi(\Theta)$ sends any
generator $a \cdot b,$ $a<b$, of the complex $\CC_*(\Theta,\mbox{odd})$
into the sum
\begin{equation}
-a\cdot b^{\ua} +a\cdot b^{\da} - b\cdot a^{\ua}+b\cdot a^{\da},
\label{altodd}
\end{equation}
and sends the generator $a \wedge b$ of the complex
$\CC_*(\Theta,\mbox{even})$ into
\begin{equation}
- a \wedge b^{\ua} +a \wedge b^{\da} +b\wedge a^{\ua}-b\wedge a^{\da}.
\label{altev}
\end{equation}
In a similar way, the operator $\Pi'(\Theta) \to \Xi(\Theta)$
in (\ref{complex})
sends the generator corresponding to the interval $a$ into
$-a \cdot a^{\ua}+a \cdot a^{\da}$ or
$-a \wedge a^{\ua}-a \wedge a^{\da}$,
depending on the parity of $n$.
\medskip

The operator $\psi$, defined by the horizontal arrow
$\|(\Theta) \to \Psi(\Theta)$,
obviously splits into the sum of
$m$ isomorphic operators $\Z^2 \to \Z^2$, any of which
corresponds to its own chord.
Any such operator is described by Fig.~\ref{difc}.
This implies immediately

\begin{lemma}
\label{lemone}
In the case of odd $n$ the top homology group
$H_2(\Theta,n)$ of the complex $($\ref{short}$)$
coincides with the kernel of the operator
$\beta: \Pi(\Theta) \to \Xi(\Theta)$.
\end{lemma}

Indeed, if the restriction of the arrow
$\Pi(\Theta) \to \|(\Theta)$
to the kernel of the arrow
$\Pi(\Theta) \to \Xi(\Theta)$
would be non-trivial, then the image of this restriction would contain
non-trivial elements of the kernel of the arrow
$\|(\Theta) \to \Psi(\Theta)$. But such elements do not exist because for
odd $n$ the matrix of coefficients in the right-hand part of
Fig. \ref{difc} is non-degenerate.
\hfill $\square$

\begin{example} \rm
\label{exnont}
It is easy to check that for the non-planar 2-chord diagram $\Theta$
some generators of the group $H_2(\Theta,\mbox{odd})$ indeed are
equal to $(1+4+5)\cdot (2+3)$ and $(1+2+5)\cdot (3+4),$
see Example \ref{exnonti}.
\end{example}

Let us discuss how the group $H_2(\Theta,3)$ is related with similar
groups for connected components of the intersection graph of $\Theta$.

\begin{proposition}
\label{decom}
Suppose that the intersection graph of the diagram $\Theta$
consists of $r$ connected components, defining subdiagrams
$\Theta_1, \dots, \Theta_r$ of $\Theta$,
consisting of $m_1, \dots, m_r$ chords respectively,
$m_1 + \dots + m_r=m.$
Then the rank of the group $H_2(\Theta,3)$ is not less than
\begin{equation}
\sum_{k=1}^r \mbox{\rm rank } H_2(\Theta_k,3) + \sum_{i<j}^r m_i m_j \ .
\end{equation}
\end{proposition}

(In \S \ref{prhomodd} we shall prove that the opposite inequality also holds,
in full conformity with Theorem \ref{homodd}.)


{\em Proof of Proposition}.
We realize first the second summand: with any pair of chords
from different components we associate an independent generator
of the group $H_2(\Theta,3)$.

If both endpoints of one of these chords lie $\R^1$ to the one
side of both endpoints of the other,
then we consider two loops of the curve
$\varphi(\R^1),$ formed by the images of the segments of $\R^1$
bounded by
endpoints of the first (respectively, the second) chord.
These two loops do not meet in $\R^3,$ therefore their linking number
is well defined; it
is expressed by the cycle of the complex $\CC_*(\Theta,3),$
equal to the sum of all products $a \cdot b,$ where $a$ (respectively, $b$)
is one of non-singular intervals placed between the endpoints of the first
(respectively, the second) chord.
If both endpoints of one chord lie between the endpoints of the other
then we take the linking number of the
loop formed by the image of the segment between the endpoints of the
``interior'' chord, and the infinite closed curve, formed by the
complement of the similar segment for the
``exterior'' one.

Now let us realize the first summand in (\ref{decom}).
For the $i$-th connected component
$\Theta_i$ of the intersection graph of the diagram $\Theta$, let us
define an injection
$\CC_2(\Theta_i,3) \to \CC_2(\Theta,3).$
Namely, with any non-singular interval, bounded by the endpoints
of chords of $\Theta_i$, we associate the sum of all non-singular
intervals bounded by endpoints of chords of $\Theta$
covered by it. Let us extend this association by multiplicativity to all
products $a \cdot b$ of such intervals, i.e. to generators of the
group $\CC_2(\Theta_i,3)$, and by linearity to entire this group.
It is easy to see that this extension sends cycles into cycles.

Let us prove that all cycles obtained in this way are linearly independent in
$\CC_2(\Theta,3).$ Any pair of different chords of diagram $\Theta$
defines a linear function on the group $H_2(\Theta,3)$.
Namely, suppose that the numbers of components of the set
$\R^1 \sm \Theta$, neighboring to the endpoints of the first
(respectively, the second) chord are $a, a+1, b$ and $b+1,$ $a<b$
(respectively, $c, c+1, d$ and $d+1$, $d>c$). Then the value of our linear
function on the element $F \in H_2(\Theta,3)$, expressed in the form
(\ref{licomi}), is equal to
$\lambda_{a+1,c+1}-\lambda_{a,c+1}-\lambda_{a+1,c}+\lambda_{a,c}.$
It is easy to check that if our two chords belong to different connected
components of the intersection graph, then this function vanishes on all
above constructed cycles, realizing the first summand in (\ref{decom}),
and also on all basis cycles, realizing the second summand, except for
exactly one such cycle, which is constructed from this pair of chords;
on the latter cycle this function takes value 1 or $-1$.
Therefore these functions define a surjective homomorphism of the group
$H_2(\Theta,3)$ onto a free Abelian group of rank
$\sum_{i<j}^r m_i m_j ;$ the kernel of this homomorphism contains all
$\sum_{k=1}^r \mbox{\rm rank } H_2(\Theta_k,3)$ cycles
constructed previously and realizing the first summand in (\ref{decom}).
It remains to prove that all these cycles (or, equivalently, the invariants
corresponding to them) are linearly independent. But they are obviously
independent already in the restriction to the set of maps $f$ of class
$\K(\Theta,3) \sm \Sigma(\Theta,3)$, satisfying conditions
1), 2) and 3) of Lemma \ref{rballs}.
\hfill $\square$ \medskip


\subsection{Proof of Theorem \ref{homeven}}
\label{prhomeven}

Let us prove first that $H_0(\Theta,\mbox{even})=0$.

Any generator $(\alpha, \beta)$ of the group $\Omega(\Theta)$,
such that the left-hand endpoint of the chord $\alpha$ is further
to the left than the left-hand endpoint of $\beta$, is
equal to the boundary of the chain
$(1, \beta) + (2,\beta)+ \dots + (a,\beta),$
where $a$ is the interval bounded from the right by the left-hand
endpoint of the chord $\alpha$, see the upper line of
Fig. \ref{bevar}.

In a similar way, any generator $(f'(u)=0)$ of the group $\Psi(\Theta)$, where
$u$ is the {\em left-hand} endpoint of the chord $\alpha$,
is equal to the boundary of the chain
$(1, \alpha) + (2,\alpha)+ \dots + (a,\alpha),$
see the upper lines of Fig. \ref{bf} and \ref{bevar}. Such a generator
for the right-hand endpoint can be reduced to the left-hand one by
any line of Fig. \ref{difc}.

\begin{lemma}
\label{lem1chet}
Any one-dimensional cycle of the complex
$($\ref{short}$)$ or, equivalently,
$($\ref{compred}$)$ with even $n$ is homologous to a cycle
lying in the subgroup $\|(\Theta)$, i.e. to a linear combination of several
cycles of the form $\alpha_\rra + \alpha_\rla$, where $\alpha$ are some
chords of our diagram. \end{lemma}

{\em Proof.}
Let us order all chords of diagram $\Theta$ with respect to the
place of their left-hand endpoints in the line $\R^1$.
Let us order lexicographically all generators $a \wedge \alpha$
of the group $\Xi(\Theta)$: first we look which generator has a
greater number of the interval $a$, and if these numbers coincide then
whose chord is of greater order. Consider an one-dimensional cycle
of the complex (\ref{compred}). Let $a \wedge \alpha$ be
the greatest in the sense of this order standard generator
of the group $\Xi(\Theta)$, participating in this cycle with a
non-zero coefficient (so that for all other such generators either
the number of the interval is smaller than $a$, or this number is
the same but the left-hand endpoint of the chord is further to the
left than that of the chord $\alpha$).

Then the $a$-th interval cannot be bounded from the right by the
{\em left-hand} endpoint of the chord $\alpha$. Indeed, if it would happen
then the boundary of the generator $a \wedge \alpha$ would contain the
element of the group $\Psi(\Theta)$, corresponding to the condition $f'=0$
at this left-hand endpoint of the chord. This contradicts to the fact
that we consider a {\em cycle} of the complex (\ref{compred}).
Indeed, this summand of the boundary of this cycle should be compensated,
i.e. participate with opposite sign in the boundary of at least one
other summand
of this cycle. Such a summand cannot be a generator of the group
$\Xi(\Theta)$ because of our assumption on the maximality of
$a \wedge \alpha$. Among the generators of $\|(\Theta)$ such summands
could be only $\alpha_\rra$ and $\alpha_\rla$, and these two summands
should participate in our cycle with different coefficients.
But then their boundary contains, also with a non-zero
coefficient, the element of the group $\Psi(\Theta)$ defined by the
condition $f'=0$ at the {\em right-hand} endpoint of the same chord
$\alpha$. This element already cannot be compensated, again because
the class $a \wedge \alpha$ is lexicographically maximal.

Now let us consider the generator $a \wedge b$ of the group
$\Pi(\Theta)$, where $b$ is the interval bounded from the
right by the left-hand endpoint of the chord $\alpha$. The boundary of
this generator consists of the element $- a \wedge \alpha$,
of generators of smaller order in the group $\Xi(\Theta)$, and maybe
one or two generators of the group $\|(\Theta)$. Therefore our lemma
follows by induction over the order of generators
$a \wedge \alpha$. \hfill $\square$ \medskip

\begin{lemma}
\label{lem2chet}
If some connected component of the intersection graph of the
chord diagram $\Theta$ defines a non-planar
chord subdiagram, then for one of its chords
$\alpha$ the cycle $\alpha_\rra + \alpha_\rla \in \|(\Theta)$ defines
an element of order no more than 2 in the group
$H_1(\Theta,\mbox{\rm even})$.
\end{lemma}

{\em Proof.}
Consider an arbitrary generic immersion $f: \R^1 \to \R^n,$
$n \ge 3$, of class $\K(\Theta,n)$. Its image $f(\R^1)$ is a
4-valent graph with two free endpoints. By a {\em cycle} of this graph
we call an arbitrary sequence of oriented edges without repetitions,
such that the end of any edge coincides with the beginning of the next one,
and either the end of the last edge coincides with the beginning of the
first one, or the first and the last edges go to the infinity.
For any two such cycles having no common edges, their {\em crossing
points} are those their common vertices at which they have no breaks.
For instance, for cycles $1+4+5$ and $2+3$ of Example \ref{exnonti} on page
\pageref{exnonti} the first self-intersection point of the curve $f(\R^1)$
is not a crossing point, while the second one is. Now we use the
following theorem by V.O.~Manturov.

\begin{theorem}[see \cite{vasya}]
\label{vassya}
If the chord diagram $\Theta$ is non-planar, then $f(\R^1)$ contains two
cycles without common edges and exactly one crossing point
$($in the previous sense$)$.
\hfill $\square$ \end{theorem}

For such a pair of cycles consider the chain $\sum a \wedge b$, where
$a$ runs over the edges of the first cycle, and $b$ of the second one.
It follows easily from (\ref{altev}) that the boundary of this chain
is equal to
$\pm 2 (\alpha_\rra+\alpha_\rla),$ where $\alpha$ is the chord
corresponding to this crossing point. The previous theorem can be
applied to any connected component of the intersection graph,
therefore in any such component defining a non-planar subdiagram of
$\Theta$
there is a chord $\alpha$
such that the cycle $\alpha_\rra+\alpha_\rla$ defines a homology class of
at most second order.
\hfill $\square$ \medskip

\begin{lemma}
\label{lem3chet}
If chords $\alpha$ and $\beta$ belong to one and the same connected
component of the intersection graph, then the class of the cycle
$(\alpha_\rra + \alpha_\rla)$ in the group $H_1(\Theta,\mbox{\rm even})$
is equal to the class of the cycle
$(\beta_\rra + \beta_\rla)$ or
$-(\beta_\rra + \beta_\rla)$.
\end{lemma}

{\em Proof.}
Suppose that the
 chords $\alpha$, $\beta$ are linked (i.e. the corresponding vertices of
the intersection graph are joined by a segment). Let $p,q$ be the numbers of
endpoints of the chord $\alpha$ among all $2m$ endpoints of chords,
and $r,s$ the numbers of endpoints of $\beta$, and $p<r<q<s$. Denote by
$R(\alpha,\beta)$ the sum of generators $a \wedge b$ of the group
$\Pi(\Theta)$, where $a,b$ are the numbers (counting from the left to
the right in $\R^1$) of all components of $\R^1 \sm \Theta$ such that
$a < b$, the $a$-th component is placed between the endpoints
of the chord $\alpha$ (i.e. $p+1 \le a \le q$), the $b$-th component
is placed between the endpoints of the
chord $\beta$ (i.e. $r+1 \le b \le s$), but these components are not placed
simultaneously between the endpoints of both $\alpha$ and $\beta$ (i.e. the
condition $r+1 \le a < b \le q$ is {\em not} satisfied).

It is easy to check that in the complex
$\CC_*(\Theta,\mbox{even})$ we have
\begin{equation}
\label{film}
\partial R(\alpha, \beta) = (\alpha_\rra + \alpha_\rla) +
 (\beta_\rra + \beta_\rla) + 2\sum (\gamma^i_\rra + \gamma^i_\rla),
\end{equation}
where $\gamma^i$ are all possible chords, both endpoints of which lie
between the marginal endpoints $p$ and $s$ of chords $\alpha, \beta$,
but do not lie simultaneously in one and the same of three intervals,
into which the points $r, q$ divide the interval $[p,s]$. The statement
of Lemma follows from this formula by the induction over the number $s-p$.
\hfill $\square$ \medskip

These lemmas \ref{lem1chet}, \ref{lem2chet}, \ref{lem3chet}
imply an upper estimate on the group
$H_1(\Theta,\mbox{even})$: any non-planar component of the intersection
graph gives us no more than
a summand $\Z_2$ of this group, and any planar
component no more than a cyclic summand. It remains to prove that
these estimates are sharp, in particular that the latter cyclic
summands are free. It will be done in \S \ref{prhomodd} for
summands $\Z_2$ and in the next Lemma \ref{lem4chet} for summands $\Z$.
However, what is already proved is enough for proving the
following statement.

\begin{corollary}
\label{coreu}
Neither the dimension of the group $H_2(\CC_*(\Theta,n) \otimes \Z_2)$,
nor the rank of the group $H_2(\Theta,n)$ for an arbitrary $n \ge 3$
is greater than the sum of ${\bnm{m}{2}}$ and the number of connected
components of the intersection graph of the diagram $\Theta$.
\end{corollary}

This follows from the previous three lemmas and reasoning on
the Euler characteristic. \hfill $\square$

\begin{lemma}
\label{lem4chet}
Suppose that some connected component of the intersection graph of the
chord diagram $\Theta$ corresponds to a planar chord subdiagram,
and $\alpha$ is an arbitrary chord of this subdiagram.
Then the cycle $\alpha_\rra + \alpha_\rla$ defines a free element of the
group $H_1(\Theta,\mbox{\rm even})$, and such free elements,
corresponding to different components of the intersection graph,
are linearly independent in this group. \end{lemma}

{\em Proof.}
The element $\alpha_\rra + \alpha_\rla$ defines the Borel--Moore
cycle of finite codimension in $\Sigma(\Theta,2k)$,
consisting of maps $f: \R^1 \to \R^{2k}$, having a self-tangency
at the self-intersection point depicted by the chord $\alpha$.
It is sufficient to prove that the linking number with this cycle
defines a free element of the group
$H^{2k-2}(\K(\Theta,2k) \sm \Sigma(\Theta,2k))$, $k \ge 2$, and
such elements corresponding to different components are linearly
independent. Let us do it. We choose the direction ``up'' in $\R^{2k}$,
not coinciding with the direction of the axis $l(\R^1)$, and denote
by $\R^{2k-1}$ the quotient space of $\R^{2k}$ by this direction.
Let $x<y$ be the endpoints of the chord $\alpha$. For the
relative Borel--Moore cycle in
$\K(\Theta,n) \mbox{ mod } \Sigma(\Theta,n)$,
spanning our cycle, we can take the set of all maps
$f: \R^1 \to \R^{2k}$ of the class $\K(\Theta,2k)$ such that the projections
of vectors $f'(x)$ and $f'(y)$ to $\R^{2k-1}$ are linearly dependent, and in
the plane spanned by these two vectors in $\R^{2k}$
there coincide the orientations
$(f'(x),f'(y))$ and $(f'(x), \mbox{direction ``up''})$.

Let $f_0: \R^1 \to \R^{2k}$ be an immersion of class
$\K(\Theta,2k) \sm \Sigma(\Theta,2k)$, described in Lemma
\ref{rballs}. This implies in particular that inside the ball,
corresponding to our component, from which all the smaller balls
are removed, the curve $f_0(\R^1)$ lies in some two-dimensional plane
$L_0$. The set of all oriented two-dimensional planes, containing the
axis of the space $\R^{2k}$, is diffeomorphic to $S^{2k-2}$.
The Euclidean structure in $\R^{2k}$ defines uniquely linear
isometries between all these planes and $L_0$, identical on the axis
$l(\R^1)$. These isometries preserve intersections
of these planes with $r$ balls described in Lemma
\ref{rballs}. Any such isometry defines a new immersion
$f: \R^1 \to \R^{2k}$, coinciding with $f_0$ outside the ball
corresponding to our component or inside smaller balls contained in it,
and obtained from $f_0$ by this isometry
in the remaining part of this ball.
The set of all such immersions $f$ also is diffeomorphic to
$S^{2k-2}$. It is easy to see that the matrix of intersection indices
of obtained spheres $S^{2k-2} \subset \K(\Theta,2k) \sm \Sigma(\Theta,2k),$
corresponding to all planar components of the intersection graph,
with the above-defined relative cycles is diagonal with $\pm 1$ on the
diagonal.  \hfill $\square$ \medskip

This proves the statement of Theorem \ref{homeven} concerning the rank
of the group $H_1(\Theta,\mbox{even})$, and the statement on the group
$H_2(\Theta,even)$ follows from it, the triviality of the group
$H_0(\Theta,even)$ and reasoning with the Euler characteristic.

It remains only to prove that the elements
$(\alpha_\rra + \alpha_\rla)$, corresponding to chords $\alpha$ from
different connected components of the intersection graph, defining
non-planar subdiagrams of $\Theta$,
give independent generators of second order in the group
$H_1(\Theta,\mbox{even})$. This will be done in the next subsection.

\subsection{Proof of Theorem \ref{homodd} and the end
of the proof of Theorem \ref{homeven}}
\label{prhomodd}

The triviality of the group $H_0(\Theta,n)$ with odd $n$
can be proved in exactly the same way as with even $n$.

To calculate $H_2(\Theta,\mbox{odd})$ and $H_1(\Theta,\mbox{odd})$,
let us assume first that the intersection graph of the diagram $\Theta$
is connected.
By Lemma \ref{lemone}, the group $H_2(\Theta,\mbox{odd})$ coincides
with the 2-dimensional homology group of the complex given by the
upper line in (\ref{complex}). Let us study this complex.

\begin{lemma}
The rank of the 1-dimensional homology group of the chain complex
defined by this line is not less than 1.
\end{lemma}

Indeed, the element $a_1 \cdot \alpha_1$ is a 1-cycle of this line.
However, none of its multiples is homologous to zero, because the
total multiplicity, with which all
elements of the form $a_1 \cdot \alpha_p$ and $a_{2m+1} \cdot \alpha_q$
occur in the boundary of any element $a_i \cdot a_j$, is equal to
zero, see (\ref{altodd}).
\hfill $\square$
\medskip

This lemma and the reasoning with the Euler characteristic implies that
the rank of the group $H_2(\Theta,\mbox{odd})$ is not less than
${\bnm{m}{2}}+1$. But it is also not greater than this number
by Corollary \ref{coreu}.

The same considerations imply now that in the case of the connected
intersection graph the rank of the group $H_1(\Theta,\mbox{odd})$
is equal to 1 and this group does not contain the 2-torsion.

In the case of an arbitrary (maybe not connected) intersection graph
we have coinciding upper and lower estimates on the rank of the group
$H_2(\Theta,\mbox{odd})$: the upper one follows from Corollary
\ref{coreu}, and the lower one from Proposition \ref{decom}
and just proved statement concerning the case of
connected graphs.

Now, again by reasoning with the Euler characteristic, we have
that the rank of the group $H_1(\Theta,\mbox{odd})$ is equal to the
number of connected components of the intersection graph.
In \S \ref{prhomeven} we have proved that the dimension of the group
$H_1(\Theta, \mbox{even}) \otimes \Z_2$ does not exceed this number
$\#(\Theta)$.
Since both our complexes coincide after reduction modulo 2,
the formula of universal coefficients implies that
$H_1(\Theta,\mbox{odd})$ has no 2-torsion, and
$H_1(\Theta,\mbox{even}) \otimes \Z_2 \simeq \Z_2^{\#(\Theta)}$.
This terminates the proof of Theorems \ref{homodd} and \ref{homeven}
(modulo the proof of Proposition \ref{diffs} on the differentials
in our complexes).
\hfill $\square$


\subsection{A characterization of non-trivial elements of the 2-torsion
of the group $H_*(\Theta,\mbox{\rm even})$}

As we shall see later, the obstructions to the existence of integer
combinatorial formulas for invariants, corresponding to
elements of the group $H_2(\Theta,3)$, lie in the 2-torsion of
$H_1(\Theta,2)$ and are trivial if the diagram $\Theta$ is planar.
Therefore it is important to recognize whether a 1-dimensional cycle of
the complex $\CC_*(\Theta,\mbox{even})$ is equal to zero in this torsion group
or not. This problem can be solved separately for any connected component
of the intersection graph, therefore we shall assume that this graph
is connected, and the chord diagram $\Theta$ is non-planar.

With any generic immersion
$\varphi : \R^1 \to \R^2$ of class $\K(\Theta,2)$
we associate a homomorphism from the group
$(\Xi(\Theta) \oplus \|(\Theta)) \otimes \Z_2$
into the group of singular 1-chains of the projective line with
coefficients in $\Z_2$. Let us define its action on the generators.
Let us fix some continuous identifications of all
$2m+1$ components of $\R^1 \sm \Theta$ with the standard
(non-closed) 1-simplex. To any standard generator
$a \wedge \alpha$ of the group $\Xi(\Theta)$,
i.e. to the pair (component $a$, chord $\alpha$),
there corresponds the
Gau\ss \ map $a \to \RP^1$, sending any point $x \in a$ to the
direction between $\varphi(x)$ and $\varphi(\alpha)$.
The composition of this map with the previous identification defines
a singular 1-simplex in $\RP^1$. Further, to any generator
$\alpha_{\rla}$ (respectively, $\alpha_{\rra}$)
of the group $\|(\Theta)$ there corresponds the singular segment in
$\RP^1$, sweeping out the directions of all rays in the angle between
the directions of the derivative of the map $\varphi$
at two endpoints of the chord
$\alpha$ (respectively, in the angle between the derivative at one of
these endpoints and minus derivative at the other).

\begin{proposition}
Suppose that the intersection graph of the chord diagram $\Theta$
is connected. Then

1. The homomorphism just constructed sends 1-dimensional cycles
of the complex $($\ref{compred}$)$ reduced modulo $2$ into cycles.

2. For any 1-dimensional cycle of the complex $($\ref{compred}$)$ mod 2,
the class of its image in $H_1(\RP^1, \Z_2)$ under this homomorphism
does not depend on the choice of the generic immersion $\varphi$.

3. These classes in $H_1(\RP^1, \Z_2)$, corresponding to
homologous cycles of this complex, coincide.

4. If the diagram $\Theta$ is non-planar, then the class of our cycle in
the group $H_1(\Theta,2)$ is non-trivial if and only if the obtained
class in $H_1(\RP^1,\Z_2)$ is not equal to 0
for some $($and then any$)$ generic immersion $\varphi$.
\label{haract}
\end{proposition}

{\em Proof.}
1. There are three types of points in $\RP^1$ at which the obtained
singular chain could be not a cycle, namely

\begin{enumerate}

\item
directions, connecting the images $\varphi(\alpha), \varphi(\beta)$
of endpoints of different chords of $\Theta$,

\item tangent directions to the curve $\varphi(\R^1)$ at its
self-intersection points depicted by chords of $\Theta$, and

\item the chosen direction of $\varphi'(\R^1)$
(or, equivalently, $l(\R^1)$) at infinity.
\end{enumerate}

Close to the points of the first (respectively, second) type the
homological condition $\partial = 0$ on the image of the cycle
follows from the triviality of the image of this cycle under the
action of the operator $\Xi(\Theta) \to \Omega(\Theta)$ (respectively,
$\Xi(\Theta) \to \Psi(\Theta)$) in the complex (\ref{complex}) reduced mod 2.
The homological condition of the third type follows from conditions of
the first type and from the fact that the boundary of the image of
any generator $a \wedge \alpha$ of the group $\Xi(\Theta)$
consists of exactly two breakpoints of the first or third type,
depending on the number of finite endpoints of the interval $a$.

2. Any two generic immersions can be connected in $\K(\Theta,2)$ by a
generic path. All local surgeries, which can happen along this path
and affect considerably the image of our cycle, are
a) jumps of the images of self-intersection points
through the images of non-singular pieces, and
b) instant self-tangencies of two branches
at self-intersection points, after which the frames formed by the tangents
of these two branches change their orientations. These two surgeries can be
considered as projections into $\R^2$ of surgeries, whose halves
(terminating by the instant of degeneration) are shown in
Fig. \ref{rmc}. In the surgery of the second type, the structure of the
image of our cycle in $\RP^1$ moves continuously, hence its homology
class does not change. In the surgery of the first type something can happen
only if our cycle contains the generator $a \wedge \alpha$, where $a$
is the preimage of the non-singular piece, and $\alpha$ is the
self-intersection, participating in the surgery. In this case the image
of the similar Gau\ss \ map into the two-fold covering space
of the projective line, i.e. into the space of oriented directions in
$\R^2$, strongly changes. Namely, to this image the fundamental cycle of
this covering space becomes added; since the covering is of multiplicity 2,
below (in $\RP^1$) the homology class mod 2 does not change.

3. We have to prove that the boundary of any generator $a \wedge b$
of the group $\Pi(\Theta)$ is mapped to zero. To do it, we need
to consider three essentially different cases:

\begin{itemize}

\item[A)]
the boundaries of intervals $a$ and $b$ in $\R^1$
do not meet and are not joined by any chord of the chord diagram $\Theta$,

\item[B)]
these two intervals are neighbors in $\R^1$, and

\item[C)]
a pair of endpoints of these intervals is joined by a chord of $\Theta$.
\end{itemize}

(Of course, there are also possible compositions of the last two versions:
two chords connecting different pairs of endpoints of these intervals,
as for intervals 2 and 4 in Fig. \ref{crossi}, or neighboring intervals,
whose remaining endpoints are joined by a chord, as for intervals
2 and 3 of the same picture. However, the study of these composite
cases reduces to the study of elementary ones.)

In the case A), if additionally the images of intervals
$\varphi(a),$ $\varphi(b)$ do not intersect in $\R^2$,
then the Gau\ss \ map, acting into $\RP^1$, is defined and continuous on the
entire closed rectangle $\bar a \times \bar b$, hence its restriction on
the boundary of this rectangle is contractible.
In the general case, this map is defined and continuous on this rectangle,
from which small discs with centers at the points, corresponding to
intersections of these intervals, are removed. The image of the
restriction of the Gau\ss \ map on the boundary of any such disc
runs over $\RP^1$ twice.

In the case B) all the same is true. Indeed, there is only one
point of the boundary of the rectangle, in which the Gau\ss \
map is not defined: it corresponds to the common point of
segments $\bar a$ and $\bar b$. It is easy to see that this map
can be extended to this point by continuity.

Finally, in the case C) the Gau\ss \ map cannot be extended
continuously to the vertex of the rectangle, corresponding to the
pair of points joined by the chord: its restriction to the boundary
of an arbitrarily small quarter disc with the center at this point
runs rapidly over the path in $\RP^1$ between the directions of
derivatives of the map $\varphi$ at these points. But exactly in this
case the boundary of the generator $a \wedge b \in \Pi(\Theta)$
contains an element of the group $\|(\Theta)$, also mapped
by our homomorphism into the same path.

4. This assertion follows from the previous ones and from the fact
that by Theorem \ref{homeven} the group $H_1(\Theta,\mbox{even})$
is isomorphic to $\Z_2$ and generated by an element of the form
$\alpha_\rra+ \alpha_\rla$, for which the degree of our map to
$\RP^1$ is obviously equal to 1. \hfill $\square$

\subsection{Homology groups of complexes $\CC_*(\Theta,n)$, obstructions to
the planarity, and identities on the extra self-intersection points
of a planar curve}

Theorem \ref{homeven} implies immediately one more (in addition to
the Manturov's Theorem \ref{vassya}) polynomial algorithm for
recognizing the planarity of a unicursal curve with a given
order of self-intersections.
Namely, we consider the chord diagram $\Theta$, encoding this order of
self-intersections, construct the corresponding complex
$\CC_*(\Theta,\mbox{even})$, whose differentials are described by
Proposition \ref{diffs}, calculate its 1-homology group, and
check whether it contains 2-torsion.

I hope that this criterion can be extended to a strong method of checking
the embeddability of different objects into Euclidean spaces and
other simple manifolds. Namely, high-dimensional homology groups of
the space of
embeddings of such an object into a space of high dimension can
depend very much on whether this object can be embedded into a space
of some critical dimension.

Two-dimensional cohomology classes of both our complexes, the even and
the odd ones, also provide (different)
obstructions in the problem on the planarity
and in the (very close to it) problem on the amphichierality of a curve with
a given type of transverse self-intersections.

The obstruction to the planarity, given by an arbitrary element of the
group $H_2(\Theta,\mbox{even}),$ can defined in the following way.
We consider an arbitrary generic immersion $f:\R^1 \to \R^2$
of class $\K(\Theta,n)$, and for any its ``superfluous'' self-intersection,
i.e. a point $f(x)=f(y)$ such that $(x,y) \not \in \Theta,$
count with appropriate sign
the coefficient, with which our cycle contains the generator
$\{x\} \wedge \{y\}$ of the group $\Pi(\Theta)$ (where
$\{x\}$ is the component of $\R^1 \sm \Theta$ containing $x$),
 and sum up all these coefficients
over all such superfluous
points. The obtained number will be the same for all
generic immersions of the class
$\K(\Theta,2)$, in particular it should vanish if there is
at least one immersion without superfluous self-intersections.
Of course, this vanishing
is only a necessary condition of the planarity, and hence
is much weaker than the presence of 2-torsion in the group
$H_1(\Theta,2)$, which is also a sufficient condition.

\begin{example} \rm
Consider an arbitrary plane curve with the self-intersection type
indicated in Fig. \ref{crossi} left, e.g. the projection to $\R^2$
of the singular knot shown in the same picture on the right.
For this type $\Theta$ the group $H_2(\Theta,2)$ with integer coefficients
is one-dimensional and generated by the class
$(1-3+5) \wedge (2-4)$. Thus, this class ignores the unique
superfluous self-intersection, shown in the same picture (in which the
3-th and 5-th intervals meet) and is unable even to prove the non-planarity
of this curve. (Fortunately, the similar obstructions mod 2 are in this
case stronger and prove the non-planarity.)

On the other hand, any of two basic invariants, generating
the group $H_2(\Theta,3)$ and considered in \S \ref{exnonti}, proves
even a much stronger assertion (``non-amphichierality''):
there does not exist any map
$f \in \K(\Theta,3) \sm \Sigma(\Theta,3)$ which
is isotopic to its own mirror image.
Indeed, the difference of values of such an
invariant on a fixed immersion $f_0$ of this class
and on its mirror image is odd, hence the same
will be true for any other immersion of class
$\K(\Theta,3) \sm \Sigma(\Theta,3)$.
\end{example}

In any case, the elements of the group $H_2(\Theta,2)$ provide
{\em identities} on the numbers of superfluous intersection points
of different smooth pieces of plane curves.
In our case this identity is obvious: it says that if we smooth down
the angles of cycles given by images of chains
$2-4$ and $1-3+5$, then the intersection index of two obtained
oriented closed curves in $\R^2$ is equal to zero.
However, the similar identities, related with invariants of the
second of higher orders, are less trivial. I hope to
discuss them in a forthcoming work.

\begin{remark} \rm
The known homological {\em deleted product} invariant
(respectively, obstruction) for embeddings into
$\R^3$ (respectively, into $\R^2$) of arbitrary graphs,
see \cite{sar}, also is in some sense a first order invariant
(respectively, obstruction).
This invariant/obstruction is defined as the image of the
fundamental cocycle of $\RP^2$ under the map
$H^2(\RP^2,O) \to H^2(B(K,2),\pm \Z),$
where $O$ is the orientation sheaf on $\RP^2$, $B(K,2)$ is the space
of unordered pairs of different points of the graph $K$,
$\pm \Z$ is the local system with fibre $\Z$, reversing its orientation
over the loops defining the transposition of points, and
the map is defined by any embedding of the graph into $\R^3$;
concerning the notion of the order for invariants of embeddings of graphs
see e.g. \cite{stan}.

Our complexes resemble
the ones calculating these invariants and obstructions, but have many
additional terms responsible for the ``differentiable'' effects
like the forbidden self-tangencies at the self-intersection points.
These terms occur in the neighborhood of the diagonal of the
configuration space (which should be deleted in the deleted product method);
therefore it is natural to call the result of our calculations
the invariants and obstructions of {\em folded product}.
\end{remark}


\section{Coorientations of strata and incidence coefficients}
\label{coors}

In this section we prove Proposition \ref{diffs}
concerning the boundary operators of the
complex (\ref{complex}), in particular fix some orientations of all
strata of the resolved discriminant, generating this complex.
Also we calculate incidence coefficients of some chains in
$\K(\Theta,n),$ providing combinatorial formulas of Polyak--Viro type
for invariants (if $n=3$) and cohomology classes of first order.

Recall that $\sigma_1(\Theta,n)$ is the space of a fiber bundle over
$B_2(\Theta)$, whose fibers are affine subspace of codimension
$n$ in $\K(\Theta,n)$. All strata of the space $B_2(\Theta),$
listed in the diagram (\ref{complex}), are cells, hence the same is true for
preimages of these cells in $\sigma_1(\Theta,n)$.
For any cell in $B_2(\Theta)$ we fix the orientation of this cell
and the {\em coorientation} (i.e. the orientation of the normal bundle)
in $\K(\Theta,n)$ of a typical fibre over it.
The incidence coefficient of cells of neighboring dimensions in
$\sigma_1(\Theta,n)$ can be calculated as the algebraic number of
entries of the smaller cell in the boundary of the greater
one, where the sign of any entry is equal to the product of a) the similar
sign for the corresponding entry of the smaller oriented cell in the
base into the boundary of the greater one, and b) the sign
$\pm 1$ depending on whether this entry defines a concordance
or discordance of fixed coorientations of fibers over these cells.

The images of cells under the projection
$\sigma_1(\Theta,n) \to \K(\Theta,n)$ have
codimensions $n-2, n-1$ or $n$ in $\K(\Theta,n)$. The
above-mentioned structure on the initial stratum in $\sigma_1(\Theta,n)$
(consisting
of the orientation of the base and coorientation of the fibre)
defines a coorientation of this image at its non-singular points.
An equivalent calculation of incidence coefficients can be carried out
in the terms of these images. Namely, we take a germ
of a manifold transversal to the smaller stratum in some its
non-singular point, consider all local branches of curves in this
manifold, along which
the bigger stratum approaches the smaller one, and count these branches with
certain coefficients equal to $1$ or $-1$. Namely, the coefficient
corresponding to such a local branch depends on
whether the orientation of the germ, defined by the chosen coorientation of
the smaller stratum, coincides with the orientation, defined at an arbitrary
point of this branch and consisting of the direction along the curve towards
the point of the smaller stratum and the coorientation of the branch, defined
by the chosen coorientation of the bigger stratum. In this section we
calculate the incidence coefficients by exactly this method, although
the calculation in the preimage often is easier.


\subsection{Orientations of discriminant strata}
\label{coor}

In this subsection we for any standard stratum in $\sigma_1(\Theta,n)$
fix some its (co)orientation, i.e. an orientation of its base in
$B_2(\Theta)$ and a coorientation of the fibre in $\K(\Theta,n).$
Also we describe the corresponding coorientations of strata in
$\K(\Theta,n).$ In all cases the choice of a positive coorientation of
a stratum is described by the indication of a positive frame in a plane
transversal to this stratum.

First of all we fix a linear coordinate system $x_1, \dots, x_n$
in the space $\R^n,$ and define the orientation of this space by the
differential form
$d x_1 \wedge \dots \wedge d x_n$ or, equivalently, by the
standard frame
$(\partial / \partial x_1, \dots , \partial / \partial x_n).$

\subsubsection{Orientation of maximal strata $a\cdot b$ or $a \wedge b$,
generating the group $\Pi(\Theta)$.}
\label{maxor}

Let $a, b$ be two non-singular intervals in $\R^1$, $a$ to
the left of $b$.
The orientation of the corresponding stratum in $B_2(\Theta)$ is
specified by the form $d t \wedge d s,$ where $t, s $ are
coordinates in intervals $a, b$ respectively, whose
orientation coincides with the standard orientation of the line
$\R^1$. The coorientation of the fibre over the point $(t,s)$
is specified by the form
$$d(x_1(s)-x_1(t)) \wedge \cdots \wedge d(x_n(s)-x_n(t)),$$
where $x_i(s), x_i(t)$ are coordinates of values $f(s), f(t)$ of the map
$f \in \K(\Theta,n)$.

The corresponding strata in $\K(\Theta,n)$ are of codimension $n-2$ there.
A generic point of such a stratum $a\cdot b$ or $a \wedge b$ can be
represented by a map $f:\R^1 \to \R^n,$
$f \in \K(\Theta,n)$, having one additional self-intersection point
$f(t_0) = f(s_0),$ $t_0 \in a, s_0 \in b,$ such that the vectors
$f'(t_0)$ and $f'(s_0)$ are not collinear (and not equal to 0).
Consider a $(n-2)$-dimensional affine plane in $\R^n$, passing through the
point $f(t_0) = f(s_0)$ and orthogonal to the plane spanned by the
vectors $f'(t_0)$ and $f'(s_0)$. Choose a frame
$(e_3, \dots, e_n)$ in it in such a way that the frame
$(f'(t_0), f'(s_0), e_3, \dots, e_n)$ defines the positive orientation
of the space $\R^n$. Then the positive coorientation of our stratum in
$\K(\Theta,n)$ is defined by the frame in (the tangent bundle of)
$\K(\Theta,n)$ at the point $f$, consisting of $n-2$ small deformations
of the map $f$, not changing this map close to the point $t_0$,
and shifting it along the vectors $e_3, \dots, e_n$, taken in this
order, close to the point $s_0$.

\subsubsection{Orientation of strata $\alpha_\rra$ and $\alpha_\rla$,
generating the group $\|(\Theta)$.}

Let $u,v \in \R^1$ be respectively the left- and right-hand endpoints
of the chord $\alpha.$ Then the orientation of the corresponding cell in
$B_2(\Theta)$ is specified by the form $d \rho,$ where $\rho$ is the scalar
ratio of the vector $f'(v)$ to the (collinear to it) vector
$f'(u)$, and the coorientation of the fibre over the point $\rho$
in $\K(\Theta,n)$ is specified by the form
$$(x'_1(v)-\rho x'_1(u))\wedge \cdots \wedge (x'_n(v)-\rho x'_n(u)).$$

\label{slft}
The codimension of corresponding strata in $\K(\Theta,n)$ is equal to
$n-1$. A generic point $f$ of such a stratum in $\Sigma(\Theta,n)$ has
a simple self-tangency at the endpoints $u, v$ of the chord $\alpha$,
in particular its derivatives $f'(u), f'(v)$ are not equal to 0.
Consider the affine hyperplane in $\R^n$, passing through the point
$f(u) = f(v)$ and orthogonal to the vector $f'(u)$. Choose a frame
$(e_2, \dots, e_n)$ in it in such a way that the frame
$(f'(u), e_2, \dots, e_n)$ defines a positive orientation of the space
$\R^n$. Then the positive coorientation of our stratum is specified by
the frame in $T_f\K(\Theta,n)$, consisting of $n-1$ small deformations
of the map $f$, not changing this map close to the point $u$,
and adding to its derivative at the point $v$ the vectors
$\varepsilon e_2, \dots, \varepsilon e_n$ respectively,
$\varepsilon \approx +0$.

\subsubsection{Orientation of strata $a \cdot \alpha$ or $a \wedge \alpha$,
where $a$ is a non-singular interval, $\alpha$ a chord, and the sign
$\cdot$ or $\wedge$ between them depends on the parity of $n$.}

These strata generate the group $\Xi(\Theta)$. The orientation of the
base is defined by the form $d t,$ $t \in a,$ and the coorientation
of the fibre by the form
$$d(x_1(\alpha)-x_1(t)) \wedge \cdots \wedge d(x_n(\alpha)-x_n(t))$$
(where $x_i(\alpha)$ means the $i$-th coordinate of the image of
both endpoints of the chord $\alpha$ under the map $f$).
\label{trip}

The codimension of corresponding discriminant strata in $\K(\Theta,n)$ also
is equal to $n-1$. A generic point of such a stratum
is a map $f \in \K(\Theta,n)$ such that $f(t_0)=f(\alpha)$
for a unique point $t_0 \in a$, and $f'(t_0) \ne 0.$
Let us consider the affine hyperplane in $\R^n$, passing through the
point $f(t_0)$ and orthogonal to the vector $f'(t_0)$.
Choose a frame $(e_2, \dots, e_n)$ in it in such a way that the frame
$(f'(t_0), e_2, \dots, e_n)$ defines the positive orientation of the
space $\R^n$. Then the positive coorientation of the stratum is defined by
the frame in $T_f\K(\Theta,n)$, consisting of $n-1$ small deformations
of the map $f$, not changing this map close to the point $t_0$,
and shifting this map close to the endpoints of the chord $\alpha$
in the direction of vectors $e_2, \dots, e_n$ respectively.

\subsubsection{Orientation of strata $\star^+(\alpha)$ and $\star^-(\alpha)$,
where $\alpha$ is a chord of $\Theta$.}

These strata
generate the group $\Psi(\Theta)$ and
consist of maps $f \in \K(\Theta,n),$ having zero derivative at the
right-hand (respectively, left-hand) endpoint of the chord
$\alpha$ in $\R^1$. A positive normal frame of such a stratum in
$\K(\Theta,n)$ consists of $n$ vectors, increasing respectively the
1-st, \dots, $n$-th coordinate of the derivative of $f$ at this
endpoint of the chord and leaving the remaining coordinates of this
derivative unchanged.

\subsubsection{Orientation of strata $\alpha \cdot \beta$ or
$\alpha \wedge \beta$, where $\alpha, \beta$ are a pair of chords
of the diagram $\Theta$.}

These strata
consist of maps gluing together all endpoints of these two chords.
We assume that $\alpha$ is the {\em smaller} of these two chords, i.e.
its left-hand endpoint is placed further to the left in $\R^1$
than that of $\beta$.
\label{twochor}

A positive normal frame to this stratum consists of $n$ deformations of
the map $f$, increasing respectively the 1-st, \dots, $n$-th coordinate
of the image of the senior chord $\beta$, and not changing neither the
remaining its coordinates nor the image of the chord $\alpha$.

\subsection{Description of the operator $\Pi(\Theta) \to \|(\Theta)$ in
(\ref{complex}).}

This operator corresponds to the degeneration shown in the middle
column of Fig. \ref{rmx}. At this degeneration both endpoints of the
broken line tend to the endpoints of some chord.
One needs to consider four cases, corresponding to four possible
mutual dispositions of the endpoints of the chord and the broken line in
$\R^1$, see Fig.~\ref{selftan0}. We describe in detail only one of them,
when both endpoints of the broken line are placed between the endpoints
of the chord before the degeneration.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(120.00,26.80)
\put(0.00,11.00){\vector(1,0){50.00}}
\bezier{140}(44.00,26.00)(40.00,6.00)(25.00,5.00)
\put(6.00,26.00){\vector(-1,4){0.20}}
\bezier{152}(25.00,5.00)(10.00,6.00)(6.00,26.00)
\put(47.00,14.00){\makebox(0,0)[cc]{$t$}}
\put(42.00,24.00){\makebox(0,0)[cc]{$s$}}
\put(12.50,11.00){\circle*{1.33}}
\put(70.00,11.00){\vector(1,0){50.00}}
\put(117.00,14.00){\makebox(0,0)[cc]{$t$}}
\put(82.50,11.00){\circle*{1.33}}
\bezier{136}(108.00,26.00)(100.67,11.00)(82.50,11.00)
\put(74.00,26.00){\vector(0,1){0.20}}
\bezier{96}(82.50,11.00)(75.00,11.00)(74.00,26.00)
\put(60.00,11.00){\makebox(0,0)[cc]{$\Longrightarrow$}}
\put(105.00,24.00){\makebox(0,0)[cc]{$s$}}
\end{picture}
\caption{The arrow $\Pi(\Theta) \to \|(\Theta)$}
\label{atoc}
\end{center}
\end{figure}

This case is shown in Fig. \ref{atoc}, where the fat point denotes the
image of endpoints of the chord, the image of the interval
$a$ with coordinate $t$ lies to the right of it on the horizontal line,
and the image of the interval $b$ with coordinate $s$ lies to the right of
it on the curve. It is sufficient to calculate the incidence coefficient
at a single non-singular point of the smaller stratum. Therefore we shall
assume, that the considered curve in the bigger stratum, approaching
this point of the smaller stratum, consists of maps
$f_\lambda$
(where the parameter $\lambda$ runs over small positive numbers and
tends to 0, when the curve approaches the smaller stratum),
which are defined in these intervals
by the conditions $t \mapsto (t, 0, \dots, 0)$ and
$s \mapsto (-s, s^2+\lambda s, 0, \dots, 0)$ respectively.
In particular, in these intervals the coordinates of the second
intersection point of $f_\lambda$ are equal to $t=\lambda$ and $s=-\lambda$.

Consider $n-2$ small deformations of the map $f_\lambda$ that leave
the first interval unmoved, while on the second one they replace the map
$s \mapsto (-s, s^2+\lambda s, 0, \dots, 0)$
by maps
$s \mapsto (-s, s^2+\lambda s, -\varepsilon s, 0, \dots, 0 )$,
$s \mapsto (-s, s^2+\lambda s, 0, -\varepsilon s, 0, \dots, 0 )$,
\dots,
$s \mapsto (-s, s^2+\lambda s, 0, 0, \dots, -\varepsilon s)$.

This set of deformations defines the {\em negative} coorientation of the
maximal stratum (see \S \ref{maxor}): indeed, the matrix of these
shifts at the deformed intersection point, completed from above
by the derivatives of the map $f_\lambda$
at the preimages of this point, is equal to

$$\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & \ldots & 0\\
-1 & -\lambda & 0 & 0 & \ldots & 0 \\
0 & 0 & \lambda \varepsilon & 0 & \ldots & 0 \\
0 & 0 & 0 & \lambda \varepsilon & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \ldots & \lambda \varepsilon
\end{array}
\right)
$$

When $\lambda$ tends to $+0$, the map $f_\lambda$ degenerates into the map
of the stratum $\alpha_\rla$, and our system of $n-2$ deformations
degenerates into the deformations of the limit map, whose derivatives
at the intersection point are equal to
$(-1,0, -\varepsilon, 0, \dots, 0),$ \dots,
$(-1,0, 0, \dots, -\varepsilon).$
Let us augment them at the beginning
with the exterior normal vector of the bigger
stratum at a point of the smaller one. This vector can be calculated
as the derivative over $-\lambda$ at the point $\lambda=0$ of the derivative
of the map
$s \mapsto (-s, s^2+\lambda s, 0, \dots, 0)$
at the point $s=0$; so it is equal to $(0,-1,0, \dots, 0)$.
By the rule of \S \ref{slft}, to calculate the sign of the orientation of the
obtained normal frame to the smaller stratum, we write above the
obtained matrix the derivative of the map $f$ at the point $t=0$.
Then we obtain the square matrix
$$
\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & \ldots & 0\\
0 & -1 & 0 & 0 & \ldots & 0 \\
0 & 0 & -\varepsilon & 0 & \ldots & 0 \\
0 & 0 & 0 & -\varepsilon & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \ldots & -\varepsilon
\end{array}
\right)
$$

The sign of this matrix is equal to $(-1)^{n-1}$. Comparing it with the
sign of the previous matrix, we obtain the desired incidence coefficient,
which is equal to $(-1)^n$.

The investigation of remaining three cases is not more complicated and
gives the coefficients indicated in Fig. \ref{selftan0}.

\subsection{Operator $\Pi(\Theta) \to \Xi(\Theta)$
in (\ref{complex}).}

Although we calculate here some twelve incidence coefficients,
described by Fig.~\ref{easiest0}, it is sufficient to
consider only four essentially different cases. Namely, the endpoint of the
broken line, tending to the endpoint of the chord,
can be the right- or left-hand endpoint of this broken line,
and it can approach the endpoint of the chord from the left
or from the right in $\R^1$. On the other hand, neither the information
whether this endpoint of the chord is its left- or right-hand
endpoint, nor the mutual disposition of remaining endpoints of the
chord and the broken line (if it is not clear from the
previous data) does not affect the answer.

Again, we consider in detail only one case, shown in Fig. \ref{atob},
see also the third term in the upper line of Fig. \ref{easiest0}.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(108.00,30.00)
\put(0.00,15.00){\vector(1,0){40.00}}
\put(11.00,15.00){\oval(16.00,10.00)[t]}
\put(21.00,15.00){\line(2,5){2.00}}
\put(23.00,20.00){\line(1,0){11.00}}
\put(34.00,20.00){\line(2,-5){2.00}}
\put(19.50,15.00){\oval(5.00,2.00)[]}
\put(68.00,15.00){\vector(1,0){40.00}}
\put(88.00,0.00){\vector(0,1){30.00}}
\put(85.50,15.00){\circle*{1.33}}
\put(105.00,18.00){\makebox(0,0)[cc]{$t$}}
\put(91.00,28.00){\makebox(0,0)[cc]{$s$}}
\put(19.00,15.00){\circle*{1.33}}
\put(24.00,11.00){\makebox(0,0)[cc]{$t$}}
\put(38.00,11.00){\makebox(0,0)[cc]{$s$}}
\end{picture}
\caption{Operator $\Pi(\Theta) \to \Xi(\Theta)$}
\label{atob}
\end{center}
\end{figure}

By the fat point in Fig. \ref{atob} we depict the endpoint of the
chord, at which the image of our curve is in some,
absolutely non-interesting for us, way intersected by another its
branch. This point $t_0$ is chosen as the origin of the coordinate
$t$, the coordinate at the other endpoint of the broken line is denoted
by $s$. For the investigated point of the smaller stratum we choose a
map, defined close to these points by conditions
$t \mapsto (t,0, \dots, 0)$ and
$s \mapsto (0,s, 0,\dots, 0)$.
We shall approach this map from the bigger stratum along the
family of maps
$f_\lambda,$ $\lambda \to +0,$ defined close to these points by
formulas
$t \mapsto (t,0, \dots, 0)$ and
$s \mapsto (\lambda,s, 0,\dots, 0)$.
We define the normal frame to the bigger stratum at a generic
point of this family by the set of deformations
$s \mapsto (0, \lambda, \varepsilon, 0, \dots, 0),$
$s \mapsto (0, \lambda, 0, \varepsilon, 0, \dots, 0),$
\dots,
$s \mapsto (0, \lambda, 0, \dots, \varepsilon)$.
The matrix of this frame, completed from above by the vectors
$f'(t_0)$ and $f'(s_0)$, is the unit one, therefore its orientation
is positive. For $\lambda \to + 0$, e.g., the last deformation of this
frame tends to {\em minus} the last vector participating in the definition of
the coorientation of the smaller stratum: indeed, in this definition
the free (i.e. the right-hand in $\R^1$) endpoint of the broken line
is considered as the first one, and the endpoint coinciding with that
of the chord as the second one, unlike the ordering of these
endpoints in the bigger stratum. In a similar way, all our
$n-2$ deformations give in limit the last $n-2$ lines of the following
matrix
$$
\left(
\begin{array}{cccccc}
0 & 1 & 0 & 0 & \ldots & 0\\
1 & 0 & 0 & 0 & \ldots & 0 \\
0 & 0 & -1 & 0 & \ldots & 0 \\
0 & 0 & 0 & -1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \ldots & -1
\end{array}
\right)
$$

The vector of the exterior normal of the bigger stratum is equal to
the derivative over $-\lambda$ at $\lambda=0$ of the vector, connecting
the image of the free endpoint (i.e. the point $(\lambda, 0, \dots, 0$))
with the image of the chord, i.e. with the origin of coordinates.
We add this vector
$\partial (-\lambda, 0, \dots, 0)/\partial(-\lambda) \equiv (1, 0, \dots, 0)$
at the beginning of this set of limit deformations and obtain a normal
frame of the smaller stratum. To calculate its sign, we add at the very
beginning the derivative of $f$ at the free endpoint of the broken line,
i.e. the vector $(0,1,0, \dots, 0)$. The sign of the obtained matrix is equal
to $(-1)^{n-1},$ this is the desired incidence coefficient in our case, and
also in essentially equivalent cases corresponding to the second term of
the first line and the second term of the last line of Fig. \ref{easiest0}.
The remaining cases are considered analogously and give all coefficients
indicated in this figure.

\subsection{Operator $\|(\Theta) \to \Psi(\Theta)$.}

Let us explain the coefficient $(-1)^n$ at the second summand in the first
line of Fig. \ref{difc}. Consider the point $f$ of the smaller stratum,
which close to the right-hand
(i.e. non-degenerating) endpoint of the chord is defined by the
formula $s \mapsto (s,0, \dots, 0),$ and at the singular one by the formula
$t \to (t^n, t^{n-1}, \dots, t^2)$.
Let us approach this point $f$ from the bigger stratum along the curve
consisting of maps $f_\lambda,$ $\lambda \to +0$, that are defined
close to the latter point by
\begin{equation}
\label{mzm}
t \mapsto (t^n+\lambda t, t^{n-1}, \dots, t^2).
\end{equation}
For any small $\lambda$ we deform $f_\lambda$ into $n-1$
maps, defined close to this point by the formulas
\begin{equation}
\label{defmzm}
\begin{array}{l}
t \mapsto f_\lambda(t)+ \lambda (0,t, 0, 0, \dots, 0), \\
t \mapsto f_\lambda(t)+ \lambda (0,0, t, 0, \dots, 0), \\
\dots \dots \\
t \mapsto f_\lambda(t)+ \lambda (0, 0, 0, \dots, 0, t).
\end{array}
\end{equation}

The sign of the obtained normal frame of the bigger stratum is equal to
$(-1)^{n-1}$. Indeed, by definition the sign of the normal frame can be
calculated as the sign of the matrix consisting of the vector
$f'(u)$ on the first place, where $u$ is the left-hand endpoint of the chord,
and of vectors
$(f+\varepsilon e_2)'(v) - (f+\varepsilon e_2)'(u),$
$(f+\varepsilon e_3)'(v) - (f+\varepsilon e_3)'(u),$
\dots,
$(f+\varepsilon e_n)'(v) - (f+\varepsilon e_n)'(u),$
where $v$ is the right-hand endpoint of the chord, on the
remaining places.

As $\lambda \to +0$ this frame tends to the frame consisting of the
last $n-1$ lines of the following matrix:

\begin{equation}
\left(
\begin{array}{cccccc}
-1 & 0 & 0 & \ldots & 0\\
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & 1
\end{array}
\right)
\label{ctf}
\end{equation}

The exterior normal vector of the bigger stratum is the derivative over
$-\lambda$ of the derivative of our map $f_\lambda$ at the point
$t=0$, i.e. the upper line of the matrix (\ref{ctf}).
The determinant of this matrix is equal to $-1$. Comparing it with the sign
$(-1)^{n-1}$ of the pre-limit orienting frame, we obtain the
coefficient $(-1)^n.$

Let us explain now the second summand in the second line of
Fig. \ref{difc}. To do it, let us approach the same point of the
smaller stratum along the similar curve (\ref{mzm}) with the same
deformations (\ref{defmzm}), but with $\lambda \to -0$. At a generic
point of this curve this frame of deformations defines the negative
coorientation of the bigger stratum. The limit matrix at
$\lambda \to -0$ is the unit one, because, unlike the previous case
providing the matrix (\ref{ctf}), the exterior normal vector
is defined by differentiation over $\lambda$, and not over $-\lambda$.

The calculation of the first summands in both lines of
Fig. \ref{difc} is just easier.

\subsection{Operator $\Xi(\Theta) \to \Omega(\Theta)$.}

This operator is described by Fig.~\ref{bevar}.
Let us discuss for example the last equality of this figure.
We assume that the point $f$ of the bigger stratum is shown in the
middle lower or left-hand lower picture of this figure, and
close to the free endpoint of the broken line this map $f$
is equal to $t \mapsto (t,0, \dots, 0)$. For the basis
$(e_2, \dots, e_n),$ used in \S \ref{trip}, we can simply
take the basis
$(\partial/\partial x_2, \dots, \partial/\partial x_n).$
When this point $f$ degenerates into the map encoded by the
right-hand lower picture of this figure, vectors of the
canonical normal basis of the
bigger stratum tend to the last $n-1$ vectors of the canonical
normal basis of the smaller stratum, taken with coefficient $-1$.  Being
completed by the exterior normal vector, these vectors give us the matrix

\begin{equation}
\left(
\begin{array}{cccccc}
\pm 1 & 0 & 0 & \ldots & 0\\
0 & -1 & 0 & \ldots & 0 \\
0 & 0 & -1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & -1
\end{array}
\right)
\label{bte}
\end{equation}
where the coefficient $\pm 1$ should be replaced by
$-1$ (respectively, by $1$) for the left-hand (respectively, middle)
picture of the lower line of Fig. \ref{bevar}.

\subsection{Operator $\Xi(\Theta) \to \Psi(\Theta)$.}

The boundary operators of this type are shown in Fig. \ref{bf} on the left.
For all four cases, the point of the bigger stratum just before the
degeneration is shown in this picture on the right.
This picture has two versions corresponding to the case of left-hand
(respectively, middle) pictures of the left-hand part of this figure
and depending on which (corresponding to the endpoint of the
chord or of the free endpoint of the broken line) preimage of the
intersection point is further to the right in $\R^1$.
The disposition
in $\R^1$ of the endpoint of the chord, which is distant from the
colliding points, does not affect the degeneration, so that the
boundary operators for both left-hand (respectively, both middle)
pictures in two lines of the left-hand part of
this figure are described in exactly the same way.
The incidence coefficients shown in this picture can be proved similarly to
the above, and moreover follow from already proved formulas and the
identity $\partial^2=0$, applied to the generators of the group
$\Pi(\Theta)$.

Proposition \ref{diffs} is completely proved.
\hfill $\square$

\subsection{Orientations and boundaries of chains defined by
non-oriented Gau\ss \  diagrams}
\label{boor}

Any oriented Gau\ss \ diagram of first order with base
$\Theta$ defines a singular chain of codimension $n-3$ in the space
$\K(\Theta,n)$. These chains are the blocks for the construction of
relative cycles in
$\K(\Theta,n) \mbox{ mod } \Sigma(\Theta,n)$,
defining combinatorial formulas for our invariants (if $n=3$)
and, more generally, for $(n-3)$-dimensional cohomology classes of first
order in the space $\K(\Theta,n) \sm \Sigma(\Theta,n)$.
To construct such formulas we need to check the homological conditions:
first, the linear combination of such chains should indeed be a relative cycle
(i.e. the boundaries of its summands, that do not lie in
$\Sigma(\Theta,n),$ should annihilate each other), and, additionally,
the boundary of this combination in $\Sigma(\Theta,n)$ should define
the desired homology class. To do it, in this subsection we
show how to calculate homological boundaries of these singular chains,
as well as of similar chains of codimension $n-2$, of which the
combinatorial formulas of $(n-2)$-dimensional cohomology classes
of first order could be constructed. As usual, we start with the
definition and choice of (co)orientations of these chains.

Let us distinguish the direction $\partial / \partial x_1$ (``up'')
in $\R^n$, and assume that it does not coincide with the direction
of the axis $l(\R^1)$ of $\R^n$. The quotient space $\R^{n-1}$ of the space
$\R^n$ by the direction ``up'' will be called the
``blackboard plane'' and oriented by the form
$dx_2 \wedge \dots \wedge dx_n.$

Consider an arbitrary oriented Gau\ss \ diagram
$\Gamma$ of first order with
base $\Theta$, so that the broken arrow participating in its picture
connects some two
points $(u_0 < v_0)$ of the set $\R^1 \sm \Theta$.
This diagram defines the submanifold $V(\Gamma)$ in
$B_2(\Theta) \times \K(\Theta,n),$
consisting of all pairs $((u,v);f)$ such that

1) the points $u,v$ are placed in the same way as $u_0,v_0$
with respect to
$\Theta$ , i.e. there is a homeomorphism of $\R^1$ to itself,
preserving the orientation and the diagram $\Theta$ and sending
$u$ to $u_0,$ and $v$ to $v_0$;

2) projections of points $f(u), f(v)$ to the blackboard plane
$\R^{n-1}$ coincide, and

3) the point $f(u)$ lies ``above'' (respectively, ``below'') the point
$f(v)$ with respect to the direction ``up'', if the
broken arrow of the diagram is oriented from
$v_0$ to $u_0$ (respectively, from $u_0$ to $v_0$).

By $\Gamma'$ (respectively, $\Gamma_0$) we denote the oriented
(respectively, non-oriented) Gau\ss \
diagram obtained from $\Gamma$ by reversing (respectively, forgetting)
the orientation, so that in particular
$\Gamma_0$ coincides with the notation
of some generator of the group $\Pi(\Theta)$.

The projection $\K(\Theta,n) \to \K(\Theta,n-1)$, induced by the
projection to the blackboard plane, sends $V(\Gamma)$ to the stratum
of the set
$\sigma_1(\Theta,n-1) \subset B_2(\Theta) \times \K(\Theta,n-1)$
defined by the diagram $\Gamma_0$. The complete preimage of this stratum under
this projection is equal to the closure of the set
$V(\Gamma) \cup V(\Gamma'),$ in particular, any of the latter two strata
coincides with this complete preimage close to any its generic point.

The canonical (co)orientation of the stratum $V(\Gamma)$ consists
of the orientation $d u \wedge d v$ of the space of pairs
$\{u < v\} \in B_2(\Theta)$ and the coorientation in $\K(\Theta,n)$
of the standard fiber over such a point, which is induced from the
canonical (defined in \S \ref{coor}) coorientation of the fibre in
$\K(\Theta,n-1).$

The boundary of any such stratum $V(\Gamma)$
contains the maximal stratum of the
resolved discriminant $\sigma_1(\Theta,n)$,
whose notation coincides with $\Gamma_0$. This part of the boundary
operator is denoted by $\partial_1;$ it is described by the formula
in the upper left-hand corner of Fig. \ref{ress}.

However, there are also other summands of the homological boundary
of $V(\Gamma)$:
they occur when the endpoints of the broken arrow tend to the
points of the chord diagram $\Theta$. Corresponding degenerations
of the curve $f(\R^1)$ are shown in Fig. \ref{rmc}. The part of the
boundary operator, consisting of these summands,
splits into the sum of the operator
$\tilde \beta,$ corresponding to the degeneration shown in the
right-hand part of Fig. \ref{rmc} and described by
Fig. \ref{easiest}, and operator $\tilde \zeta$, related with
degenerations shown in the left-hand part of Fig. \ref{rmc} and
described by Fig. \ref{selftan}.

Again, these summands of the boundary of $V(\Gamma)$ lie in
$B_2(\Theta) \times \K(\Theta,n),$ and any of them is an open domain in
the complete preimage of some non-maximal stratum of the
resolved discriminant $\sigma_1(\Theta,n-1)$ under the projection
$B_2(\Theta) \times \K(\Theta,n) \to B_2(\Theta) \times \K(\Theta,n-1).$
In particular, the canonical (co)orientations of these domains
are induced from (co)orientations of strata $\sigma_1(\Theta,n-1),$
defined in \S \ref{coor}. The incidence coefficients between these
strata and manifolds $V(\Gamma)$ repeat the incidence coefficients for
corresponding strata in $\sigma_1(\Theta,n-1)$, in particular we obtain
the formulas indicated in Figs. \ref{easiest}, \ref{selftan}.
In any of three lines of Fig. \ref{easiest} we can simultaneously
reverse the orientations of all five arrows. The pictures in
right-hand parts of formulas of Fig. \ref{easiest} mean that the image of the
intersection point depicted by a chord lies in
$\R^n$ above or below (depending on the orientation of the
arrow) the image of some point of the corresponding non-singular
interval.

Additional expressions, written under the right-hand parts of equations
of Fig. \ref{selftan} and consisting of a couple of arrows with numbers,
denote the following conditions on the derivatives of the map
$f$ at the preimages of the intersection point depicted by the chord.
The subscript under the first (respectively, second) picture means that
the projections of these derivatives into the blackboard plane
$\R^{n-1}$ are co-directed, but the derivative of $f$ at the right-hand
endpoint goes above (respectively, below) that at the left-hand one,
i.e. this derivative is a positive linear combination of the derivative
at the other point and the direction ``up'' (respectively, ``down'').
The subscript under the third (respectively, fourth) picture means that
the projections of these derivatives are counter-directed in
$\R^{n-1}$, and the direction ``up'' is a negative (respectively,
positive) linear combination of these two derivatives.


\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(126.00,43.00)
\put(0.00,37.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(11.00,37.00){\line(0,1){4.00}}
\put(11.00,41.00){\line(-1,0){6.00}}
\put(5.00,41.00){\vector(0,-1){4.00}}
\put(3.00,37.00){\line(1,0){10.00}}
\put(16.00,37.00){\makebox(0,0)[cc]{=}}
\put(27.00,37.00){\line(0,1){4.00}}
\put(27.00,41.00){\line(1,0){6.00}}
\put(33.00,41.00){\vector(0,-1){4.00}}
\put(35.00,37.00){\line(-1,0){10.00}}
\put(21.00,37.50){\makebox(0,0)[cc]{$- \partial_1$}}
\put(38.00,37.00){\makebox(0,0)[cc]{=}}
\put(43.00,37.00){\line(0,1){4.00}}
\put(43.00,41.00){\line(1,0){6.00}}
\put(49.00,41.00){\line(0,-1){4.00}}
\put(51.00,37.00){\line(-1,0){10.00}}
\put(63.00,37.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(71.00,37.00){\oval(6.00,8.00)[t]}
\put(78.00,37.00){\line(0,1){6.00}}
\put(78.00,43.00){\line(-1,0){7.00}}
\put(71.00,43.00){\vector(0,-1){2.00}}
\put(66.00,37.00){\line(1,0){14.00}}
\put(83.00,37.00){\makebox(0,0)[cc]{=}}
\put(88.00,37.50){\makebox(0,0)[cc]{$- \partial_1$}}
\put(97.00,37.00){\oval(6.00,8.00)[t]}
\put(104.00,43.00){\line(-1,0){7.00}}
\put(92.00,37.00){\line(1,0){14.00}}
\put(97.00,41.00){\line(0,1){2.00}}
\put(117.00,37.00){\oval(6.00,8.00)[t]}
\put(124.00,43.00){\line(-1,0){7.00}}
\put(112.00,37.00){\line(1,0){14.00}}
\put(117.00,41.00){\line(0,1){2.00}}
\put(124.00,43.00){\line(0,-1){6.00}}
\put(104.00,43.00){\vector(0,-1){6.00}}
\put(109.00,37.00){\makebox(0,0)[cc]{=}}
\put(0.00,7.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(9.00,7.00){\oval(8.00,8.00)[t]}
\put(3.00,7.00){\line(1,0){12.00}}
\put(23.00,7.50){\makebox(0,0)[cc]{$-\partial_1$}}
\put(42.00,7.00){\makebox(0,0)[cc]{=}}
\put(33.00,7.00){\oval(8.00,8.00)[t]}
\put(51.00,7.00){\oval(8.00,8.00)[t]}
\put(27.00,7.00){\line(1,0){12.00}}
\put(45.00,7.00){\line(1,0){12.00}}
\put(18.00,7.00){\makebox(0,0)[cc]{=}}
\put(69.00,7.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(78.00,7.00){\oval(8.00,8.00)[t]}
\put(72.00,7.00){\line(1,0){12.00}}
\put(92.00,7.50){\makebox(0,0)[cc]{$-\partial_1$}}
\put(111.00,7.00){\makebox(0,0)[cc]{=}}
\put(102.00,7.00){\oval(8.00,8.00)[t]}
\put(120.00,7.00){\oval(8.00,8.00)[t]}
\put(96.00,7.00){\line(1,0){12.00}}
\put(114.00,7.00){\line(1,0){12.00}}
\put(87.00,7.00){\makebox(0,0)[cc]{=}}
\put(29.00,3.00){\vector(3,1){6.00}}
\put(29.00,3.00){\vector(3,-1){6.00}}
\put(38.00,5.00){\makebox(0,0)[cc]{1}}
\put(38.00,1.00){\makebox(0,0)[cc]{2}}
\put(5.00,3.00){\vector(3,1){6.00}}
\put(5.00,3.00){\vector(3,-1){6.00}}
\put(14.00,5.00){\makebox(0,0)[cc]{2}}
\put(14.00,1.00){\makebox(0,0)[cc]{1}}
\put(47.00,4.50){\vector(1,0){6.00}}
\put(47.00,3.00){\vector(1,0){6.00}}
\put(116.00,4.50){\vector(1,0){6.00}}
\put(122.00,3.00){\vector(-1,0){6.00}}
\put(78.00,2.00){\vector(2,1){4.00}}
\put(78.00,2.00){\vector(-2,1){4.00}}
\put(72.00,4.00){\makebox(0,0)[cc]{2}}
\put(84.00,4.00){\makebox(0,0)[cc]{1}}
\put(102.00,5.00){\vector(2,-1){4.00}}
\put(102.00,5.00){\vector(-2,-1){4.00}}
\put(96.00,2.00){\makebox(0,0)[cc]{1}}
\put(108.00,2.00){\makebox(0,0)[cc]{2}}
\put(63.00,22.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(75.00,22.00){\oval(6.00,8.00)[t]}
\put(68.00,22.00){\line(0,1){6.00}}
\put(75.00,28.00){\line(-1,0){7.00}}
\put(75.00,28.00){\vector(0,-1){2.00}}
\put(66.00,22.00){\line(1,0){14.00}}
\put(83.00,22.00){\makebox(0,0)[cc]{=}}
\put(88.00,22.50){\makebox(0,0)[cc]{$- \partial_1$}}
\put(101.00,22.00){\oval(6.00,8.00)[t]}
\put(101.00,28.00){\line(-1,0){7.00}}
\put(92.00,22.00){\line(1,0){14.00}}
\put(101.00,26.00){\line(0,1){2.00}}
\put(121.00,22.00){\oval(6.00,8.00)[t]}
\put(121.00,28.00){\line(-1,0){7.00}}
\put(112.00,22.00){\line(1,0){14.00}}
\put(121.00,26.00){\line(0,1){2.00}}
\put(114.00,28.00){\line(0,-1){6.00}}
\put(94.00,28.00){\vector(0,-1){6.00}}
\put(109.00,22.00){\makebox(0,0)[cc]{=}}
\put(0.00,22.50){\makebox(0,0)[cc]{$\partial_1$}}
\put(17.00,22.00){\makebox(0,0)[cc]{=}}
\put(22.00,22.50){\makebox(0,0)[cc]{$- \partial_1$}}
\put(40.00,22.00){\makebox(0,0)[cc]{=}}
\put(8.00,22.00){\oval(8.00,8.00)[t]}
\put(31.00,22.00){\oval(8.00,8.00)[t]}
\put(48.00,22.00){\oval(8.00,8.00)[t]}
\put(3.00,22.00){\line(1,0){11.00}}
\put(26.00,22.00){\line(1,0){11.00}}
\put(43.00,22.00){\line(1,0){11.00}}
\put(8.00,22.00){\vector(0,1){4.00}}
\put(31.00,26.00){\vector(0,-1){4.00}}
\put(48.00,22.00){\line(0,1){4.00}}
\end{picture}
\caption{Killing the discriminant strata}
\label{ress}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(90.00,79.00)
\bezier{200}(0.00,48.00)(20.00,63.00)(0.00,78.00)
\bezier{36}(7.90,71.20)(10.00,74.00)(15.00,78.00)
\bezier{168}(0.00,3.00)(15.00,18.00)(0.00,33.00)
\bezier{64}(15.00,33.00)(8.25,28.00)(7.75,20.00)
\bezier{64}(15.00,3.00)(7.60,11.00)(7.60,16.00)
\put(7.00,45.00){\vector(0,-1){10.00}}
\bezier{128}(15.00,48.00)(0.00,59.00)(7.00,70.00)
\put(7.50,55.50){\circle*{1.00}}
\put(7.50,16.00){\circle*{1.00}}
\put(36.00,71.00){\line(1,-1){23.00}}
\put(36.00,56.00){\line(1,1){23.00}}
\put(49.00,48.00){\line(0,1){9.00}}
\put(49.00,59.00){\line(0,1){9.00}}
\put(49.00,70.00){\line(0,1){9.00}}
\put(43.50,63.50){\circle*{1.00}}
\put(67.00,79.00){\line(1,-1){23.00}}
\put(90.00,70.00){\line(-1,-1){22.00}}
\put(77.00,48.00){\line(0,1){8.00}}
\put(77.00,58.00){\line(0,1){10.00}}
\put(77.00,70.00){\line(0,1){9.00}}
\put(83.00,63.00){\circle*{1.00}}
\put(52.00,28.00){\line(1,-1){23.00}}
\put(74.00,27.00){\line(-1,-1){22.00}}
\put(63.50,16.50){\circle*{1.00}}
\put(63.50,3.00){\line(0,1){12.50}}
\put(63.50,17.50){\line(0,1){12.50}}
\put(53.00,47.00){\vector(1,-2){6.50}}
\put(73.00,47.00){\vector(-1,-2){6.50}}
\end{picture}
\end{center}
\caption{Reidemeister degenerations of self-intersecting curves}
\label{rmc}
\end{figure}


\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(134.66,11.00)
\put(0.00,6.50){\makebox(0,0)[cc]{$\tilde \beta$}}
\put(8.33,6.00){\oval(8.00,6.00)[t]}
\put(17.33,11.00){\vector(0,-1){5.00}}
\put(21.34,6.00){\makebox(0,0)[cc]{=}}
\put(26.00,6.50){\makebox(0,0)[cc]{$-\tilde \beta$}}
\put(53.00,6.50){\makebox(0,0)[cc]{$-\tilde \beta$}}
\put(79.00,6.50){\makebox(0,0)[cc]{$\tilde \beta$}}
\put(36.34,6.00){\oval(8.00,6.00)[t]}
\put(62.33,6.00){\oval(8.00,6.00)[t]}
\put(86.33,6.00){\oval(8.00,6.00)[t]}
\put(123.66,6.00){\oval(8.00,6.00)[t]}
\put(44.34,11.00){\vector(0,-1){5.00}}
\put(71.33,11.00){\vector(0,-1){5.00}}
\put(96.33,11.00){\vector(0,-1){5.00}}
\put(132.66,11.00){\vector(0,-1){5.00}}
\put(73.33,6.00){\line(-1,0){17.00}}
\put(98.33,6.00){\line(-1,0){17.00}}
\put(134.66,6.00){\line(-1,0){17.00}}
\put(48.34,6.00){\makebox(0,0)[cc]{=}}
\put(75.66,6.00){\makebox(0,0)[cc]{=}}
\put(100.66,6.00){\makebox(0,0)[cc]{=}}
\put(3.33,6.00){\line(0,1){5.00}}
\put(3.33,11.00){\line(1,0){14.00}}
\put(67.33,6.00){\line(0,1){5.00}}
\put(67.33,11.00){\line(1,0){4.00}}
\put(89.83,6.00){\oval(3.00,2.00)[]}
\put(66.83,6.00){\oval(3.00,2.00)[]}
\put(32.84,6.00){\oval(3.00,2.00)[]}
\put(3.83,6.00){\oval(3.00,2.00)[]}
\put(123.66,9.00){\line(0,1){2.00}}
\put(123.66,11.00){\line(1,0){9.00}}
\put(44.34,11.00){\line(-1,0){11.00}}
\put(33.34,11.00){\line(0,-1){5.00}}
\put(96.33,11.00){\line(-1,0){7.00}}
\put(89.33,11.00){\line(0,-1){5.00}}
\put(110.00,6.50){\makebox(0,0)[cc]{$(-1)^{n-1}$}}
\put(2.00,6.00){\line(1,0){17.00}}
\put(30.00,6.00){\line(1,0){16.00}}
\end{picture}


\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(126.00,11.00)
\put(27.00,6.50){\makebox(0,0)[cc]{$-\tilde \beta$}}
\put(42.00,6.00){\oval(8.00,6.00)[t]}
\put(45.00,11.00){\vector(0,-1){5.00}}
\put(48.00,6.00){\line(-1,0){17.00}}
\put(106.00,6.00){\makebox(0,0)[cc]{=}}
\put(45.50,6.00){\oval(3.00,2.00)[]}
\put(1.00,6.50){\makebox(0,0)[cc]{$\tilde \beta$}}
\put(13.00,6.00){\oval(8.00,6.00)[t]}
\put(18.00,11.00){\vector(0,-1){5.00}}
\put(20.00,6.00){\line(-1,0){17.00}}
\put(22.00,6.00){\makebox(0,0)[cc]{=}}
\put(4.00,6.00){\line(0,1){5.00}}
\put(4.00,11.00){\line(1,0){14.00}}
\put(17.50,6.00){\oval(3.00,2.00)[]}
\put(126.00,6.00){\line(-1,0){17.00}}
\put(120.00,6.00){\oval(8.00,6.00)[t]}
\put(111.00,6.00){\line(0,1){5.00}}
\put(111.00,11.00){\line(1,0){9.00}}
\put(120.00,11.00){\vector(0,-1){2.00}}
\put(55.00,6.50){\makebox(0,0)[cc]{$-\tilde \beta$}}
\put(70.00,6.00){\oval(8.00,6.00)[t]}
\put(76.00,6.00){\line(-1,0){17.00}}
\put(65.83,6.00){\oval(3.00,2.00)[]}
\put(83.00,6.50){\makebox(0,0)[cc]{$\tilde \beta$}}
\put(96.00,6.00){\oval(8.00,6.00)[t]}
\put(103.00,6.00){\line(-1,0){17.00}}
\put(79.00,6.00){\makebox(0,0)[cc]{=}}
\put(92.50,6.00){\oval(3.00,2.00)[]}
\put(45.00,11.00){\line(-1,0){12.00}}
\put(33.00,11.00){\line(0,-1){5.00}}
\put(50.00,6.00){\makebox(0,0)[cc]{=}}
\put(87.00,6.00){\line(0,1){5.00}}
\put(87.00,11.00){\line(1,0){6.00}}
\put(93.00,11.00){\vector(0,-1){5.00}}
\put(60.00,6.00){\line(0,1){5.00}}
\put(60.00,11.00){\line(1,0){5.00}}
\put(65.00,11.00){\vector(0,-1){5.00}}
\end{picture}


\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(134.00,10.00)
\put(0.00,6.50){\makebox(0,0)[cc]{$(-1)^{n-1} \tilde \beta$}}
\put(27.00,6.00){\line(-1,0){17.00}}
\put(29.00,6.00){\makebox(0,0)[cc]{=}}
\put(38.33,6.50){\makebox(0,0)[cc]{$(-1)^n \tilde \beta$}}
\put(70.00,6.50){\makebox(0,0)[cc]{$-\tilde \beta$}}
\put(97.33,6.50){\makebox(0,0)[cc]{$\tilde \beta$}}
\put(62.67,6.00){\line(-1,0){17.00}}
\put(91.33,6.00){\line(-1,0){17.00}}
\put(116.33,6.00){\line(-1,0){17.00}}
\put(64.67,6.00){\makebox(0,0)[cc]{=}}
\put(93.33,6.00){\makebox(0,0)[cc]{=}}
\put(119.33,6.00){\makebox(0,0)[cc]{=}}
\put(113.83,6.00){\oval(3.00,2.00)[]}
\put(89.16,6.00){\oval(3.00,2.00)[]}
\put(48.17,6.00){\oval(3.00,2.00)[]}
\put(11.50,6.00){\oval(3.00,2.00)[]}
\put(19.00,6.00){\oval(14.00,8.00)[t]}
\put(54.50,6.00){\oval(13.00,8.00)[t]}
\put(83.00,6.00){\oval(14.00,8.00)[t]}
\put(107.00,6.00){\oval(12.00,8.00)[t]}
\put(11.00,6.00){\line(0,-1){5.00}}
\put(11.00,1.00){\line(1,0){8.00}}
\put(19.00,1.00){\vector(0,1){5.00}}
\put(49.00,6.00){\line(0,-1){5.00}}
\put(49.00,1.00){\line(1,0){5.00}}
\put(54.00,1.00){\vector(0,1){5.00}}
\put(89.00,6.00){\line(0,-1){5.00}}
\put(89.00,1.00){\line(-1,0){6.00}}
\put(83.00,1.00){\vector(0,1){5.00}}
\put(114.67,6.00){\line(0,-1){5.00}}
\put(114.67,1.00){\line(-1,0){8.00}}
\put(106.67,1.00){\vector(0,1){5.00}}
\put(128.00,6.00){\oval(9.00,8.00)[t]}
\put(128.00,10.00){\vector(0,-1){4.00}}
\put(122.00,6.00){\line(1,0){12.00}}
\end{picture}
\end{center}
\caption{Simplest degeneration of a Gau\ss \ diagram}
\label{easiest}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(86.00,30.00)
\put(45.00,26.00){\line(1,0){18.00}}
\put(24.00,26.00){\makebox(0,0)[cc]{=}}
\put(0.00,26.50){\makebox(0,0)[cc]{$\tilde \zeta$}}
\put(35.00,26.50){\makebox(0,0)[cc]{$(-1)^{n-1}\tilde \zeta$}}
\put(3.00,26.00){\line(1,0){18.00}}
\put(66.00,26.00){\makebox(0,0)[cc]{=}}
\put(78.00,26.00){\oval(12.00,8.00)[t]}
\put(86.00,26.00){\line(-1,0){16.00}}
\put(45.00,11.00){\line(1,0){18.00}}
\put(24.00,11.00){\makebox(0,0)[cc]{=}}
\put(0.00,11.50){\makebox(0,0)[cc]{$\tilde \zeta$}}
\put(35.00,11.50){\makebox(0,0)[cc]{$(-1)^{n-1}\tilde \zeta$}}
\put(3.00,11.00){\line(1,0){18.00}}
\put(66.00,11.00){\makebox(0,0)[cc]{=}}
\put(78.00,11.00){\oval(12.00,8.00)[t]}
\put(86.00,11.00){\line(-1,0){16.00}}
\put(60.00,11.00){\oval(4.00,2.00)[]}
\put(48.00,11.00){\oval(4.00,2.00)[]}
\put(6.00,11.00){\oval(4.00,2.00)[]}
\put(18.00,11.00){\oval(4.00,2.00)[]}
\put(6.00,26.00){\oval(4.00,2.00)[]}
\put(18.00,26.00){\oval(4.00,2.00)[]}
\put(48.00,26.00){\oval(4.00,2.00)[]}
\put(60.00,26.00){\oval(4.00,2.00)[]}
\put(11.00,26.00){\oval(12.00,8.00)[t]}
\put(7.00,26.00){\line(0,-1){4.00}}
\put(7.00,22.00){\line(1,0){12.00}}
\put(19.00,22.00){\vector(0,1){4.00}}
\put(55.00,26.00){\oval(12.00,8.00)[t]}
\put(59.00,26.00){\line(0,-1){4.00}}
\put(59.00,22.00){\line(-1,0){12.00}}
\put(47.00,22.00){\vector(0,1){4.00}}
\put(11.00,11.00){\oval(12.00,8.00)[t]}
\put(55.00,11.00){\oval(12.00,8.00)[t]}
\put(47.00,11.00){\line(0,-1){4.00}}
\put(47.00,7.00){\line(1,0){12.00}}
\put(59.00,7.00){\vector(0,1){4.00}}
\put(19.00,11.00){\line(0,-1){4.00}}
\put(19.00,7.00){\line(-1,0){12.00}}
\put(7.00,7.00){\vector(0,1){4.00}}
\put(75.00,22.00){\vector(3,-1){6.00}}
\put(75.00,22.00){\vector(3,1){6.00}}
\put(84.00,20.00){\makebox(0,0)[cc]{1}}
\put(84.00,24.00){\makebox(0,0)[cc]{2}}
\put(75.00,7.00){\vector(3,-1){6.00}}
\put(75.00,7.00){\vector(3,1){6.00}}
\put(84.00,9.00){\makebox(0,0)[cc]{1}}
\put(84.00,5.00){\makebox(0,0)[cc]{2}}
\end{picture}

\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(88.00,25.00)
\put(3.00,21.00){\line(1,0){18.00}}
\put(24.00,21.00){\makebox(0,0)[cc]{=}}
\put(35.00,21.50){\makebox(0,0)[cc]{$(-1)^{n-1} \tilde \zeta$}}
\put(0.00,21.50){\makebox(0,0)[cc]{$\tilde \zeta$}}
\put(45.00,21.00){\line(1,0){18.00}}
\put(67.00,21.00){\makebox(0,0)[cc]{=}}
\put(80.00,21.00){\oval(12.00,8.00)[t]}
\put(88.00,21.00){\line(-1,0){16.00}}
\put(3.00,6.00){\line(1,0){18.00}}
\put(24.00,6.00){\makebox(0,0)[cc]{=}}
\put(35.00,6.50){\makebox(0,0)[cc]{$(-1)^{n-1} \tilde \zeta$}}
\put(0.00,6.50){\makebox(0,0)[cc]{$\tilde \zeta$}}
\put(45.00,6.00){\line(1,0){18.00}}
\put(67.00,6.00){\makebox(0,0)[cc]{=}}
\put(80.00,6.00){\oval(12.00,8.00)[t]}
\put(88.00,6.00){\line(-1,0){16.00}}
\put(54.00,21.00){\oval(14.00,8.00)[t]}
\put(49.00,21.00){\line(0,-1){4.00}}
\put(49.00,17.00){\line(1,0){10.00}}
\put(59.00,17.00){\vector(0,1){4.00}}
\put(12.00,21.00){\oval(10.00,8.00)[t]}
\put(19.00,21.00){\line(0,-1){4.00}}
\put(19.00,17.00){\line(-1,0){14.00}}
\put(5.00,17.00){\vector(0,1){4.00}}
\put(80.00,19.00){\vector(3,-1){6.00}}
\put(80.00,19.00){\vector(-3,-1){6.00}}
\put(72.00,17.00){\makebox(0,0)[cc]{1}}
\put(88.00,17.00){\makebox(0,0)[cc]{2}}
\put(54.00,6.00){\oval(14.00,8.00)[t]}
\put(12.00,6.00){\oval(10.00,8.00)[t]}
\put(59.00,6.00){\line(0,-1){4.00}}
\put(59.00,2.00){\line(-1,0){10.00}}
\put(49.00,2.00){\vector(0,1){4.00}}
\put(5.00,6.00){\line(0,-1){4.00}}
\put(5.00,2.00){\line(1,0){14.00}}
\put(19.00,2.00){\vector(0,1){4.00}}
\put(80.00,1.00){\vector(2,1){6.00}}
\put(80.00,1.00){\vector(-2,1){6.00}}
\put(72.00,4.00){\makebox(0,0)[cc]{1}}
\put(88.00,4.00){\makebox(0,0)[cc]{2}}
\put(18.00,6.00){\oval(4.00,2.00)[]}
\put(6.00,6.00){\oval(4.00,2.00)[]}
\put(48.00,6.00){\oval(4.00,2.00)[]}
\put(60.00,6.00){\oval(4.00,2.00)[]}
\put(48.00,21.00){\oval(4.00,2.00)[]}
\put(60.00,21.00){\oval(4.00,2.00)[]}
\put(6.00,21.00){\oval(4.00,2.00)[]}
\put(18.00,21.00){\oval(4.00,2.00)[]}
\end{picture}
\end{center}
\caption{Birth of a self-tangency of the projection}
\label{selftan}
\end{figure}

Now let us study the boundaries of strata participating in the
right-hand parts of Figs.~\ref{easiest}, \ref{selftan}. First of all,
the boundary of any such stratum contains some stratum of
$\sigma_1(\Theta,n)$.
The corresponding part $\partial_1$ of the
boundary operator is described by the remaining five
(but the one shown in the upper left-hand corner) formulas of
Fig. \ref{ress}. Other components of boundaries are defined
by degenerations of configurations of points in $\R^1$, participating in
the definition of the stratum. In Fig. \ref{verrt}
(respectively, \ref{coproj}) we show what is happening at such degenerations,
when the endpoint of the arrow, connecting a chord with a
non-singular piece of $\R^1$, tends to some endpoint of the same
(respectively, other) chord. These degenerations ``cover'' the
boundary operators in $\sigma_1(\Theta,n-1)$, corresponding to the operators
$\Xi(\Theta) \to \Psi(\Theta)$ (respectively,
$\Xi(\Theta) \to \Omega(\Theta)$), in particular they have the same
matrix coefficients. In any line of Fig. \ref{coproj} one can
reverse simultaneously the directions of all three arrows.


\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(71.00,58.00)
\put(1.00,37.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(13.00,37.00){\oval(10.00,8.00)[t]}
\put(13.00,41.00){\line(0,1){2.00}}
\put(13.00,43.00){\line(-1,0){7.00}}
\put(7.00,37.00){\oval(4.00,2.00)[]}
\put(4.00,37.00){\line(1,0){16.00}}
\put(23.00,37.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,37.50){\makebox(0,0)[cc]{$-$}}
\put(41.00,37.00){\oval(12.00,8.00)[t]}
\put(36.00,37.00){\oval(4.00,2.00)[]}
\put(33.00,37.00){\line(1,0){16.00}}
\put(52.00,37.00){\makebox(0,0)[cc]{$=$}}
\put(63.00,37.00){\oval(12.00,8.00)[t]}
\put(55.00,37.00){\line(1,0){16.00}}
\put(1.00,7.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(40.00,7.00){\oval(10.00,8.00)[t]}
\put(40.00,13.00){\line(1,0){7.00}}
\put(47.00,13.00){\line(0,-1){6.00}}
\put(46.00,7.00){\oval(4.00,2.00)[]}
\put(49.00,7.00){\line(-1,0){16.00}}
\put(23.00,7.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,7.50){\makebox(0,0)[cc]{$-$}}
\put(12.00,7.00){\oval(12.00,8.00)[t]}
\put(17.00,7.00){\oval(4.00,2.00)[]}
\put(20.00,7.00){\line(-1,0){16.00}}
\put(52.00,7.00){\makebox(0,0)[cc]{$=$}}
\put(63.00,7.00){\oval(12.00,8.00)[t]}
\put(55.00,7.00){\line(1,0){16.00}}
\put(30.00,37.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(30.00,7.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(1.00,52.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(13.00,52.00){\oval(10.00,8.00)[t]}
\put(13.00,58.00){\line(-1,0){7.00}}
\put(7.00,52.00){\oval(4.00,2.00)[]}
\put(4.00,52.00){\line(1,0){16.00}}
\put(23.00,52.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,52.50){\makebox(0,0)[cc]{$-$}}
\put(41.00,52.00){\oval(12.00,8.00)[t]}
\put(36.00,52.00){\oval(4.00,2.00)[]}
\put(33.00,52.00){\line(1,0){16.00}}
\put(52.00,52.00){\makebox(0,0)[cc]{$=$}}
\put(63.00,52.00){\oval(12.00,8.00)[t]}
\put(55.00,52.00){\line(1,0){16.00}}
\put(1.00,22.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(40.00,26.00){\line(0,1){2.00}}
\put(40.00,28.00){\line(1,0){7.00}}
\put(46.00,22.00){\oval(4.00,2.00)[]}
\put(49.00,22.00){\line(-1,0){16.00}}
\put(23.00,22.00){\makebox(0,0)[cc]{$=$}}
\put(26.50,22.50){\makebox(0,0)[cc]{$-$}}
\put(17.00,22.00){\oval(4.00,2.00)[]}
\put(20.00,22.00){\line(-1,0){16.00}}
\put(52.00,22.00){\makebox(0,0)[cc]{$=$}}
\put(55.00,22.00){\line(1,0){16.00}}
\put(30.00,52.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(30.00,22.50){\makebox(0,0)[cc]{$\tilde \phi$}}
\put(40.00,22.00){\oval(10.00,8.00)[t]}
\put(12.00,22.00){\oval(12.00,8.00)[t]}
\put(63.00,22.00){\oval(12.00,8.00)[t]}
\put(41.00,56.00){\vector(-1,-1){4.00}}
\put(6.00,58.00){\line(0,-1){6.00}}
\put(13.00,58.00){\vector(0,-1){2.00}}
\put(57.00,48.00){\makebox(0,0)[cc]{$1 \uparrow$}}
\put(6.00,43.00){\vector(0,-1){6.00}}
\put(37.00,37.00){\vector(1,1){4.00}}
\put(57.00,33.00){\makebox(0,0)[cc]{$1 \downarrow$}}
\put(47.00,28.00){\vector(0,-1){6.00}}
\put(16.00,22.00){\vector(-1,1){4.00}}
\put(69.00,18.00){\makebox(0,0)[cc]{$2 \uparrow$}}
\put(12.00,11.00){\vector(1,-1){4.00}}
\put(40.00,13.00){\vector(0,-1){2.00}}
\put(69.00,3.00){\makebox(0,0)[cc]{$2 \downarrow$}}
\end{picture}
\end{center}
\caption{Birth of a vertical tangent}
\label{verrt}
\end{figure}

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(97.00,30.00)
\put(1.00,24.00){\makebox(0,0)[cc]{$\tilde \epsilon$}}
\put(6.00,30.00){\line(1,0){12.00}}
\put(18.00,30.00){\line(0,-1){2.00}}
\put(18.00,24.00){\oval(8.00,8.00)[t]}
\put(24.00,24.00){\line(-1,0){20.00}}
\put(13.00,24.00){\oval(10.00,8.00)[b]}
\put(7.00,24.00){\oval(4.00,2.00)[]}
\put(34.00,24.00){\makebox(0,0)[cc]{$-\tilde \epsilon$}}
\put(51.00,30.00){\line(0,-1){2.00}}
\put(51.00,24.00){\oval(8.00,8.00)[t]}
\put(57.00,24.00){\line(-1,0){20.00}}
\put(40.00,24.00){\oval(4.00,2.00)[]}
\put(45.00,24.00){\oval(12.00,8.00)[b]}
\put(41.00,30.00){\line(1,0){10.00}}
\put(28.00,24.00){\makebox(0,0)[cc]{$=$}}
\put(61.00,24.00){\makebox(0,0)[cc]{$=$}}
\put(74.00,24.00){\oval(12.00,8.00)[b]}
\put(78.00,24.00){\oval(12.00,8.00)[t]}
\put(66.00,24.00){\line(1,0){20.00}}
\put(10.00,5.00){\oval(8.00,8.00)[t]}
\put(10.00,9.00){\line(0,1){2.00}}
\put(10.00,11.00){\line(1,0){12.00}}
\put(15.00,5.00){\oval(10.00,8.00)[b]}
\put(21.00,5.00){\oval(4.00,2.00)[]}
\put(4.00,5.00){\line(1,0){20.00}}
\put(43.00,5.00){\oval(8.00,8.00)[t]}
\put(43.00,9.00){\line(0,1){2.00}}
\put(54.00,5.00){\oval(4.00,2.00)[]}
\put(37.00,5.00){\line(1,0){20.00}}
\put(49.00,5.00){\oval(12.00,8.00)[b]}
\put(43.00,11.00){\line(1,0){10.00}}
\put(1.00,5.00){\makebox(0,0)[cc]{$\tilde \epsilon$}}
\put(34.00,5.00){\makebox(0,0)[cc]{$-\tilde \epsilon$}}
\put(28.00,5.00){\makebox(0,0)[cc]{$=$}}
\put(61.00,5.00){\makebox(0,0)[cc]{=}}
\put(70.00,5.00){\makebox(0,0)[cc]{$(-1)^{n-1}$}}
\put(89.00,5.00){\oval(12.00,8.00)[b]}
\put(85.00,5.00){\oval(12.00,8.00)[t]}
\put(97.00,5.00){\line(-1,0){20.00}}
\put(78.00,28.00){\vector(-1,-2){4.00}}
\put(41.00,30.00){\vector(0,-1){6.00}}
\put(6.00,30.00){\vector(0,-1){6.00}}
\put(85.00,9.00){\vector(1,-2){4.00}}
\put(53.00,11.00){\vector(0,-1){6.00}}
\put(22.00,11.00){\vector(0,-1){6.00}}
\end{picture}
\end{center}
\caption{Collision of projections of chords}
\label{coproj}
\end{figure}

\section{Obstructions to the representability of elements of
$H^{n-3}_{(1)}(\K(\Theta,n) \sm \Sigma(\Theta,n))$
by integral linear combinations of oriented Gau\ss \ diagrams}
\label{obstcof}

In this section we prove Theorems \ref{predsti},
\ref{predsti2} and \ref{necconi}, but first we give the exact
definition of the notion of representability, participating
in these theorems, in the case of an arbitrary $n$.

\begin{definition} \rm
A class $v \in H^{n-3}(\K(\Theta,n) \sm \Sigma(\Theta,n))$
is {\em representable} by a linear combination of oriented Gau\ss \
diagrams $\Gamma_j$ of rank 1, if the corresponding
linear combination of chains $V(\Gamma_j)$ (defined in \S \ref{boor}),
being projected from
$B_2(\Theta) \times \K(\Theta,n)$ to $\K(\Theta,n)$, becomes a
relative cycle in
$\K(\Theta,n) \mbox{ mod } \Sigma(\Theta,n),$ whose boundary
corresponds to our class $v$ via the Alexander isomorphism
(\ref{evr}).
\end{definition}

If the class $v$ is representable by a linear combination of such diagrams
$V(\Gamma),$ then its value on the class
$Q \in H_{n-3}(\K(\Theta,n) \sm \Sigma(\Theta,n))$
can be calculated as the intersection number. Namely, we
realize this class by a generic cycle consisting of smooth
singular simplices in $\K(\Theta,n) \sm \Sigma(\Theta,n)$,
then any of these simplices has only finitely many
transversal intersection points with the chains $V(\Gamma).$
The coorientation of these chains, defined in \S \ref{boor},
allows us to calculate the intersection indices of these
simplices and chains; the rest is standard.
For $n=3$ this description coincides with the one given in
the Introduction for the value which a linear combination of oriented
Gau\ss \ diagrams takes on a 0-dimensional simplex, i.e. on a
single point.

\begin{remark} \rm
\label{remta}
The problem on the representation of cohomology classes by linear
combinations of oriented Gau\ss \ diagrams
is equivalent to the following one. We have the line fiber bundle
over the space $B_2(\Theta) \subset I(\Theta)$, orthogonal to the
tautological bundle over $I(\Theta)$ in $F(\Theta)$, see \S \ref{tavtol}.
Let $T$ be the local system of groups with fiber $\Z$ over $B_2(\Theta)$,
watching for the orientation of this bundle. Consider an element
$\Delta$ of the group $\bar H_2(B_2(\Theta), T^{\otimes n})$ (i.e.
simply of the group
$\bar H_2(B_2(\Theta),\Z)$ if $n$ is even, or of the group
$\bar H_2(B_2(\Theta),T)$ if $n$ is odd).
The element $\Delta$ can be uniquely represented by a linear
combination of maximal cells in $B_2(\Theta)$ with the same coefficients
$\Z$ or $T$. Let us identify $B_2(\Theta)$ with the zero section
of this line bundle.
We need to span the cycle $\Delta$ by a film (i.e. a Borel--Moore
chain) with the same coefficients $\Z$ or $T$
in the total space of this line bundle.

If the bundle would be trivial, then it would be possible to do it in two
ways, multiplying the cycle $\Delta$ by an arbitrary half of the fibre
and taking the obtained chain with one or other sign. However the
bundle is non-trivial, therefore, starting to span the cycle $\Delta$
by one of these ways over some maximal cell in $B_2(\Theta)$ and
extending this spanning by continuity to the neighboring cells,
we, after traversing some loop in the base, can come to the
opposite way over the initial cell.

On the other hand, if we apply both these ways of spanning simultaneously,
i.e. take the sum of both such spanning films over any cell, then after
traversing any path we come back to the same way and thus obtain a
well defined chain. The unique disadvantage is that the boundary of the
obtained chain will be equal to $2\Delta$, and not to $\Delta$ as desired.
Multiplying this chain by $1/2$ (for which, strictly speaking,
we have to multiply the system of coefficients by
$\Z[1/2]$ or some other group with division by 2), we obtain
the desired spanning chain with half-integer coefficients.
\end{remark}

On the language of our standard chains in $B_2(\Theta) \times \K(\Theta,n)$
this procedure can be described in the following way.
\medskip

{\em Proof of Theorem \ref{predsti}.}
1. Let $\Delta$ be an arbitrary two-dimensional cycle of the complex
$\CC_*(\Theta,n)$. It is defined by a linear combination of non-oriented
Gau\ss \ diagrams, generating the group $\Pi(\Theta)$.
Let us denote by the same letter $\Delta$ the corresponding cycle in
$\sigma_1(\Theta,n)$, and by the letter $\delta$ its projection into
$\Sigma(\Theta,n)$.
The desired linear combination $G$ of chains $V(\Gamma_i)$ in
$B_2(\Theta) \times \K(\Theta,n),$ defined by
oriented Gau\ss \ diagrams $\Gamma_i$ with basis
$\Theta$, should satisfy two conditions:

($*$) the projection of this linear combination to $\K(\Theta,n)$
defines a relative Borel--Moore cycle modulo $\Sigma(\Theta,n),$ and

($**$) the boundary of this cycle is equal to $\delta$.

It is easy to see that this condition is equivalent to the analogous
condition in the preimage: our linear combination should be a
relative cycle in $B_2(\Theta) \times \K(\Theta,n)$ modulo
$\sigma_1(\Theta,n),$
and its boundary there should be equal to $\Delta$. Let us prove
that there always exists such a linear combination with half-integer
coefficients. We associate with any element $a \cdot b$ or
$a \wedge b,$ $a <b,$ of the group $\Pi(\Theta)$ the
{\em half-difference} of two oriented
Gau\ss \ diagrams with basis $\Theta$,
the first of which is expressed by the broken line oriented from the
interval $b$ to $a$, and the second by the broken line oriented from
$a$ to $b$. Extending this association by linearity, we obtain an
embedding of the group $\Pi(\Theta)$ into the group of formal
linear combinations of oriented Gau\ss \ diagram of order one
with half-integer coefficients.

\begin{lemma}
If $\Delta$ is a two-dimensional cycle of the complex
$\CC_*(\Theta,n)$, then this embedding sends
$\Delta$ into a Borel--Moore chain with half-integer coefficients in
$B_2(\Theta) \times \K(\Theta,n)$, whose boundary is equal to
$\Delta$.
\end{lemma}

Indeed, the formula in the upper left-hand corner of Fig. \ref{ress} implies
that in the intersection with $\sigma_1(\Theta,n)$ the boundary of
this chain is equal to $\Delta$. Comparing the formulas of figures
\ref{easiest}, \ref{selftan} with figures \ref{easiest0}, \ref{selftan0},
we obtain immediately, that all the other pieces of the summands of
this chain should annihilate each other, provided that the
corresponding pieces of boundaries of cells in $\Delta$,
from which these summands occur, did it.
\hfill $\square$
\medskip

This proves the first statement of Theorem \ref{predsti}.
\medskip

2. Now let us try to construct an {\em integer} linear combination
$G$ of chains $V(\Gamma)$ with properties $(*)$, $(**)$ for an
integral 2-cycle $\Delta$.
In total we have $2 {\bnm{2m+1}{2}}$ Gau\ss \ diagrams $\Gamma$
for this, two for
any generator of the group $\Pi(\Theta)$, in correspondence with
two possible orientations of the non-oriented Gau\ss \ diagram
$\Gamma_0$ encoding
such a generator. The condition $(**)$ implies immediately
${\bnm{2m+1}{2}}$ independent linear restrictions
on the coefficients at these
chains $V(\Gamma)$ in the desired linear combination $G$.
Namely, for any such non-oriented diagram
$\Gamma_0$, two chains $V(\Gamma)$ and $V(\Gamma')$, whose
diagrams occur by orientating this diagram, should participate in the
desired chain $G$ with coefficients, whose difference is equal to
the coefficient at the chain $V(\Gamma_0)$ in the linear combination
$\Delta$. This follows from the upper left-hand equality in
Fig. \ref{ress} and from the fact that any generators
$V(\Gamma_0)$ of the group $\Pi(\Theta)$ can occur in the
boundaries of exactly two such chains $V(\tilde \Gamma)$ only.

Let us choose an {\em arbitrary} linear combination $\tilde G$
of sums of chains $V(\Gamma_i)$, satisfying this condition $(**)$.
For instance, we can simply orient from the right to the left
in $\R^1$ all broken lines participating in notations of
all summands of the linear combination
$\Delta$. The intersection of the boundary of this chain
$\tilde G$ with $\sigma_1(\Theta,n)$ will be equal to $\Delta$,
however aside of $\sigma_1(\Theta,n)$ this boundary can have
some extra components.

To replace this preliminary solution $\tilde G$ by a correct one, $G$,
we need to find at least one solution of the following equation
on the discrepancy $\tilde G - G$:
\begin{equation}
\label{homeq}
\partial (\tilde G - G) = \partial \tilde G - \Delta,
\end{equation}
i.e. outside $\sigma_1(\Theta,n)$ the boundary of the chain $\tilde G - G$
should coincide with $\partial \tilde G,$ and on
$\Sigma(\Theta,n)$ this boundary should be zero. The latter condition
is equivalent to the following one: any two oriented
Gau\ss \ diagrams, that differ by
the orientation of arrows, participate in the linear combination
$\tilde G - G$ with equal coefficients, so that it remains
to solve the first condition in the class of such linear combinations.

For any map $f: \R^1 \to \R^n$ we define the map
$f_1: \R^1 \to \R^{n-1}$ as the composition of $f$ and the
projection of $\R^n$ to the ``blackboard plane'' $\R^{n-1}$.
It is obvious that this transformation $f \mapsto f_1$ projects
$\K(\Theta,n)$ onto $\K(\Theta,n-1)$ with fiber $\K(\Theta,1)$.

\begin{proposition}
The right-hand part $\partial \tilde G - \Delta$ of the equation
$($\ref{homeq}$)$ is a linear combination of chains, in whose
definition only the conditions on the behavior of the map
$f_1$ at certain points participate, namely the conditions of the
triple self-intersection of the image of this map
$($defining some generators of the group $\Xi(\Theta)$ of the complex
$\CC_*(\Theta,n-1))$ or self-tangencies
$($defining generators of the group $\|(\Theta)$ of the same complex$)$.
\label{maprop}
\end{proposition}

{\em Proof.} By construction, the chain $\partial \tilde G - \Delta$
is a linear combination of strata defined by the diagram $\Theta$
with one additional condition like ones shown in the right-hand parts
of equalities of Figs. \ref{easiest} and \ref{selftan}. Also, it is a
cycle in the entire space $B_2(\Theta) \times \K(\Theta,n)$.
This implies that together with any summand shown in the right-hand part of
either of equalities of Fig. \ref{easiest} it should contain with the same
coefficient the similar summand with the reversed arrow. Indeed, otherwise
the homological condition $\partial (\partial \tilde G - \Delta)=0$
is not satisfied close to the common piece of the boundaries of these
strata, which is defined by the similar picture with non-oriented broken
line. This follows from first two lines of Fig. \ref{ress} and from the fact
that this their common boundary does not participate in the boundary
of any other manifold occurring as a standard piece of the boundary
of the stratum $V(\Gamma)$ defined by an oriented
Gau\ss \ diagram of first order.

But the sum of two such summands is defined in
$B_2(\Theta) \times \K(\Theta,n)$
by the condition that the value of the map $f_1$ at some point of a
certain non-singular interval coincides with the value of this map
at the endpoints of some chord of the diagram $\Theta$.

In the same way, by the lower line of Fig. \ref{ress}, any summand,
shown in the right-hand part of Fig. \ref{selftan}, should be matched by
the similar summand, in which the subscript like the one under the first
(respectively, second, third, fourth) formula is replaced by the subscript
under the second (respectively, first, fourth, third) formula of the same
figure. The sum of these two summands can be defined by a single
condition that the derivatives in $\R^{n-1}$ of the map $f_1$ at
the endpoints of the chord should be co- or counter-directed.
\hfill $\square$ \medskip

So, the chain $\partial \tilde G - \Delta$ can be described as
the direct product of the space $\K(\Theta,1)$ and some one-dimensional
chain of the complex $\CC_*(\Theta,n-1).$ It is natural to denote
the latter chain by
$\frac{\partial \tilde G - \Delta}{\K(\Theta,1)}.$
Since $\partial \tilde G - \Delta$ is a cycle in
$B_2(\Theta,n) \times \K(\Theta,n)$, this chain also is a cycle.

On the other hand, the desired chain $\tilde G - G$
in the equation (\ref{homeq}) should be a linear
combination of oriented Gau\ss \  diagrams, in which any diagram participates
simultaneously (and with the same coefficient) with the same diagram
with reversed orientation. Such pairs of diagrams are in a
one-to-one correspondence with generators of the group
$\Pi(\Theta)$ of the complex $\CC_*(\Theta,n-1)$,
so that the chain $\tilde G - G$ corresponds to a certain
element of this group. The chain $\tilde G - G$ satisfies the
equation (\ref{homeq}) if and only if the boundary of the
corresponding element $\frac{\tilde G - G}{\K(\Theta,1)}$
of this group $\Pi(\Theta)$
is equal to the element
$\frac{\partial \tilde G - \Delta}{\K(\Theta,1)}$
of the group
$\Xi(\Theta) \oplus \|(\Theta)$,
that corresponds to the cycle
$\partial \tilde G - \Delta$
by Proposition \ref{maprop}.

So, the unique obstruction to the existence of such a chain
$\tilde G - G$ is the class of the element
$\frac{\partial \tilde G - \Delta}{\K(\Theta,1)}$
in the group $H_1(\Theta,n-1)$. The proof of part 1 of Theorem
\ref{predsti}
implies that this class is either equal to zero or of order 2.
The cycle $\delta$ can be represented by
an integer combinatorial formula, consisting of oriented
Gau\ss \ diagrams, if and only if this class is zero. By Theorems
\ref{homodd} and \ref{homeven} the latter condition is surely satisfied if
$n-1$ is odd or if $n-1$ is even but the chord diagram $\Theta$ is
planar: indeed, in these cases the group $H_1(\Theta,n-1)$ has no
2-torsion.
\hfill $\square$ \medskip

\begin{example} \rm
Suppose that $n=3$. Let us investigate the second invariant of
order four of (non-singular) long knots, see \cite{knsp}, \cite{fasis},
and the corresponding first order invariant of long curves with three
self-intersections.
Since our obstruction to the existence of an integer combinatorial formula
lies in the 2-torsion group, we drop the hard work of accounting orientations
and carry out all calculations modulo 2 only.

The principal part of the invariant is shown in Fig. \ref{mainp}.
Its first differential defines first order invariants for all long
curves $\R^1 \to \R^3$ with three self-intersections.

\begin{figure}
\begin{center}
\unitlength 1.00mm
\linethickness{0.4pt}
\begin{picture}(111.00,44.00)
\put(13.00,39.00){\oval(16.00,6.00)[t]}
\put(17.00,39.00){\oval(16.00,6.00)[b]}
\put(21.00,39.00){\oval(16.00,10.00)[t]}
\put(25.00,39.00){\oval(16.00,10.00)[b]}
\put(3.00,39.00){\line(1,0){32.00}}
\put(38.00,39.00){\makebox(0,0)[cc]{$+$}}
\put(53.00,39.00){\oval(20.00,10.00)[t]}
\put(53.00,39.00){\oval(12.00,6.00)[t]}
\put(61.00,39.00){\oval(12.00,6.00)[b]}
\put(61.00,39.00){\oval(20.00,10.00)[b]}
\put(73.00,39.00){\line(-1,0){32.00}}
\put(76.00,39.00){\makebox(0,0)[cc]{$+$}}
\put(87.00,39.00){\oval(12.00,8.00)[t]}
\put(95.00,39.00){\oval(12.00,6.00)[b]}
\put(95.00,39.00){\oval(20.00,10.00)[b]}
\put(103.00,39.00){\oval(12.00,8.00)[t]}
\put(111.00,39.00){\line(-1,0){32.00}}
\put(11.00,23.00){\oval(12.00,6.00)[t]}
\put(17.00,23.00){\oval(16.00,6.00)[b]}
\put(21.00,23.00){\oval(16.00,10.00)[t]}
\put(27.00,23.00){\oval(12.00,10.00)[b]}
\put(35.00,23.00){\line(-1,0){32.00}}
\put(38.00,23.00){\makebox(0,0)[cc]{$+$}}
\put(51.00,23.00){\oval(16.00,6.00)[b]}
\put(55.00,23.00){\oval(16.00,6.00)[t]}
\put(61.00,23.00){\oval(20.00,10.00)[b]}
\put(61.00,23.00){\oval(12.00,10.00)[t]}
\put(73.00,23.00){\line(-1,0){32.00}}
\put(76.00,23.00){\makebox(0,0)[cc]{$+$}}
\put(89.00,23.00){\oval(16.00,6.00)[t]}
\put(95.00,23.00){\oval(20.00,6.00)[b]}
\put(95.00,23.00){\oval(12.00,10.00)[t]}
\put(101.00,23.00){\oval(16.00,10.00)[b]}
\put(111.00,23.00){\line(-1,0){32.00}}
\put(0.00,23.00){\makebox(0,0)[cc]{$+$}}
\put(3.00,7.00){\line(1,0){32.00}}
\put(15.00,7.00){\oval(20.00,6.00)[t]}
\put(15.00,7.00){\oval(12.00,6.00)[b]}
\put(21.00,7.00){\oval(16.00,10.00)[t]}
\put(25.00,7.00){\oval(16.00,10.00)[b]}
\put(0.00,7.00){\makebox(0,0)[cc]{$+$}}
\end{picture}
\caption{Principal part of the second invariant of fourth order}
\label{mainp}
\end{center}
\end{figure}

In particular, for curves with self-intersection type given by the
3-chord diagram
\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(54.00,13.50)
\put(23.00,6.00){\oval(16.00,9.00)[t]}
\put(23.00,6.00){\oval(32.00,15.00)[t]}
\put(35.00,6.00){\oval(24.00,12.00)[b]}
\put(54.00,6.00){\line(-1,0){52.00}}
\put(4.00,3.00){\makebox(0,0)[cc]{1}}
\put(11.00,3.00){\makebox(0,0)[cc]{2}}
\put(19.00,3.00){\makebox(0,0)[cc]{3}}
\put(27.00,3.00){\makebox(0,0)[cc]{4}}
\put(35.00,3.00){\makebox(0,0)[cc]{5}}
\put(43.00,9.00){\makebox(0,0)[cc]{6}}
\put(51.00,9.00){\makebox(0,0)[cc]{7}}
\put(9.00,13.50){\makebox(0,0)[cc]{$\alpha$}}
\put(15.00,10.00){\makebox(0,0)[cc]{$\beta$}}
\put(23.50,0.50){\makebox(0,0)[cc]{$\gamma$}}
\end{picture}
}
\end{equation}
this first differential $\Delta$ is equal to

\begin{equation}
\label{firdif}
(1+6+7)\cdot (3+4).
\end{equation}

Let us kill its six summands by oriented Gau\ss \
diagrams, constituting the preliminary
solution $\tilde G$ and shown in the left-hand parts of the following
formulas (\ref{d1-1})---(\ref{d1-6}). Here the killed summands of
(\ref{firdif}) are the first summands of the right-hand parts of these
equalities, and the remaining ones form the discrepancy
$\partial \tilde G - \Delta.$ Killing the non-oriented Gau\ss \ diagrams,
it is convenient to use the {\em rule of left endpoint}:
we replace the non-oriented
broken line, participating in the notation of the
diagram, by the broken arrow directed to the right
or to the left depending on the parity of the number of the
component of $\R^1 \sm \Theta$ containing the left-hand endpoint of the
broken line.  Thanks to this rule, many summands of the discrepancy
annihilate immediately.

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(117.00,26.00)
\put(13.00,21.00){\oval(16.00,10.00)[t]}
\put(13.00,21.00){\oval(8.00,6.00)[t]}
\put(19.00,21.00){\oval(12.00,8.00)[b]}
\put(3.00,21.00){\line(1,0){24.00}}
\put(30.00,21.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,21.00){\oval(16.00,10.00)[t]}
\put(103.00,21.00){\oval(16.00,10.00)[t]}
\put(48.00,6.00){\oval(16.00,10.00)[t]}
\put(73.00,21.00){\oval(8.00,6.00)[t]}
\put(103.00,21.00){\oval(8.00,6.00)[t]}
\put(48.00,6.00){\oval(8.00,6.00)[t]}
\put(79.00,21.00){\oval(12.00,8.00)[b]}
\put(109.00,21.00){\oval(12.00,8.00)[b]}
\put(54.00,6.00){\oval(12.00,8.00)[b]}
\put(63.00,21.00){\line(1,0){24.00}}
\put(93.00,21.00){\line(1,0){24.00}}
\put(38.00,6.00){\line(1,0){24.00}}
\put(0.00,21.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,21.00){\makebox(0,0)[cc]{$+$}}
\put(35.00,6.00){\makebox(0,0)[cc]{$+$}}
\put(4.00,21.00){\line(0,-1){4.00}}
\put(4.00,17.00){\line(1,0){7.00}}
\put(11.00,17.00){\vector(0,1){4.00}}
\put(73.00,26.00){\vector(-1,-3){1.67}}
\put(39.00,6.00){\line(0,-1){4.00}}
\put(39.00,2.00){\line(1,0){6.00}}
\put(45.00,2.00){\vector(1,2){3.50}}
\put(94.00,21.00){\line(0,-1){6.00}}
\put(94.00,15.00){\line(1,0){15.00}}
\put(109.00,15.00){\vector(0,1){2.00}}
\put(43.00,21.00){\oval(16.00,10.00)[t]}
\put(43.00,21.00){\oval(8.00,6.00)[t]}
\put(49.00,21.00){\oval(12.00,8.00)[b]}
\put(33.00,21.00){\line(1,0){24.00}}
\put(34.00,21.00){\line(0,-1){4.00}}
\put(34.00,17.00){\line(1,0){7.00}}
\put(41.00,17.00){\line(0,1){4.00}}
\put(60.00,21.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-1}
\end{equation}

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(117.00,26.00)
\put(13.00,21.00){\oval(16.00,10.00)[t]}
\put(13.00,21.00){\oval(8.00,6.00)[t]}
\put(19.00,21.00){\oval(12.00,8.00)[b]}
\put(3.00,21.00){\line(1,0){24.00}}
\put(30.00,21.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,21.00){\oval(16.00,10.00)[t]}
\put(103.00,21.00){\oval(16.00,10.00)[t]}
\put(47.00,5.00){\oval(16.00,10.00)[t]}
\put(73.00,21.00){\oval(8.00,6.00)[t]}
\put(103.00,21.00){\oval(8.00,6.00)[t]}
\put(47.00,5.00){\oval(8.00,6.00)[t]}
\put(79.00,21.00){\oval(12.00,8.00)[b]}
\put(109.00,21.00){\oval(12.00,8.00)[b]}
\put(53.00,5.00){\oval(12.00,8.00)[b]}
\put(63.00,21.00){\line(1,0){24.00}}
\put(93.00,21.00){\line(1,0){24.00}}
\put(37.00,5.00){\line(1,0){24.00}}
\put(0.00,21.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,21.00){\makebox(0,0)[cc]{$+$}}
\put(34.00,5.00){\makebox(0,0)[cc]{$+$}}
\put(4.00,21.00){\line(0,-1){4.00}}
\put(4.00,17.00){\line(1,0){7.00}}
\put(11.00,17.00){\vector(1,1){4.00}}
\put(64.00,21.00){\line(0,-1){6.00}}
\put(64.00,15.00){\line(1,0){15.00}}
\put(79.00,15.00){\vector(0,1){2.00}}
\put(94.00,21.00){\line(0,-1){4.00}}
\put(94.00,17.00){\line(1,0){6.00}}
\put(100.00,17.00){\vector(1,2){3.50}}
\put(47.00,10.00){\vector(1,-3){1.67}}
\put(43.00,21.00){\oval(16.00,10.00)[t]}
\put(43.00,21.00){\oval(8.00,6.00)[t]}
\put(49.00,21.00){\oval(12.00,8.00)[b]}
\put(33.00,21.00){\line(1,0){24.00}}
\put(34.00,21.00){\line(0,-1){4.00}}
\put(34.00,17.00){\line(1,0){7.00}}
\put(41.00,17.00){\line(1,1){4.00}}
\put(60.00,21.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-2}
\end{equation}

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(123.00,36.01)
\put(13.00,31.00){\oval(16.00,10.00)[t]}
\put(13.00,31.00){\oval(8.00,6.00)[t]}
\put(19.00,31.00){\oval(12.00,8.00)[b]}
\put(3.00,31.00){\line(1,0){24.00}}
\put(30.00,31.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,31.00){\oval(16.00,10.00)[t]}
\put(103.00,31.00){\oval(16.00,10.00)[t]}
\put(47.00,15.00){\oval(16.00,10.00)[t]}
\put(73.00,31.00){\oval(8.00,6.00)[t]}
\put(103.00,31.00){\oval(8.00,6.00)[t]}
\put(47.00,15.00){\oval(8.00,6.00)[t]}
\put(79.00,31.00){\oval(12.00,8.00)[b]}
\put(109.00,31.00){\oval(12.00,8.00)[b]}
\put(53.00,15.00){\oval(12.00,8.00)[b]}
\put(63.00,31.00){\line(1,0){24.00}}
\put(93.00,31.00){\line(1,0){24.00}}
\put(37.00,15.00){\line(1,0){24.00}}
\put(0.00,31.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,31.00){\makebox(0,0)[cc]{$+$}}
\put(34.00,15.00){\makebox(0,0)[cc]{$+$}}
\put(11.00,31.00){\line(0,-1){6.00}}
\put(11.00,25.00){\line(1,0){12.00}}
\put(23.00,25.00){\vector(0,1){6.00}}
\put(71.00,31.00){\line(0,-1){6.00}}
\put(71.00,25.00){\line(1,0){8.00}}
\put(79.00,25.00){\vector(0,1){2.00}}
\put(101.00,31.00){\vector(1,3){1.67}}
\put(77.00,15.00){\oval(16.00,10.00)[t]}
\put(77.00,15.00){\oval(8.00,6.00)[t]}
\put(83.00,15.00){\oval(12.00,8.00)[b]}
\put(67.00,15.00){\line(1,0){24.00}}
\put(64.00,15.00){\makebox(0,0)[cc]{$+$}}
\put(109.00,15.00){\oval(16.00,10.00)[t]}
\put(109.00,15.00){\oval(8.00,6.00)[t]}
\put(115.00,15.00){\oval(12.00,8.00)[b]}
\put(99.00,15.00){\line(1,0){24.00}}
\put(95.00,15.00){\makebox(0,0)[cc]{$+$}}
\put(53.00,11.00){\vector(1,1){4.00}}
\put(77.00,18.00){\line(0,1){3.00}}
\put(77.00,21.00){\line(1,0){10.00}}
\put(87.00,21.00){\vector(0,-1){6.00}}
\put(99.00,6.00){\vector(2,1){6.00}}
\put(99.00,6.00){\vector(2,-1){6.00}}
\put(108.00,3.00){\makebox(0,0)[cc]{6}}
\put(108.00,9.00){\makebox(0,0)[cc]{3}}
\put(43.00,31.00){\oval(16.00,10.00)[t]}
\put(43.00,31.00){\oval(8.00,6.00)[t]}
\put(49.00,31.00){\oval(12.00,8.00)[b]}
\put(33.00,31.00){\line(1,0){24.00}}
\put(41.00,31.00){\line(0,-1){6.00}}
\put(41.00,25.00){\line(1,0){12.00}}
\put(53.00,25.00){\line(0,1){6.00}}
\put(60.00,31.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-3}
\end{equation}

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(117.00,31.00)
\put(13.00,26.00){\oval(16.00,10.00)[t]}
\put(13.00,26.00){\oval(8.00,6.00)[t]}
\put(19.00,26.00){\oval(12.00,8.00)[b]}
\put(3.00,26.00){\line(1,0){24.00}}
\put(30.00,26.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,26.00){\oval(16.00,10.00)[t]}
\put(103.00,26.00){\oval(16.00,10.00)[t]}
\put(48.00,12.00){\oval(16.00,10.00)[t]}
\put(73.00,26.00){\oval(8.00,6.00)[t]}
\put(103.00,26.00){\oval(8.00,6.00)[t]}
\put(48.00,12.00){\oval(8.00,6.00)[t]}
\put(79.00,26.00){\oval(12.00,8.00)[b]}
\put(109.00,26.00){\oval(12.00,8.00)[b]}
\put(54.00,12.00){\oval(12.00,8.00)[b]}
\put(63.00,26.00){\line(1,0){24.00}}
\put(93.00,26.00){\line(1,0){24.00}}
\put(38.00,12.00){\line(1,0){24.00}}
\put(0.00,26.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,26.00){\makebox(0,0)[cc]{$+$}}
\put(35.00,12.00){\makebox(0,0)[cc]{$+$}}
\put(78.00,12.00){\oval(16.00,10.00)[t]}
\put(78.00,12.00){\oval(8.00,6.00)[t]}
\put(84.00,12.00){\oval(12.00,8.00)[b]}
\put(68.00,12.00){\line(1,0){24.00}}
\put(65.00,12.00){\makebox(0,0)[cc]{$+$}}
\put(11.00,26.00){\line(0,-1){6.00}}
\put(11.00,20.00){\line(1,0){15.00}}
\put(26.00,20.00){\vector(0,1){6.00}}
\put(71.00,26.00){\line(0,-1){6.00}}
\put(71.00,20.00){\line(1,0){8.00}}
\put(79.00,20.00){\vector(0,1){2.00}}
\put(109.00,22.00){\line(0,-1){2.00}}
\put(109.00,20.00){\line(1,0){7.00}}
\put(116.00,20.00){\vector(0,1){6.00}}
\put(48.00,15.00){\line(0,1){3.00}}
\put(48.00,18.00){\line(1,0){13.00}}
\put(61.00,18.00){\vector(0,-1){6.00}}
\put(74.00,2.00){\vector(2,1){6.00}}
\put(74.00,2.00){\vector(-2,1){6.00}}
\put(66.00,5.00){\makebox(0,0)[cc]{6}}
\put(82.00,5.00){\makebox(0,0)[cc]{3}}
\put(43.00,26.00){\oval(16.00,10.00)[t]}
\put(43.00,26.00){\oval(8.00,6.00)[t]}
\put(49.00,26.00){\oval(12.00,8.00)[b]}
\put(33.00,26.00){\line(1,0){24.00}}
\put(41.00,26.00){\line(0,-1){6.00}}
\put(41.00,20.00){\line(1,0){15.00}}
\put(56.00,20.00){\line(0,1){6.00}}
\put(60.00,26.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-4}
\end{equation}

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(121.00,28.00)
\put(13.00,23.00){\oval(16.00,10.00)[t]}
\put(13.00,23.00){\oval(8.00,6.00)[t]}
\put(19.00,23.00){\oval(12.00,8.00)[b]}
\put(3.00,23.00){\line(1,0){24.00}}
\put(30.00,23.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,23.00){\oval(16.00,10.00)[t]}
\put(103.00,23.00){\oval(16.00,10.00)[t]}
\put(47.00,10.00){\oval(16.00,10.00)[t]}
\put(73.00,23.00){\oval(8.00,6.00)[t]}
\put(103.00,23.00){\oval(8.00,6.00)[t]}
\put(47.00,10.00){\oval(8.00,6.00)[t]}
\put(79.00,23.00){\oval(12.00,8.00)[b]}
\put(109.00,23.00){\oval(12.00,8.00)[b]}
\put(53.00,10.00){\oval(12.00,8.00)[b]}
\put(63.00,23.00){\line(1,0){24.00}}
\put(93.00,23.00){\line(1,0){24.00}}
\put(37.00,10.00){\line(1,0){24.00}}
\put(0.00,23.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,23.00){\makebox(0,0)[cc]{$+$}}
\put(34.00,10.00){\makebox(0,0)[cc]{$+$}}
\put(77.00,10.00){\oval(16.00,10.00)[t]}
\put(107.00,10.00){\oval(16.00,10.00)[t]}
\put(77.00,10.00){\oval(8.00,6.00)[t]}
\put(107.00,10.00){\oval(8.00,6.00)[t]}
\put(83.00,10.00){\oval(12.00,8.00)[b]}
\put(113.00,10.00){\oval(12.00,8.00)[b]}
\put(67.00,10.00){\line(1,0){24.00}}
\put(97.00,10.00){\line(1,0){24.00}}
\put(64.00,10.00){\makebox(0,0)[cc]{$+$}}
\put(94.00,10.00){\makebox(0,0)[cc]{$+$}}
\put(23.00,23.00){\line(-4,-3){4.00}}
\put(19.00,20.00){\vector(-4,3){4.00}}
\put(79.00,19.00){\vector(-1,1){4.00}}
\put(103.00,28.00){\vector(1,-3){1.67}}
\put(57.00,10.00){\line(0,1){6.00}}
\put(57.00,16.00){\line(-1,0){10.00}}
\put(47.00,16.00){\vector(0,-1){3.00}}
\put(87.00,10.00){\vector(-1,-1){4.00}}
\put(102.00,1.00){\vector(2,1){6.00}}
\put(102.00,1.00){\vector(-2,1){6.00}}
\put(94.00,4.00){\makebox(0,0)[cc]{6}}
\put(110.00,4.00){\makebox(0,0)[cc]{3}}
\put(43.00,23.00){\oval(16.00,10.00)[t]}
\put(43.00,23.00){\oval(8.00,6.00)[t]}
\put(49.00,23.00){\oval(12.00,8.00)[b]}
\put(33.00,23.00){\line(1,0){24.00}}
\put(53.00,23.00){\line(-4,-3){4.00}}
\put(49.00,20.00){\line(-4,3){4.00}}
\put(60.00,23.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-5}
\end{equation}

\begin{equation}
\unitlength 1.00mm
\linethickness{0.4pt}
\mbox{
\begin{picture}(117.00,32.00)
\put(13.00,27.00){\oval(16.00,10.00)[t]}
\put(13.00,27.00){\oval(8.00,6.00)[t]}
\put(19.00,27.00){\oval(12.00,8.00)[b]}
\put(3.00,27.00){\line(1,0){24.00}}
\put(30.00,27.00){\makebox(0,0)[cc]{$=$}}
\put(73.00,27.00){\oval(16.00,10.00)[t]}
\put(103.00,27.00){\oval(16.00,10.00)[t]}
\put(48.00,12.00){\oval(16.00,10.00)[t]}
\put(73.00,27.00){\oval(8.00,6.00)[t]}
\put(103.00,27.00){\oval(8.00,6.00)[t]}
\put(48.00,12.00){\oval(8.00,6.00)[t]}
\put(79.00,27.00){\oval(12.00,8.00)[b]}
\put(109.00,27.00){\oval(12.00,8.00)[b]}
\put(54.00,12.00){\oval(12.00,8.00)[b]}
\put(63.00,27.00){\line(1,0){24.00}}
\put(93.00,27.00){\line(1,0){24.00}}
\put(38.00,12.00){\line(1,0){24.00}}
\put(0.00,27.00){\makebox(0,0)[cc]{$\partial$}}
\put(90.00,27.00){\makebox(0,0)[cc]{$+$}}
\put(35.00,12.00){\makebox(0,0)[cc]{$+$}}
\put(78.00,12.00){\oval(16.00,10.00)[t]}
\put(78.00,12.00){\oval(8.00,6.00)[t]}
\put(84.00,12.00){\oval(12.00,8.00)[b]}
\put(68.00,12.00){\line(1,0){24.00}}
\put(65.00,12.00){\makebox(0,0)[cc]{$+$}}
\put(26.00,27.00){\line(0,-1){6.00}}
\put(26.00,21.00){\line(-1,0){11.00}}
\put(15.00,21.00){\vector(0,1){6.00}}
\put(79.00,23.00){\vector(-1,1){4.00}}
\put(116.00,27.00){\line(0,-1){6.00}}
\put(116.00,21.00){\line(-1,0){7.00}}
\put(109.00,21.00){\vector(0,1){2.00}}
\put(61.00,12.00){\line(0,1){6.00}}
\put(61.00,18.00){\line(-1,0){13.00}}
\put(48.00,18.00){\vector(0,-1){3.00}}
\put(68.00,4.00){\vector(2,1){6.00}}
\put(68.00,4.00){\vector(2,-1){6.00}}
\put(77.00,1.00){\makebox(0,0)[cc]{6}}
\put(77.00,7.00){\makebox(0,0)[cc]{3}}
\put(43.00,27.00){\oval(16.00,10.00)[t]}
\put(43.00,27.00){\oval(8.00,6.00)[t]}
\put(49.00,27.00){\oval(12.00,8.00)[b]}
\put(33.00,27.00){\line(1,0){24.00}}
\put(56.00,27.00){\line(0,-1){6.00}}
\put(56.00,21.00){\line(-1,0){11.00}}
\put(45.00,21.00){\line(0,1){6.00}}
\put(60.00,27.00){\makebox(0,0)[cc]{+}}
\end{picture}
}
\label{d1-6}
\end{equation}


Denote by (a;b) the b-th summand in the right-hand part of the equation
(a), then we have
$(\ref{d1-1};2)+(\ref{d1-3};3)= 3\wedge \alpha , $
$(\ref{d1-3};4)+(\ref{d1-5};5)= 6 \wedge \gamma, $
$(\ref{d1-3};5)+(\ref{d1-5};4)= 6 \wedge \beta , $
$(\ref{d1-4};3)+(\ref{d1-6};3)= 7 \wedge \gamma, $
$(\ref{d1-4};4)+(\ref{d1-6};4)= 7 \wedge \beta , $
while the remaining summands of the sum of all these right-hand parts
either annihilate or coincide with the killed summands of the cycle
(\ref{firdif}).

So, these six oriented
Gau\ss \ diagrams provide a homology between the cycle
(\ref{firdif}) and the cycle
$3\wedge \alpha +6 \wedge \beta + 6 \wedge \gamma + 7 \wedge \beta
+ 7 \wedge \gamma \in \Xi(\Theta)$
(strictly speaking, multiplied by $\K(\Theta,1)$).

Using the algorithm from the proof of Lemma \ref{lem1chet},
we reduce this cycle to the group $\|(\Theta)$:
$$\partial (3 \wedge 7) = 3 \wedge \gamma + 7 \wedge \gamma + 7 \wedge \beta
+ \gamma_\rla \ ;$$
$$\partial (3 \wedge 6) = 3 \wedge \gamma + 3 \wedge \alpha +
6 \wedge \gamma + 6 \wedge \beta + \gamma_\rra \ .$$
So, our cycle is homologous to the element
$\gamma_\rra + \gamma_\rla$, and by Theorem \ref{homeven}
is not equal to zero in the group
$H_1(\Theta,2) \simeq \Z_2.$
\end{example}

{\em Proof of Theorem \ref{necconi}.}
\label{prneccon}

1. Let $f$, $f_1$ be the same as before the formulation of this theorem.
Any two generic immersions of class $\K(\Theta,3)$ can be connected
in $\K(\Theta,3)$ by a sequence of

a) smooth isotopies that do not change the topological types of both
the curve $f(\R^1) \subset \R^3$ and the curve
$f_1(\R^1) \subset \R^2$;

b) local surgeries not changing the topological type of the curve
$f(\R^1) \subset \R^3$ but changing the type of its projection
$f_1(\R^1)$;

c) local surgeries of the curve $f(\R^1)$, occurring at traversing
the instant additional self-intersection of this curve and described
in the terms of the planar diagram as replacing an overcrossing
by undercrossing or vice versa.

The moves a) and c) obviously do not change the parity of the
sum (\ref{maprep}). Local moves b) can be of four types:

1) the image of the $a$-th component of $\R^1 \sm \Theta$ can pass over or
under the intersection point, depicted by the chord $\alpha$ (see the
right-hand part of Fig. \ref{rmc} on page \pageref{rmc})

2), 3) the curve $f_1(\R^1)$
can experience an instant self-tangency at some its
self-intersection point depicted by the chord $\alpha$
(see the left-hand part of the same figure), while the derivatives
of the map $f_1$ at this point can be co-directed or counter-directed;

4) the curve $f_1(\R^1)$ can experience an instant self-tangency at some
its non-singular pieces, after which the number of its double
points increases or decreases by 2; cf. \cite{arnold}.

It is obvious that the surgery 4) does not change the parity of the
sum (\ref{maprep}). It follows immediately from the description of
differentials in the
complex $\CC_*(\Theta,3)$ that the increase of this sum
at the surgery of first (respectively, second or third) type
is equal modulo 2 to the coefficient, with which the element
$a \cdot \alpha \in \Xi(\Theta)$ (respectively, elements
$\alpha_\rra \in \|(\Theta)$ or $\alpha_\rla \in \|(\Theta)$) participate
in the boundary of our cycle $F$. Since the boundary of the cycle
is equal to zero, statement 1 of the Theorem is proved.

2. For an arbitrary immersion $f: \R^1 \to \R^3$ of class $\K(\Theta,3)$
denote by $\tilde f$ the result of its rotation on the angle
$180^\circ$ around the axis $l(\R^1)$  of $\R^3$.

Suppose that our cycle $F$ is represented by an integer linear combination of
oriented Gau\ss \ diagrams.
The number $\lambda_{a,b}$ from the description (\ref{licomi})
of this cycle is equal up to a sign (and hence also modulo 2)
to the difference between the coefficients, with which this linear
combination contains two oppositely oriented Gau\ss \ diagrams, whose
broken arrows connect
the $a$-th and $b$-th non-singular intervals. This implies immediately,
that for an arbitrary generic immersion $f$ of class $\K(\Theta,3)$
the difference of values of our combinatorial formula on $f$ and $f_1$
coincides modulo 2 with the number (\ref{maprep}). Since
$f$ and $f_1$ are obviously isotopic to one another, this difference
should be even. \hfill $\square$ \medskip

{\em Proof of Theorem \ref{predsti2}.}
Let us take a pair of cycles of the graph $f(\R^1)$, whose existence
is asserted in the Manturov's Theorem \ref{vassya}.
Let us show that for the desired element of the group
$H^{n-3}_{(1)}(\K(\Theta,n)\sm \Sigma(\Theta,n))$
we can take the element, corresponding by Proposition \ref{justa}
to the product of these two cycles, i.e. to
the sum of all elements $a \cdot b \in \Pi(\Theta)$,
where $a$ is an interval of the first cycle, and
$b$ is an interval of the second one. Indeed, the
intersection index of projections of these two cycles to the plane $\R^2$ is
obviously equal to zero. This zero consists of the number (equal to 1)
of crossing points considered in Theorem \ref{vassya},
and the number of (counted with signs) double points of
these projections, considered in Theorem \ref{necconi}.
Therefore the number of double points of the latter type
is odd, and Theorem \ref{predsti2} follows from Theorem
\ref{necconi}. \hfill $\square$

\section{Realization of groups $H_1(\Theta,n)$ with odd $n$}
\label{realodd}

\begin{theorem}
\label{realiz}
Suppose that $n$ is odd, $n \ge 3,$ and $\Theta$ is a $m$-chord diagram.
Then for any chord $\alpha$ of $\Theta$ the linear combination
\begin{equation}
\label{basone}
1 \cdot \alpha + 2 \cdot \alpha + \dots + (2m+1) \cdot \alpha
\end{equation}
is a cycle of the complex $\CC_*(\Theta,n)$.
Such cycles, corresponding to chords $\alpha$
from different connected components of the intersection graph
of the diagram $\Theta$, are linearly independent and, taken one for
any such component, define a basis of the group
$H_1(\Theta,\mbox{\rm odd})$.
\end{theorem}

{\em Proof.} The fact that (\ref{basone}) is a cycle follows immediately
from the description of boundary operators of our complex,
see Figs. \ref{bf}, \ref{bevar}. Let us choose arbitrarily one such cycle
for any connected component of the intersection graph of $\Theta$.
By Theorem \ref{homeven} the rank of the group $H_1(\Theta,\mbox{odd})$
is equal to the number of these cycles. Let us construct a collection
of elements in $H_{n-2}(\K(\Theta,n)\sm \Sigma(\Theta,n)),$
Alexander dual to these cycles.

Consider the immersion $\varphi: \R^1 \to \R^n$
and collection of $\#(\Theta)$ balls in $\R^n$, described in Lemma
\ref{rballs} on page \pageref{rballs}. We associate with any such ball
a submanifold in $\K(\Theta,n) \sm \Sigma(\Theta,n)$,
diffeomorphic to $S^{n-2}$, all whose points are maps $\R^1 \to \R^n$
coinciding with $\varphi$ outside this
ball. For instance, for $n=3$ this manifold is obtained from
$\varphi$ by all possible rotations of this ball around the axis
$l(\R^1)$ of the space $\R^n$. In the general case we consider the space
$S^{n-2}$ of all possible oriented two-dimensional planes in
$\R^n,$ containing this axis, and the fiber bundle of oriented flags
over $S^{n-2}$, whose fiber over such a plane consists of all possible
oriented three-dimensional subspaces in $\R^n$, containing this plane.
Since $n$ is odd, this flag bundle has a smooth section. This section
allows us to identify all these three-dimensional subspaces to
one another in such a way that any such identification transforms
the affine structure, Euclidean metric, intersection of the subspace with
the ball, the axis and the two-dimensional plane into the analogous
objects of the other subspace. Without loss of generality we can
assume that inside the ball the curve $\varphi(\R^1)$ lies in one of
such three-dimensional subspaces. Using our identifications, we obtain
a family of similar curves, that coincide with one another outside the ball,
while inside the ball any of them lies in its own three-dimensional
subspace; this family is naturally diffeomorphic to the base
$S^{n-2}$ of the flag bundle. Denote this family by $S^{n-2}_j$,
where $j$ is the number of our connected component of the intersection graph.

Let us extend somehow our partial ordering on the set of connected
components of the intersection graph to a complete one, and order
correspondingly both these $(n-2)$-dimensional submanifolds
$S^{n-2}_j$ in the space of generic curves of class $\K(\Theta,n)$,
and the cycles (\ref{basone}) in $\Sigma(\Theta,n)$, corresponding to
some collection of chords $\alpha_j$, chosen one from each
such connected component.

\begin{proposition}
\label{rea1}
For an appropriate orientation of submanifolds $S^{n-2}_j$,
the matrix of linking numbers of these submanifolds with
Borel--Moore cycles in $\Sigma(\Theta,n)$, corresponding to the elements
$\alpha_j$ of the group $H_1(\Theta,\mbox{\rm odd})$, is triangular
with ones on the diagonal, while all non-zero non-diagonal elements
lie only in the cells $(i,j)$, where $j$ is the number of a component
subordinate to the component with number $i$. \end{proposition}

{\em Proof.}
First we construct a relative Borel--Moore cycle in
$\K(\Theta,n)$ mod $\Sigma(\Theta,n)$,
whose boundary is equal to the cycle (\ref{basone}). It will be
a linear combination of $2m+1$ chains similar to ones shown in the
left-hand parts of the second, the third, and the fourth formulas
of Fig. \ref{ress}: they can be obtained from the summands
$1 \cdot \alpha, \dots , (2m+1) \cdot \alpha$ of the chain (\ref{basone}),
shown in the right-hand parts of these formulas, by appropriate
orienting of the broken lines participating in their notation.
Namely, if $p < q$ are the numbers of
endpoints of the chord $\alpha$, then in the first $p$ and last $2m+1-q$
of these summands the broken arrow should be oriented from the point of
the non-singular interval to the center of the chord $\alpha$ and
taken with coefficient $1$, while in the middle $q-p$ ones the arrow
should be oriented from the center of the chord $\alpha$ to the
non-singular interval and taken with coefficient $-1$. The formulas of
Figs. \ref{ress}, \ref{verrt}, \ref{coproj} imply immediately that the
cycle (\ref{basone}) is the boundary of this linear combination.

Let us calculate the intersection index of this relative cycle with the sphere
$S^{n-2}_j$, where $j$ is the number of the component, containing $\alpha$.
Let us draw the affine hyperplane in $\R^n$, passing through the point
$\varphi(\alpha)$ and orthogonal to the axis of the space $\R^n.$
Without loss of generality we can assume that the curve
$\varphi(\R^1)$ lies arbitrarily close to a fixed two-dimensional plane
containing this axis, and that all intersection points of
$\varphi(\R^1)$ with this hyperplane lie in this two-plane.
Then the desired intersection index is equal to the number of
such intersection points of our hyperplane and $\varphi(\R^1),$
lying in the ball, corresponding to the $j$-th component of the
intersection graph, and taken with the following signs.
The points, whose preimages in $\R^1$
lie between the endpoints of the chord
$\alpha$, are counted for with the sign $-$
(respectively, +) if at this point $\varphi$ intersects the hyperplane
from the left to the right (respectively, from the right to the left); for
points, whose preimages lie on one side of both endpoints of the chord
$\alpha$, the accounting of signs is opposite.

The obtained weighted number of intersection
points is equal to the intersection index of our hyperplane and the
infinite curve $\R^1 \to \R^n$, coinciding with
\begin{itemize}
\item[a)]
the axis of the space $\R^n$ outside our ball,
\item[b)]
the curve $\varphi(\R^1)$ inside this ball but outside
the image of the segment between the endpoints of the chord
$\alpha$, and
\item[c)]
the same curve traversed in the opposite direction in the
image of this segment.
\end{itemize}
(Indeed, calculating this intersection index we count
with the same signs all the same points of the curve
$\varphi(\R^1)$ plus, maybe, additionally twice but with opposite
signs the point $\varphi(\alpha)$ itself.) It is obvious
that this intersection index is equal to $1$. The assertion of
Proposition \ref{rea1} concerning the non-diagonal elements follows
immediately from the construction.
\hfill $\square$

Theorem \ref{realiz} follows immediately from this proposition.
\hfill $\square$

\begin{remark} \rm
In the case of even $n$ the similar construction generally does not
work because the even-dimensional sphere has no everywhere
non-zero vector fields. The unique case when this vector field is not
needed is the case of a planar diagram $\Theta$; see the proof of Lemma
\ref{lem4chet} and Theorem \ref{homeven}.
\end{remark}

\begin{thebibliography}{99}

\bibitem{ar70}
{\it Arnold V.I.}, On some topological invariants of algebraic functions,
Transact. (Trudy) Moscow Math. Society 1970, {\bf 21}, 27--46.

\bibitem{arnold}
{\it Arnold V.I.},
Plane curves, their invariants, perestroikas and classifications.
In: Singularities and
Bifurcations. Providence, RI: AMS,
1994. 33--91. (Adv. in Sov. Math., 21)

\bibitem{BN} {\it Bar-Natan, D.} On the Vassiliev knot invariants,
{\it Topology}, {\bf 34}, (1995) 423--472.

\bibitem{bw} {\it Birman J.} New points of view in knot theory.
{\it Bull. AMS (N.S.)}, {\bf 28}:3 (1993), 253--287.

\bibitem{CE} {\it Cairns G., Elton D.,} The planarity problem for
signed Gauss words, {\em Journal of Knot Theory and Its Ramifications},
{\bf 2}, No.4. (1993), pp. 359-367, and {\bf 5}, No.2. (1996),
pp. 137-144.

\bibitem{CDL} {\it Chmutov S.V., Duzhin S.V., Lando S.K.,}
Vassiliev knot invariants. II. Intersection graph conjecture for trees.
In: Singularities and
Bifurcations. Providence, RI: AMS,
1994. 127--134. (Adv. in Sov. Math., 21)

\bibitem{GPV} {\it Goussarov M., Polyak M., and Viro O.} Finite type
invariants of classical and virtual knots, {\it Topology} {\bf 39}:5 (2000),
1045--1068.

\bibitem{kauf} {\it Kauffman, L.H.}
Virtual Knot Theory, {\it European Journal of Combinatorics,}
{\bf 20}:7 (1999), 662--690.

\bibitem{LM} {\it Lov\'asz L., Marx M.,} A forbidden substructure
characterization of Gauss codes, {\it  Acta Sci.  Math.} (Szeged) 38 (1976),
no.  1--2, 115--119, short version: {\it Bull.  Amer. Math. Soc.} 82 (1976),
no.  1, 121--122.

\bibitem{vasya} {\it Manturov V.O.}
Proof of the V.A.Vassiliev's conjecture on the planarity of singular links,
{\it Izvestiya RAN}, 2005.

\bibitem{PV} {\it Polyak M. and Viro O.}
Gau\ss \ diagram formulas for Vassiliev invariants,
{\it Internat. Math. Res. Notes} {\bf 11} (1994), 445--453.

\bibitem{sar} {\it Sarkaria K.S.,} A one-dimensional Whitney trick and
Kuratowski's graph planarity criterion // {\it Israel J. Math.} 1991.
V. 73. P. 79--89.

\bibitem{stan} {\it Stanford T.,} Finite type invariants of knots, links,
and graphs. {\it Topology}, 1996, 35(4), 1027--1050.

\bibitem{tyurina} {\it Tyurina S.D.} (1999) On the Lannes and Viro--Polyak
type formulas for finite type invariants, {\it Mat. Zametki
(Math. Notes)} {\bf 66}, No.3--4, 525--530.

\bibitem{knsp} {\it Vassiliev V.A.} (1990)
Cohomology of knot spaces, in:
{\it Theory of Singularities and its Applications} (V.~I.~Arnold, ed.),
Advances in Soviet Math. Vol. 1, 23--69 (AMS, Providence, RI).

\bibitem{b94} {\it Vassiliev V.A.} (1994)
{\it Complements of discriminants of smooth maps: topology and
applications, Revised ed.},
Translations of Math. Monographs 98, AMS, Providence RI.

\bibitem{fasis} {\it Vassiliev V.A.} (1997)
{\it Topology of complements of discriminants,}
Phasis, M., 552 p. (In Russian)

\bibitem{kotor} {\it Vassiliev V.A.} (1999)
Topological order complexes and
resolutions of discriminant sets,
{\it Publications de l'Institut
Math\'ematique Belgrade, Nouvelle s\'erie} {\bf 66(80)}, 165--185.

\bibitem{tt} {\it Vassiliev V.A.} (2001)
On combinatorial formulas for cohomology of spaces of knots,
{\it Moscow Math. J.} {\bf 1}:1, 91--123.

\bibitem{bryden} {\it Vassiliev V.A.},
Combinatorial formulas for cohomology
of spaces of knots, in: Advances in Topological Quantum
Field Theory (J.~Bryden, ed.), Springer-Verlag, 2004,

\bibitem{combcomp} {\it Vassiliev V.A.}
Combinatorial computation of combinatorial formulas for knot invariants,
{\it Transact. (Trudy) of Moscow Math. Society, N 66 (2005), 3--94.

\end{thebibliography}

\bigskip
{\sc Steklov Mathematical Institute and Poncelet Laboratory
(UMI 2615 of CNRS and Independent University of Moscow)}

\end{document}
