\documentclass[11pt]{article}

\title{Thom polynomials for Lagrange, Legendre,\\ 
and critical point function singularities}

\author{Maxim Kazarian\thanks{Partially supported by 
the grants RFBR 98-01-00612  and
NWO-047.008.005}\\
Steklov Mathematical Institute\\
Independent Moscow University\\
\em kazarian@mccme.ru}
\date{}

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\begin{document}

\maketitle

\begin{abstract} 
We define Thom polynomials for Lagrange, Legendre and
critical point function singularities. Our approach is based on the
notion of classifying space of singularities. This approach provides a
universal method of computing Thom polynomials.  Characteristic
classes of complex Lagrange and Legendre singularities of small
codimensions are computed. These expressions reduced modulo 2 agree
with those obtained by Vassiliev for the real case. 
\end{abstract}




\section{Introduction}

The natural way to study the global properties of isolated hypersurface
singularities is to include the hypersurface into a generic family.
Formally this can be described as follows.
Consider a smooth embedded hypersurface in the total space of a
smooth locally trivial complex analytic fibration:
   $$\begin{array}{ccccc} 
  H&\hookrightarrow&W&\\
  &&\downarrow\lefteqn{\pi}\\ 
  &&B\end{array} $$ 
(The case when $\pi$ is the trivial bundle is already interesting enough.) 
We consider $H$ as a family of (possibly singular) hypersurfaces
$H_b\subset W_b$, $W_b=\pi^{-1}(b)$, $H_b=H\cap W_b$, $b\in B$. Let
$M\subset H$ be the union of all singular points of $H_b$'s.
Generically $M$ is smooth and has the codimension $n=\dim W-\dim B$ in
$H$. It can be identified with the zero locus of a certain section
of some vector bundle. Namely, the bundle is $\Hom(V, I)$, where $V\subset
TW$ is the subbundle of vectors tangent to the fibres of
$\pi$ and $I$ is the normal bundle of $H$. The section is given by the
natural projection $V\subset T_HW\to T_HW/TH=I$. The
genericity condition means that this section is transversal to the
zero section of the same bundle. This condition is open but in
complex situation it is not necessary dense. Similarly below
by genericity for some smooth map we mean the transversality of its
jet extension to a certain stratification on the jet space.

Let $\O$
be any class of isolated hypersurface singularities (an algebraic 
subvariety in some jet space of function germs $\C^n,0\to\C,0$ which
is invariant with respect to the group of left-right changes of
variables). Define the locus $\O(H)\subset M$ as the locus
consisting of the points at which the hypersurface $H_b$
belongs to the given singularity class $\O$. According to the general
principle of Thom the cohomology class Poincar\'e dual to the locus
$\O(H)$ is independent on $H$ (provided $H$ is generic) and can be
expressed as a universal polynomial in Chern classes of $W,B,H$. We
claim that this polynomial can be expressed in terms of some particular
combinations of these classes. Namely, let $u=c_1(I)=c_1(TW-TH)$ be
the restriction to $M$ of the class of the divisor $H$. Denote
$c_i=c_i(V)=c_i(TW-\pi^*TB)$, and define classes 
$a_i=c_i(V^*{\otimes} I-V)$ as homogeneous components in the expansion 
\begin{equation}
1+a_1+a_2+\dots=
\frac{(1+u)^n-(1+u)^{n-1}c_1+(1+u)^{n-2}c_2-\dots\pm c_n}
{1+c_1+c_2+\dots+ c_n}.
\label{atoc}
\end{equation} 
These classes satisfy relations 
\begin{equation}
(1+a_1+a_2+\dots)
\(1-\frac{a_1}{1+u}+\frac{a_2}{(1+u)^2}-\dots\)=1,
\end{equation}
(following from the identity $U+U^*{\otimes} I=0$, where $U$ is the formal
difference $U=V^*{\otimes} I-V$). These relations allow to expand the
squares of classes $a_i$ and hence any polynomial in
$u,a_1,a_2,\dots$ can be expressed as a linear combination of
monomials $u^{i_0}a_1^{i_1}a_2^{i_2}\dots$ with $i_0\ge0$,
$i_k\in\{0,1\}$ $(k>0)$.

\th Theorem 1. {For any isolated hypersurface singularity class $\O$ the
cohomology class in $H^*(M)$ Poincar\'e dual to the locus $\O(H)$
can be expressed as a universal polynomial $P_\O$ in
$u,a_1,a_2,\dots$. This polynomial \(called {\em Thom polynomial}\)
is independent on $n$ \(we use the same letter $\O$ for the
class of function germs $\C^{n'},0\to\C,0$, $n'\ne n$, stably
equivalent to the functions from $\O$\).

 For the singularity classes of codimension not greater
than $6$ the Thom polynomials are represented in
Table~{\rm\ref{thompol}}.}
\begin{table}[h]
\caption{Thom polynomials of isolated hypersurface singularities of
$\codim\le6$}\label{thompol}
\small
$$\begin{array}{@{}ccl@{}}
%\begin{eqnarray*}
A_2&=&{\bf a_1}\\
A_3&=&{\bf 3a_2}+ua_1\\
A_4&=&{\bf 3a_1a_2+6a_3}+4ua_2+u^2a_1\\
D_4&=&{\bf a_1a_2-2a_3}-ua_2\\
A_5&=&{\bf 27a_1a_3+6a_4}+u(16a_1a_2-12a_3)-4u^2a_2+u^3a_1\\
D_5&=&{\bf 6a_1a_3-12a_4}+u(4a_1a_2-14a_43-4u^2a_2\\
A_6&=&{\bf 87a_2a_3+54a_1a_4+78a_5}+\\ &&\hfill u(127a_1a_3-53a_4)+
   u^2(59a_1a_2-126a_3)-41u^3a_2+u^4a_1\\
D_6&=&{\bf 12a_2a_3-24a_5}+u(14a_1a_3-40a_4)+
   u^2(8a_1a_2-30a_3)-8u^3a_2\\
E_6&=&{\bf 9a_2a_3-12a_1a_4+6a_5}+3ua_4+u^2(3a_1a_2-6a_3)-3u^3a_2\\
A_7&=&{\bf 135a_1a_2a_3+465a_2a_4+264a_1a_5+522a_6}+
   u(516a_2a_3-16a_1a_4+485a_5)+%\kern-1em
   \\ &&\hfill 
   u^2(305a_1a_3-70a_4)+u^3(190a_1a_2-440a_3)-165u^4a_2+u^5a_1\\
D_7&=&{\bf 24a_1a_2a_3-24a_2a_4+48a_1a_5-144a_6}+
   u(8a_2a_3+44a_1a_4-224a_5)+\\ &&\hfill 
   u^2(48a_1a_3-172a_4)+u^3(20a_1a_2-88a_3)-20u^4a_2\\
E_7&=&{\bf 9a_1a_2a_3+6a_2a_4-42a_1a_5+36a_6}+
   u(21a_2a_3-61a_1a_4+80a_5)+\\\samepage &&\hfill 
   u^2(43a_4-6a_1a_3)+u^3(7a_1a_2-8a_3)-7u^4a_2\\
P_8&=&{\bf a_1a_2a_3-6a_2a_4+6a_1a_5-4a_6}+\\ &&\hfill 
   u(7a_1a_4-4a_2a_3-10a_5)+
   u^2(2a_1a_3-8a_4)-2u^3a_3
\end{array}$$
%\end{eqnarray*}}
\end{table}

To determine the cohomology class dual to the locus $\O(H)$ in $H$ or
in $W$ we apply the push-forward formula $i_*(i^*a\;b)=a\;i_*(b)$ to
the embeddings $M\stackrel i\hookrightarrow H\stackrel
j\hookrightarrow W$ and the classes $a=P_\O$, $b=1$.  For instance,
since $i_*(1)=[M]=c_n(\Hom(V,I))=u^n-u^{n-1}c_1+\dots\pm c_n$, we get
that the dual of $\O(H)$ considered as a locus in $H$ is equal to 
$$[\O(H)]=(u^n-u^{n-1}c_1+\dots\pm c_n)\;P_\O
\in H^*(H).$$
Similarly, the homomorphism $j_*:H^*(H)\to H^*(W)$ on the class above 
is given by the multiplication by $u$.



\medskip
To prove Theorem~1 we relate the problem to the theory of Lagrange and
Legendre singularities and their characteristic classes. Namely, we
consider the hypersurface $H$ as the `generating family' for the
Legendre immersion $M\to PT^*B$. 

In the simplest case when the bundle $I$ is trivial (and hence $u=0$)
the problem is reduced to the study of Lagrange singularities.
Lagrange singularities are those of the projection of a Lagrange
submanifold to the base of the cotangent bundle. Singularity loci of
this projection could define cohomology classes on this manifold. The
simplest example is Arnold-Maslov class which is dual to the total
critical set of the projection. The theory of characteristic classes
related to the {\em real\/} Lagrange singularities was developed by
V.Vassiliev. In his book~\cite{V} a cochain complex (so called
Vassiliev universal complex of singularity classes) was constructed
whose generators correspond to the singularity classes. The
cohomology groups of this complex are well defined characteristic
classes. Vassiliev has computed the cohomology of this complex in the
codimension not exceeded $6$ and found the expressions for all these
classes (except $A_7$) in terms of Stiefel-Whitney classes.

In the paper~\cite{K2} we suggested an approach to this problem
based on the study of classifying space of Lagrange singularities.
This has led to understanding the geometrical meaning of Vassiliev
complex and to introducing new characteristic classes. In this paper
we develop this approach. In particular, we complete computing
characteristic classes dual to singularity classes in terms of
multiplicative generators the ring of characteristic classes (i.e.\
Thom polynomials) and describe also the complex version of the
theory.

Any classification problem in singularity theory can be considered as
a problem of classifying orbits for an action of some Lie group $G$ on some
vector space $V$. For instance, for left-right equivalence of maps
$V$ is the space of map germs $(\R^n,0)\to(\R^m,0)$ (or jets of maps
of fixed order) and $G$ is the group of the left-right changes which
is homotopy equivalent to ${\rm GL}(n,\R)\times{\rm GL}(m,\R)$. The
theory of characteristic classes of this classification problem is
therefore the theory of characteristic classes of the given Lie group $G$.
The {\em classifying space} $BV$ of this classification problem is
defined using standard Borel's construction, $BV=V\times_G BE=(V\times
BE)/G$, where $BE\to BG$ is the classifying principle $G$-bundle. The
classifying space $BV$ is homotopy equivalent to $BG$ (since the
bundle $BV\to BG$ has contractible fibres). Any invariant algebraic
subset $\S\subset V$ gives rise to a subset $B\S=\S\times_G BE\subset
V$ of the same codimension. With this approach the `Theorem about
existence of Thom polynomials' is evident; Thom polynomial of $\S\subset
V$ is just the element represented by the fundamental cycle of $\S$
in the equivariant cohomology group $H^*_G(V)\cong H^*_G(\pt)\cong
H^*(BG)$, or, which is equivalent, the element, represented by the
fundamental cycle of $B\S$ in the usual cohomology group
$H^*(BV)\cong H^*(BG)$.

The classifying spaces $BG$ and $BV$ have infinite dimensions, but
they always have very nice finite dimensional approximations that can
be used as well for `stable' problems, where the maps of manifolds of
different dimensions are considered. For example, for the theory of
singularities of $(\R^*,0)\to (\R^{*+k},0)$ the classifying space is
the space of germs (or jets of high order) of $n$-manifolds in
$(\R^N,0)$, $N\gg n\gg0$. This space is homotopy equivalent to the
usual Grassmannian $G_{n,N}$ and stratified according to
singularities of the projection to the fixed coordinate
$(n+k)$-subspace. In a similar way, for the classifying space of
Lagrange singularities one can take the space of all germs
(or jets of high order) of Lagrange submanifolds in the symplectic
space $(\R^{2n},0)$, $n\gg0$. The same constructions can be used for
studying complex spaces and holomorphic maps.

Note that if $\S\subset V$ is an orbit then $B\S=\S\times_G EG\cong
BG_\S$ is the classifying space of the `symmetry group' $G_\S$ of
the singularity $\S$ (the stationary subgroup of any point $x\in\S$). A
similar description of $B\S$ exists even if $\S$ consists of many
orbits.

In~\cite{SR} Sz\"ucs and Rim\'anyi used an alternative
approach to the definition of the classifying space of singularities
based on Sz\"ucs's idea of gluing the classifying spaces of symmetry
groups of singularities. They considered only simple singularities,
and very clear topology of the classifying space does not follow from
their construction. It should be noticed nevertheless that their
construction works as well for the case of multisingularities, 
see~\cite{Sz,R,R1}
for some applications. It is an interesting problem to
find an {\em a priori} construction for the classifying space of
multisingularities and to describe its topology (the work~\cite{SR}
implies that it should be related to cobordism theory).


The group of Lagrange characteristic classes is the cohomology group
of Lagrange Grassmannian $\L^\R$ (or its complex analogue $\L^\C$). The
cohomological information about adjacencies of singularities is
translated into properties of the spectral sequence constructed by
the filtration on the classifying space by the codimension of
singularities~\cite{K1}. Let us describe this spectral sequence for
the classification of (complex) Lagrange singularities~\cite{K2}. 
The initial term of this sequence is $E^{*,*}_2=\bigoplus_\S
H^*(BG_\S)$, where $G_\S$ is the symmetry group of the singularity $\S$.
The symmetry groups of all singularities of small codimensions are
well known. They are all finite (some extension of $S_3$ for the
singularity $D_4$ and cyclic groups for all other singularities). In
fact, Vassiliev proved recently that the symmetry group of any
critical point singularity of finite multiplicity is finite, see the
Russian translation of~\cite{V}. This gives the complete description
of the groups $E^{p,q}_2\cong E^{p,q}_\infty$ for small $p$. They are
all torsion groups for $q>0$ and the free generators of the raw
$E^{*,0}_2$ correspond to the fundamental cycles of singularity
classes. Hence we immediately arrive without any calculation to the
following conclusion.

\th Theorem. The classes of complex Lagrange singularities
$A_2,A_3,\dots,E_7,P_8$ form a basis in the group of Lagrange
characteristic classes $H^{\le12}(\L^\C,\Q)$. In case of integer
coefficients these classes generate freely subgroups of finite indices
$1,3,12,360,\dots$ respectively in $H^{2d}(\L^\C,\Z)$, $d\le 6$.

Of course, this result follows also from the explicit form of these
classes represented in Table~\ref{thompol} (one should set $u=0$; the
corresponding terms are marked in boldface).
The presents of even numbers in the sequence $1,3,12,360,...$ implies that
in case of real singularities there are some relations between these
classes. These relations have been found by Vassiliev
in~\cite{V}: $A_4-D_4=D_5=D_6=A_6-E_6=D_7=E_7-P_8=0\pmod2$.


We describe two methods for computing Thom polynomials, both based
on the concept of the classifying space. The idea of the first method is
the following. Consider a cellular partition of the classifying space
such that both Schubert cycles and singularity classes are some
combinations of cells. Then the problem is reduced to linear algebra
in some finitely generated cochain complex. To obtain such a partition
one can consider a classification of critical point singularities with
respect to a smaller group of equivalence consisting of diffeomorphism
germs with identical linear terms. Such classification has no simple
singularities but the number of modules is always finite. The main
property of this classification is that the symmetry group of every its
singularity class is trivial, and hence, the induced partition of
the classifying space is a cellular partition.

A realization of the program above is possible though it requires a
great deal of computations. To reduce the amount of computations we
use as a kind of compromise another detailed classification of
singularities with smaller but not trivial symmetry groups. The
corresponding classes are called {\em marked\/} singularity classes.
A number of relations between these marked singularities is sufficient to
reduce the problem of finding Thom polynomials to those singularity
classes for which this problem can be solved by classical methods
like resolutions of singularities. To describe these relations we
introduce the notion of {\em adjacency exponent} for a pair of
singularities of (complex) neighbour codimensions which is an analogue
of the incidence coefficient in real case, see~\cite{V}.

For the simplicity we present the computations of Thom polynomials for
the classes of function singularities. All steps of our computations
can easily be reformulated in terms of Lagrange (or Legendre)
singularities.

Another method of finding Thom polynomials uses basically the idea
of R.Rim\'anyi. Any example when both Chern classes and the class
represented by singularity locus can be computed produces some relations
between the coefficients of Thom polynomial. When the number of
computed examples is high enough these relations could be sufficient
to determine Thom polynomial completely. Rim\'anyi noticed that a
lot of examples can be produced by considering tubular neighbourhoods
of singularity loci in the classifying space. To see when this could
give the result let us look again at the characteristic spectral
sequence converging to the group of characteristic classes. Its
second term is $E^{*,*}_2=\bigoplus_\S H^*(BG_\S)$. In complex case
all topology is often concentrated in even dimensions and the spectral
sequence converges in the second term. This means that any
characteristic class is completely determined by the collection of
its images in the groups $H^*(BG_\S)$. In fact, these images belong not
to the group $H^*(BG_\S)=H^*(B\S)$ itself but to the (isomorphic to
it) cohomology group $H^*(T\S)$ of Thom space $T\S$ of the normal
bundle to the submanifold $B\S$ of the classifying space of
singularities. Therefore, to apply Rim\'anyi's method we need that the
homomorphisms $\f:H^*(T\S)\to H^*(B\S)$ induced by inclusion of the zero
section of the normal bundle to its Thom space would be monomorphic.

If $G_\S$ is trivial then the homomorphism $\f$ (given by the
multiplication by the top Chern class of the bundle) is trivial. In a
similar way $\f$ is trivial if $G_\S$ is finite and we consider
the cohomology with coefficients in a field of characteristic zero.
Therefore {\em Rim\'anyi's method of finding Thom polynomials can be
applied only when every singularity has a continuous group of
symmetry}. This is not the case for Lagrange singularities.
Nevertheless this is true in case of {\em Legendre} or {\em twisted
Lagrange} singularities. In the classification of Legendre singularities
the right equivalence of functions is replaced by $V$- (or $K$-
depending on the terminology) classification when one allows to
multiply a function by another nowhere vanishing function. In this
classification any quasihomogeneous singularity has an obvious
$U(1)$-symmetry. Thus the Rim\'anyi's method gives Thom polynomials of
all Legendre singularities in small codimensions and as a particular
case Thom polynomials of Lagrange singularities.

Reducing the coefficients modulo two we get real Legendre
characteristic classes (Thom polynomials for them were not
computed in~\cite{V}).


The paper is organised as follows. In Section~2 we present two
independent methods of computing Thom polynomials of
Table~\ref{thompol}. Their existence is proved in Section~3 where the
characteristic classes of Lagrange and Legendre singularities are
studied.

I appreciate the hospitality of the I.~Newton Institute, Cambridge, where
the work on the paper was completed.


\section{Resolutions and adjacencies of function singularities}

In this section we compute the Thom polynomials listed in Theorem~1
for the case of bundle map problem considered here.  The proof of
their existence is postponed until Section~\ref{LegCl}. Everywhere in this
section we use the same notations for the singularity loci and for the
cohomology classes represented by these loci.

\subsection{Bundle map problem}\label{bmap}

Before computing Thom polynomials of Theorem~1 we formulate a 
slightly different but, in fact, an equivalent problem. 
Consider two complex vector bundles
$V,I$ of ranks $\rank V=n$, $\rank I=1$ over a smooth base $M$.
We do not assume any complex structure on the base $M$. Consider a
smooth bundle map 
  $$\begin{array}{ccc}V&\too f& I\\
   &\hbox to 0pt{\hss$\searrow\quad\swarrow$\hss}\\
   &M\end{array}$$ 
whose restriction $f_w:V_w\to I_w$, $w\in M$ to each fibre is a
complex polynomial of some fixed degree $N\gg0$ with a critical point
at the origin. We may think of $f$ as a generic section
$f=f_{(2)}+f_{(3)}+\dots$ of the vector bundle
$S^2V^*{\otimes}I\oplus S^3V^*{\otimes}I\oplus\cdots$. With any
function singularity class $\O$ we associate the locus $\O(f)\subset
M$ consisting of the points $w\in M$ such that the polynomial $f_w$ has
the prescribed singularity type $\O$ at the origin. Theorem~1 is
a particular cases of the following more general one.

\th Theorem~2. For any generic $f$ the cohomology class Poincar\'e
dual to the locus $\O(f)$ can be expressed as a universal polynomial
$P_\O$ in classes $u=c_1(I)$ and $a_i=c_i(V^*{\otimes} I-V)$. For
singularity classes of codimension $\le6$ these polynomials are those
listed in Theorem~{\rm1}.

\spa{Proof~of~Theorem~1}. 
Let as explain how the problem of studying the hypersurface
singularities discussed at the introduction can be reduced to a
bundle map problem considered here. Let the diagram $H\subset W
\stackrel\pi\to B$  and the critical set $M\subset H$ be as in
the introduction. Assume that the divisor $H$ is given locally by the
equation $g=0$ where $g$ is some function given in a neighbourhood of
the given point $w\in M$. Denote by $f_w$ the $N$-jet of the
restriction of $g$ to the fibre of $\pi$ through $w$. These function jets
$f_w$ form together a section in the bundle $J$ whose structure group
is the Lie group formed by the $N$-jets of right changes in $\C^n,0$ and
multiplications by the $N$-jets of non-vanishing functions. 
This group is contractible to the subgroup
$\GL(n,\C)\times\GL(1,\C)$ of linear changes. Hence, the structure
group of $J$ can be reduced to this subgroup. The possibility of such
reduction means the possibility of introducing the complex linear
structures `up to the order $N$' on the germs $W_w$,
$C^\infty$-smoothly depending on the point $w\in M$. 
After such reduction the
$N$-jets $f_w$ can be considered
as polynomial maps $f_w:V_w\to I_w$, where $V_w=T_wW_w$,
$I_w=T_wW/T_wH$. By Tugeron's finite determinacy theorem the singularity
type of the hypersurface $H_w$ is that of the function germ $f_w$.
Therefore the partition on $M$ by different singularity loci of $H_w$
coincides with that of the polynomial bundle map $f:V\to I$.\QED




\subsection{The Gysin homomorphism}\label{gysin}

Some part of Thom polynomials of Theorem~1 is computed using the
method of resolutions of singularities. The computations of this
method use a formula for the Gysin homomorphism proved in~\cite{K3}. 
Consider vector bundles
$V,I\to M$ of ranks $n,1$ respectively and a {\em quadratic\/}
bundle map $f_{(2)}:V\to I$. The map $f_{(2)}$ is considered as a
section of the bundle $S^2V^*\otimes I$ or as a linear self-adjoined
bundle map $f_{(2)}:V\to V^*{\otimes} I$. Denote by $\F_r=\F_r(V)$,
$r\le n$ the flag bundle over $M$ whose total space  is formed by all
possible flags
 $$F^1_w\subset F^2_w\subset\dots F^r_w\subset V_w,
           \qquad\dim F^i_w=i,\quad w\in M,$$
in the fibres of $V$. The cohomology ring of $\F_r$ is generated by
the cohomology of $M$ and by the classes
$t_i=-c_1(F^i/F^{i-1})=c_1((F^i/F^{i-1})^*)$, where $F^i$ are the
corresponding tautological vector bundles. Denote by $Z_r\subset \F_r$ the
locus defined by the condition $F^r_w\subset \ker f_{(2)w}$, $w\in M$.
Generically $Z_r$ is smooth and $\dim_\C Z_r=\dim_\C M-r$. We study the
Gysin (or push-forward or transfer) homomorphism 
      $$p_{r*}:H^*(Z_r)\to H^{*+2r}(M)$$ 
corresponding to the natural projection $p_r:Z_r\to M$.

\th Theorem \rm(\cite{K3}). Assume that the bundle $I$ is trivial. 
Then for any monomial $t_1^{s_1}\cdots t_r^{s_r}$ we have
 $$p_{r*}(t_1^{s_1}\cdots t_r^{s_r})=\s_{s_1+1,\dots,s_r+1}\in
 H^{2\sum(s_i+1)}(M),$$
where $\s_{\l_1,\dots,\l_r}=\s_{\l_1,\dots,\l_r}(a_1,a_2,\dots)$ are 
polynomials in classes $a_k=c_k(V^*-V)$ defined as follows:
\begin{itemize}
\item
for $r=1$ we have $\s_k=a_k$;
\item
for $r=2$ we set
 $$\s_{k,l}=-\s_{l,k}=\sum_{i=1}^\infty(-1)^i 
 \left|\begin{array}{cc} a_{k+i}& a_{k-i}\\
 a_{l+i}&a_{l-i}\end{array}\right|;$$
\item 
for any even $r\ge4$ we set
$$\s_{\l_1,\dots,\l_r}=\Pf|\s_{\l_i,\l_j}|_{1\le i,j\le r};$$
\item
for any odd $r\ge3$ we set
$$\s_{\l_1,\dots,\l_r}=\sum_{k=1}^r(-1)^{k-1}a_{\l_k}
\s_{\l_1,\dots,\l_{k-1},\l_{k+1},\dots\l_r}.$$
\end{itemize}

Here $\Pf$ is the Pfaffians of a skew-symmetric matrix. Remark that in
the infinite sum for $\s_{k,l}$ only first $\max(k,l)$ terms may be
different from zero.

The computation of the homomorphism $p_{r*}$ in generic case can be
reduced to the case considered above.

\th Corollary. The formula of previous Theorem holds true in the
generic case if it is applied to the classes $\hat
t_i=-c_1(F^i/F^{i-1})+u/2$ instead of $t_i$ and to the classes $\hat
a_k$ instead of $a_k$, where $u=c_1(I)$ and
 $$1+\hat a_1+\hat a_2+\dots=
 \frac{1-\frac{c_1}{1+u/2}+\frac{c_2}{(1+u/2)^2}-\dots}
 {1+\frac{c_1}{1-u/2}+\frac{c_2}{(1-u/2)^2}+\dots},\quad c_i=c_i(V).$$
 
\spa{Proof}. We use the following trick borrowed from \cite{HT}. Consider
first the case when $I=J^{\otimes2}$, where $J$ is another line bundle
with $c_1(J)=c_1(I)/2=u/2$.
Then $S^2V^*{\otimes} I=S^2(V{\otimes} J^*)^*$ so $f_{(2)}$ can be treated
as a self-adjoined bundle map $V\otimes J^*\to(V\otimes J^*)^*$ and
we can apply previous Theorem to the classes $\hat
t_i=-c_1((F^i/F^{i-1}){\otimes} J^*)=-c_1(F^i/F^{i-1})+u/2$ and
$\hat a_k=c_k((V{\otimes} J^*)^*-V{\otimes} J^*)$. The formulas obtained in
this way can be applied to any line bundle $I$ since they are
universal, cf.~\cite{HT,K3}.

\ex Remark. The classes $\hat a_i$ are related to the integer 
classes $a_i=c_i(V^*-V{\otimes} I)$ via
  $$1+\hat a_1+\hat a_2+\dots=
  1+\frac{a_1}{1-u/2}+\frac{a_2}{(1-u/2)^2}+\dots.$$
The formula of Corollary above uses the division by powers of
$2$. Of course the direct image under $p_{r*}$ of any monomial in
$t_i=c_1(F_i/F_{i-1})$ is an integer class and so it can be expressed 
as a universal polynomial with integer coefficients in
$u,a_1,a_2,\dots$. The Corollary allows to find these polynomials 
for any particular monomial but these expressions have no nice closed form
similar to that in case $I=\C$.

\ex Example. The image $p_1(Z_1)\subset M$ is the closure of the locus
$A_2(f)$. Therefore,
 $$A_2=p_{1*}(1)=\s_1=a_1.$$
 This is the first formula of Table~\ref{thompol}. 



\subsection{Marked singularities and their resolutions}

In this section we compute the Thom polynomials for the classes 
$A_2$, $A_3$, $D_4$, $D_5$, $E_6$, $P_8$ using their resolutions. The
other classes from the list of Theorem~1 have no good resolutions
but the partial resolution of these classes simplifies the
computation of Thom polynomials for them as well.

Let $f:V\to I$ be a polynomial bundle map such that the restrictions
$f_w:V_w\to I_w$ have a critical point at the origin for any $w\in M$
as in Section~\ref{bmap}.  The second differential $f_{(2)w}=d^2f_w:V_w\to
I_w$ is a well defined at any point $w\in M$. We
consider $f_{(2)}$ as a twisted self-adjoined map of vector 
bundles $f_{(2)}:V\to V^*{\otimes} I$ on $M$. 

Let $\F_1=P(V)$ be the projectivisation of the vector bundle $V\to M$.
Consider the submanifold $Z_1\subset \F_1$ formed by the pairs of the form
(a point $w\in M$, a line $l\subset K_w$, where $K_w=\ker f_{(2)w}$ is
the kernel of
the second differential of $f_w$. We would like to represent the
cycles of fibre singularities of $f$ as the direct images of certain
cycles on $Z_1$ under the natural projection
  $$p_1:Z_1\to M.$$

\ex Definition. Let $g:\C^n,0\to\C,0$ be a germ of a critical point
singularity and $K=\ker d^2g\subset T_0\C^n$ be the kernel of its
second differential. The line $l\subset K$ is called {\em
distinguished\/} if the cubic form $g_{(3)}$ given by the third-order
terms in the Taylor expansion of $g$ vanishes on $l$. A {\em marked
critical point singularity\/} is a pair $(g,l)$, where $g$ is a
function germ and $l$ is a distinguished direction. 

Remark that the cubic form $g_{(3)}$ is well defined on $K$ (in a sense
that it is independent on the choice of coordinates on $\C^n,0$).
Consider the classification of marked critical point singularities. 

If $g$ has a singularity of type $A_k$, $k\ge3$, or $E_k$, $k=6,7,8$,
then the distinguished direction is unique. The corresponding marked
singularity is denoted by $\At_k$, $\Et_k$ respectively.

In case of singularity $D_k$, the distinguished direction may be
either a simple or a double zero of the cubic form $g_{(3)}|_K$. We
denote the two cases by $\Dt_k'$ and $\Dt_k''$
respectively. So for $k=4$ there are $3$ directions of $\Dt_4'$-type and
for $k>4$ there are two distinguished directions, one $\Dt_k'$ and one
$\Dt_k''$.

In case of singularity $P_8$ the distinguished directions form a
cubic curve in $\C P^2=P(K)$. We use the notation $\Pt_8'$ if this is
a generic point of the cubic and $\Pt_8''$ if this is one of the
$9$ inflection points on it.


We get the following classification of marked function
singularities.

$$
\begin{array}{lc}
&\codim\\
\At_3                              &1\\
\At_4,\Dt_4'                       &2\\
\At_5,\Dt_5',\Dt_5''               &3\\
\At_6,\Dt_6',\Dt_6'',\Et_6,\Pt_8' &4\\
\At_7,\Dt_7',\Dt_7'',\Et_7,\Pt_8''&5\\
\qquad\dots
\end{array}
$$

The codimension is counted in the space of function germs
$g:\C^n,0\to\C,0$ such that $dg(0)=0$ and $\ker d^2g$ contains some
fixed direction $l_0\subset T_0\C^n$.

The classification of marked singularities produces the corresponding
classification of the points on the manifold $Z_1$.
Namely, we say that the point $(w,l)\in Z_1\subset P(V)$ has marked
singularity $\widetilde\O$ if the function germ $f_w$ has
singularity type $\O$ and the line $l\subset \ker f_{(2)w}$ is
distinguished.

\th Lemma. The cohomology classes on $M$ dual to the singularity loci
of codimension $\le6$ can be determined as the direct images of the
following cycles on $Z_1$: 
\begin{eqnarray*}
 A_k&=&p_{1*}\At_k,\quad (k\ge3);\\
 D_4&=&\frac13p_{1*}\Dt_4';\qquad 
 D_k\;=\;p_{1*}\Dt_k'\;=\;p_{1*}\Dt_k'',\quad (k\ge5);\\
 E_k&=&p_{1*}\Et_k,\quad(k=6,7,8);\\
 P_8&=&\frac19p_{1*}\Pt_8''\;=\;\frac13p_{1*}t_1\Pt_8'.
\end{eqnarray*}
 
 \spa{Proof}.  Just note that every point $w\in M$ such that the
singularity type of $f_w$ is $A_k$, $E_k$, or $D_{>4}$ has a unique
preimage in the corresponding cycle on $Z_1$. For the singularity
$D_4$ there are $3$ such preimages (corresponding to three zero lines
of $f_{(3)w}$ on $K_w$) and for the singularity $P_8$
there are $9$ preimages on $\Pt_8''$ (corresponding to $9$
inflection points of the cubic $f_{(3)w}=0$ on $P(K)$). The
restriction of $t_1=-c_1(F^1)$ to any fibre of $P(V)\to M$ is the
class of a hyperplane. Hence the three
preimages of a generic point $w\in P_8$ on a cycle representing $t_1 \Pt_8'$ 
correspond to $3$ points of the intersection of the cubic $f_{(3)w}=0$
with a generic projective line in $P(K_w)$.\QED

Since the homomorphism $p_{1*}$ is known by results of
Section~\ref{gysin}, 
the last lemma reduces the problem to the computation of cohomology
classes represented by certain cycles on $Z_1$.

\ex Example. The third differential $d^3f_w=f_{(3)w}:F^1\to I$ is a
section of the line bundle $(F^1)^{*\otimes3} \otimes I$ on $Z_1$, 
where $F_1$ 
is the tautological line bundle on $Z_1\subset P(V)$. 
The closure of the cycle $\At_3$ is the zero locus of this section.  
Therefore, $\At_3=c_1((F^1)^{*\otimes3} \otimes I)=3t_1+u$ and
 $$A_3=p_{1*}(3t_1+u),\qquad t_1=-c_1(F_1), \quad u=c_1(I).$$
which gives after applying the formulas of Section~\ref{gysin} 
the Thom polynomial for $A_3$.
 
Similar arguments can be used for some other marked singularity
classes if we construct resolutions with help of flags instead of
projective spaces. The following Lemma uses notations of
Section~\ref{gysin}.

\th Lemma. The following relations hold for the Gysin homomorphism
$p_{1*}:H^*(Z_1)\to M^{*+2}$.
\begin{equation}
\begin{array}{rcl}
p_{1*}(\At_3)&=&p_{1*}(3t_1+u)\\
p_{1*}(\Dt'_4)&=&p_{2*}(3t_1+u)\\
p_{1*}(\Dt''_5)&=&p_{2*}((3t_1+u)(2t_1+t_2+u))\\
p_{1*}(\Dt'_5)&=&p_{2*}((3t_1+u)2(t_1+2t_2+u))\\
p_{1*}(\Et_6)&=&p_{2*}((3t_1+u)(2t_1+t_2+u)(t_1+2t_2+u))\\
p_{1*}(\Pt'_8)&=&p_{3*}((3t_1+u)(2t_1+t_2+u))\\
p_{1*}(\Pt''_8)&=&p_{3*}((3t_1+u)(2t_1+t_2+u)(t_1+2t_2+u))
\end{array}
\label{gys}
\end{equation}

Applying formulas for the homomorphisms $p_{r*}:H^*(Z_r)\to M$
described in Section~\ref{gysin} we complete the computations
of the Thom polynomials for singularities $A_3,D_4,D_5,E_6,P_8$.

\spa{Proof}. The case of $\At_3$ is considered above. In the case of
singularities $\Dt'_4,\dots, \Et_6$ the kernel $K_w=\ker f_{(2)w}$ has
dimension $2$. We can associate with any of these cycles the
corresponding cycles $\widetilde\Dt'_4,\dots,\widetilde\Et_6\subset
Z_2\subset\F_2$ by letting $F^2_w$ to be the kernel $K_w$. Then we get
$p_{1*}\Dt'_4=p_{2*}\widetilde\Dt'_4$ and the same for
$\Dt'_5,\Dt''_5,\Et_6$. The third differential
$f_{(3)w}$ can be written as
  $$f_{(3)w}=a_0 x^3+a_1x^2y+a_2xy^2+a_3y^3$$
with an appropriate choice of coordinates $(x,y)$ on $F^2$ such that
the line $F^1$ is given by $y=0$. The coefficients $a_i$ are
globally defined sections of the bundles $(F^1)^{*\otimes(3-i)}\otimes
(F^2/F^1)^{*\otimes i}\otimes I$ with the first Chern classes
$(3-i)t_1+it_2+u$, $i=0,1,2,3,4$ (every section in this
sequence is defined only on the zero locus of the previous one). The
closures of the considered cycles are given by the equations
\begin{eqnarray*}
\widetilde\Dt'_4:  &&a_0=0,\\
\widetilde\Dt''_5: &&a_0=a_1=0,\\
\widetilde\Et_6:   &&a_0=a_1=a_2=0,\\
\widetilde\Dt'_5:  &&a_0=a_2^2-4a_1a_3=0\
\end{eqnarray*}
respectively which gives the formulas of Lemma (the expression
$a_2^2-4a_1a_3$ is a well defined section of $((F^1)^*\otimes
(F^2/F^1)^{*\otimes2}\otimes I)^{\otimes 2}$
on the zero locus of $a_1$).

The singularities $\Pt'_8,\Pt''_8$ are resolved in a similar way using
flags $F^1_w\subset F^2_w\subset F^3_w$, where $F^3_w=K_w$, $PF^2_w$ is the tangent
line to the cubic $f_{(3)w}|{F^3_w}=0$. The tangency condition is
equivalent to the condition that $f_{(3)w}|{F^2_w}$ has a double zero on
$F^1_w$ and the condition that the tangent line is a point of
inflection is equivalent to the condition that $f_{(3)w}|{F^2_w}$ has a
triple zero on $F^1_w$. Arguing as above we arrive to the expressions
for the direct images of $\widetilde\Pt'_8,\widetilde\Pt''_8$.\QED
 
 
\subsection{Adjacency exponents}\label{adjacency}

 The Thom polynomials for classes not considered in previous Section
 are computed using relations between them. These relations are
 applied not to the classes themselves but to the classes of their
 partial resolutions.
 
 \th Lemma (basic relations). The cohomology classes of cycles 
 $\At_3,\At_4,\dots \Pt_8''\subset
 Z_1$ are subject to the following relations.
 \begin{equation}
 \begin{array}{rcl}
   (4t_1+u)\;\At_3  &=& \At_4   -\Dt_4' \\
   (5t_1+u)\;\At_4  &=& \At_5  -2\Dt_5''\\
  (6t_1+2u)\;\Dt_4' &=&-\Dt_5' +4\Dt_5''\\
   (6t_1+u)\;\At_5  &=& \At_6  -2\Dt_6''-3\Et_6 \\
  (8t_1+3u)\;\Dt_5' &=&-\Dt_6'+12\Et_6  +4\Pt_8'\\
   (4t_1+u)\;\Dt_5''&=& \Dt_6'' -\Pt_8' \\
   (7t_1+u)\;\At_6  &=& \At_7  -2\Dt_7''-5\Et_7 \\
 (10t_1+4u)\;\Dt_6' &=&-\Dt_7'+16\Et_7  \\
   (5t_1+u)\;\Dt_6''&=& \Dt_7'' -\Et_7  \\
   (4t_1+u)\;\Et_6  &=& \Et_7   -\Pt_8''
 \end{array}
\label{rel}
 \end{equation}
 
 \spa{Final~computations~of~polynomials~of~Theorem~1}. 
A part of Thom polynomial are computed in previous
section. Using relations~(\ref{gys}) and~(\ref{rel}) 
we get subsequently
\begin{eqnarray*}
\At_4&=&(4t_1+u)\At_3+\Dt_4',\\
p_{1*}(\At_4)&=&p_{1*}((4t_1+u)(3t_1+u))+
p_{2*}(3t_1+u);\\
\At_5&=&(5t_1+u)\At_4+2\Dt_5'',\\
p_{1*}(\At_5)&=&p_{1*}((5t_1+u)(4t_1+u)(3t_1+u))+
p_{2*}((9t_1+2t_2+3u)(3t_1+u));
\end{eqnarray*}
and so on. Continuing this way we get expressions for the direct
images of all cycles on $Z_1$ of codimension less or equal to~$5$.
Applying formulas for the Gysin homomorphisms from
Section~\ref{gysin} we obtain all Thom polynomials of Theorem~1. Note that
the expressions for $p_{1*}(\Dt_k')$ are not necessary for computing Thom
polynomials. We use these expressions only to verify our
computations.\QED

 
 The reason for the basic relations is as follows. Consider a germ
of some marked singularity, say $\At_k$. In some coordinate system it 
can be written as 
       $$z=x^{k+1}+Q,$$ 
where $Q$ is a nondegenerate quadratic form in the remaining
variables. Such a coordinate system is not unique. Another choice of
the coordinate system results in the multiplication of the tangent
vector $\D/\D x$ to the distinguished line by some
complex number $c$ and the simultaneous multiplication of the tangent
vector $\D/\D z$ to the target space by $c^{k+1}$.
Therefore the tensor $s=dx^{\otimes(k+1)}\otimes\D/\D z$
is invariantly defined. We obtain that the bundle
$(F^1)^{*\otimes(k+1)}\otimes I$ restricted to the cycle $\At_k\subset
Z_1$ admits a canonical nowhere vanishing section $s$.

\th Lemma. The restrictions of the line bundles listed in the table
below to the corresponding singularity loci in $Z_1$ are trivial.
$$\begin{array}{c|c|c|c}
& f & {\rm bundle}~Y&c_1(Y)\\\hline
\At_k&x^{k+1}&(F^1)^{*\otimes{k+1}}\otimes I&(k+1)t_1+u\\
\Dt_k'&y^{k-1}+yx^2&(F^1)^{*\otimes(2k-2)}\otimes I^{\otimes(k-2)}&
(2k-2)t_1+(k-2)u\\
\Dt_k''&x^{k-1}+xy^2&(F^1)^{*\otimes(k-1)}\otimes I&(k-1)t_1+u\\
\Et_6&x^4+y^3&(F^1)^{*\otimes{4}}\otimes I&4t_1+u\\
\Et_7&x^3y+y^3&(F^1)^{*\otimes{9}}\otimes I^{\otimes2}&9t_1+2u\\
\Et_8&x^5+y^3&(F^1)^{*\otimes{5}}\otimes I&5t_1+u
\end{array}$$


 The canonical sections for the bundles from this lemma are chosen so
that for the (marked) functions in the normal form above the
coordinate of this section is equal to $1$ (the distinguished line is
the $x$-axis).

The proof for all cases is the same as for the case of singularity
$\At_k$. The symmetry group of all these singularities acts on the
lines $F^1$ and $I$ by quasi-homogeneous homoteties. Therefore, the
required bundle can be chosen in the form
$(F^1)^{*\otimes\alpha}\otimes I^{\otimes\beta}$, where $\alpha/\beta$
is equal to the quotient of quasi-homogeneous weights of the function
and of the variable $x$ respectively.\QED


Let $\O\subset Z_1$ be a cycle of some marked singularity from the
last Lemma, $Y\to Z_1$ be the corresponding line bundle whose
restriction to $\O$ is trivialised. This
trivialisation of $Y$ can not be extended to the closure of $\O$
since the bundle $Y$ is not trivial. Let $\Theta\subset Z_1$ be a
singularity class of neighbouring complex codimension, $\codim
\Theta=\codim\O +1$. Choose some point $w\in\Theta$ and a
$(\codim\Theta)$-dimensional transversal slice $T$ to $\Theta$ at this
point. The singularity locus $\O$ cut out a number of curves
$\gamma_1,\gamma_2,...$ on $T$. Let $(\C,0)\to (T,w)$ be a
normalisation ($=$ parameterisation) of one of these curves $\g_i$.
Then the canonical section $s$ of $Y$ on $\g_i\subset\O$ can be
written (using some local trivialisation of $Y$ near $w$) in the form
$s=\t^{k_i}h_i$ where $\t$ is a parameter on the curve and $h_i$ is a
germ of a holomorphic non-vanishing function.

\ex Definition. The {\em adjacency exponent\/} $[\O,\Theta]$ is
the sum of the exponents $k_i$ over all curves $\g_i$ of singularity
$\O$ in the transversal to the singularity $\Theta$.

\th Lemma. The following equality holds in the cohomology of $Z_1$,
$$c_1(Y)\,\O=\sum[\O,\Theta]\;\Theta,$$
where the sum is taken over all classes of marked singularities
$\Theta\subset Z_1$ with $\codim\Theta=\codim\O+1$.
 
\spa{Proof}. Let $Q\subset Z_1$ be a test compact cycle of real dimension
$2(\codim_\C\O+1)$. Without loss of generality we assume that $Q$
intersects $\O$ transversally so that $D=Q\cap \O$ is a real
surface without some finite set $\Sing D$ of points corresponding to
intersections of $Q$ with singularity classes of (complex) codimension
$\codim_\C\O+1$. Then
 $$(c_1(Y)\;\O,Q)=(c_1(Y),\overline D).$$
The last number can be computed using the restriction of the section
$s$ to the cycle $D$. It is equal to the sum of indices $\ind_x(s)$
of this section over all points of $\Sing D$. But every such index 
$\ind_x(s)$ is equal, by definition, to the adjacency exponent of
singularities $\O$ and $\Theta\ni x$. Hence, the sum of the indices is equal
to the intersection number of $Q$ with the linear combination of
cycles $\sum[\O,\Theta]\;\Theta$ over all classes $\Theta\subset
Z_1$ with $\codim\Theta=\codim\O+1$. 

Thus both sides of the equality of Lemma take the same values on the
elements of homology group of the complement dimension. This proves
the equality of Lemma modulo torsion. In fact, this equality holds
for any group of coefficients since the group of characteristic
classes of complex vector bundles is torsion free.\QED

To complete the proof of basic relations~(\ref{rel}) 
we need to compute the adjacency
exponents for marked singularity classes.
Finding adjacency exponents is a part of the proof which really
requires a lot of computations. One should find all possible
adjacencies of classes of neighbour codimensions and to compute
the adjacency exponents. In these computations the methods and results
from~\cite{A,V,S} are used.

\th Lemma. The following lists and the comments below exhaust
all possible adjacencies of marked singularity classes of
$\codim\le6$. \(The distinguished direction in the lists below is the
diection of the $x$-axis. The functions of the family corresponding
to the adjacency $\Theta\to\O$ have singularity $\O$ at the origin for
all parameter values $\t\ne0$ and $\Theta$ for $\t=0$. In the table below 
$s$ is the canonical section, $c\in\C$ is a constant.\) 
$$\def\~{\hskip1em}
\begin{array}{r|l|l|r|c}
\Theta\to\O\quad&\kern7em f&s&[\O:\Theta]&notes\\\hline
\At_{k+1}\to\At_k    &x^{k+2}+\t x^{k+1}             &c\t      &1 \~\\
\Dt'_{k+1}\to\Dt'_k  &y^k+x^2y+\t y^{k-1}            &c\t^{-1} &-1  \~\\
\Dt''_{k+1}\to\Dt''_k&x^k+xy^2+\t x^{k-1}            &c\t      & 1 \~\\ 
\Dt'_4\to\At_3       &y^3+x^2y+\t y^2                &c\t^{-1} &-1  \~\\
\Dt''_5\to\Dt'_4     &x^4+xy^2+\t x^2y               &c\t^{4}  &4  \~\\
\Dt''_5\to\At_4      &(x^2+\t y)^2+xy^2              &c\t^{-2} &-2   \~\\
\Dt''_6\to\At_5      &x^5-xy^2+\t(y\pm x^2)^2        &c\t^{-1} &-2  \~&(i)\\
\Et_6\to\At_5        &y^3+(x^2+\t y)^2               &c\t^{-3} &-3  \~\\
\Et_6\to\Dt'_5       &x^4+y(y+\t x)^2                &c\t^{12} &12\~\\
\Et_6\to\Dt''_5      &x^4+(y+\t x)y^2                &c        &0  \~\\
\Pt'_8\to\Dt'_5      &x^2y+z^3-zy^2+\t z^2           &c\t      &4 \~&(ii,iii)\\
\Pt'_8\to\Dt''_5     &y^2x+z^3-zx^2+\t z^2           &c\t^{-1} &-1  \~&(iii)\\
\Dt''_7\to\At_6      &x^6+xy^2-\t^2x^5+(\t^2x^2+\t y)^2&c\t^{-2} &-2  \~\\
\Et_7\to\At_6        &y^3+(x^2-4\t y)(xy+\t x^2-4\t^2y) &c\t^{-5}&-5\~\\
\Et_7\to\Dt'_6       &x^3y+(y+\t x)^2y+\t x^4          &c\t^{16}&16\~\\
\Et_7\to\Dt''_6      &x^3y+(y+\t x)y^2               &c\t^{-1}      &-1  \~\\
\Et_7\to\Et_6        &x^3y+\t y^4+y^3                &c\t &1 \~\\
\Pt''_8\to\Et_6      &y^3+x^2z+xz^2+\t z^2           &c\t^{-1}      &11  \~&(iii)
\end{array}
$$


\ex Comments. {(i) There are $2$ curves realizing the adjacency
$\Dt_6''\to\At_5$ corresponding to the two possible signs of $\pm$. For both
of them we have $s=c\t^{-1}$ so $[\At_5:\Dt_6'']=-2$.

(ii) There are $4$ curves realizing the adjacency $\Pt_8'\to\Dt_5'$ (see
below). For all of them we have $\l=c\t$ so $[\Dt'_5:\Pt_8']=4$.

(iii) The singularity $\Pt''_8$ has a module (of a plane cubic).
Similarly $\Pt'_8$ has two modules (a plane cubic and a point on it). In
the formulas above we used some particular values of modules. The
adjacency exponent does not depend on the choice of modules.

(iv) The critical point function singularity $P_8$ is not adjacent
neither to $D_6$, nor to $A_6$, see~\cite{S}.

(v) There are no adjacencies $\Dt'_{k+1}\to\At_k$ ($k>3$),
$\Dt''_{k+1}\to\Dt'_k$ ($k>4$), $\Dt'_{k+1}\to\Dt''_k$ .

(vi) There is no adjacency $\Pt'_8\to\At_5$.
}

\spa{Proof}. The method of finding the adjacencies is described in
details in~\cite{V}.  The condition that a function has a
singularity of certain type is reformulated as a system of algebraic
equations on the coefficients of the Taylor expansion of the
function. These equations may be explicitly solved which gives the
formulas above. Most of these formulas (except those related to
adjacencies of `new' singularity type $P'_8$, see below) are taken
from~\cite{A,V,S} (sometimes with a minor change of variables). 

Essentially new part of our calculations is finding the adjacency
exponents. As an example we show the computation of the asymptotic
$s=c\t^{-5}$ for the adjacency $\Et_7\to\At_6$ above. 
Consider the family of function germs
$$f(x,y;\t)=y^3+(x^2-4\t y)(xy+\t x^2-4\t^2y).$$
For $\t\ne0$ the partial derivative 
$$f_y=32\t^3y-8\t^2x^2-8\t xy+3y^2+x^3$$ 
has no critical point at the origin. Therefore by parametric Morse
Lemma this function is stably equivalent to its restriction to the
smooth curve $f_y=0$. This equation defines implicitly $y$ as a
function in $x$
$$y=\frac{x^2}{4\t}+\frac{x^3}{32\t^3}+\frac{x^4}{512\t^5}
-\frac{x^5}{1024\t^7}+o(x^5).$$
After substitution to $f$ we get
$$f|_{f_y=0}=\frac{x^7}{512\t^5}+o(x^7).$$
This means that the singularity type of $f$ is $A_6$ for $\t\ne0$ and 
   $$s=\frac1{512\t^5};\qquad [\At_6:\Et_7]=-5.$$
The cases of other singularities are treated in a similar way.

\mdskip
Let us describe in more details adjacencies of the singularity
$\Pt'_8$. The function $f_0$ realizing this singularity is a cubic form
in three variables $x,y,z$. The distinguished direction $P\in\C P^2$
of the $x$-axis  belongs to the cubic $C\in\C P^2$ given by $f_0=0$.
The codimension of the class $\Pt'_8$ is $3$. A possible
transversal is given by the family $f_0+Q$ where 
$Q=\l_1y^2+2\l_2yz+\l_3z^2$ 
is the family of quadratic forms having the direction
$\D/\D x$ in the kernel. By homogeneity all functions of the family 
$$
f=f_0+\t Q_0
$$
are right equivalent to each other for $\t\ne0$ for any fixed
quadratic form $Q_0$.  Hence any adjacency is realized by a family of
this type. The function $f$ has the singularity $D_5$ iff $Q_0=l^2$ where
$l=0$ is the equation of the tangent to the cubic $C$. If the
tangency point is $P$ then the distinguished  direction $P$ is of
$\Dt_5''$ type in our classification. It is possible also that the line
$l=0$ passes through $P$ and is tangent to $C$ at another point. Then
the distinguished direction $P$ has $\Dt_5'$ type. Generically there
are $4$ such lines. So there are $4$ curves of singularity
$\Dt_5'$ in the transversal to $\Pt_8'$.

\mdskip Now let us prove the equality $[\At_5:\Pt_8'']=0$. It is
sufficient to show that $f_0=xy^2+z^3-zx^2$ is not adjacent to
$\At_5$.  This would imply that no singularity of type $\Pt_8'$ close
to $f_0$ is adjacent to $\At_5$ and neither are $\Pt_8'$-singularities from a
Zarisski open set in the space of modules and hence $[\At_5:\Pt_8'']=0$.
So assume that a function of the form $f_0+Q_0$ with $Q_0=\l_1
y^2+2\l_2yz+\l_3z^2$ has singularity $A_5$ at the origin. Then
the form $Q_0$ is non-degenerate and $f$ is stably equivalent to its
restriction $f|_{f_y=f_z=0}$. Resolving the system $f_y=f_z=0$ we get
 $$f|_{f_y=f_z=0}=c_4x^4+c_5x^5+o(x^5),\qquad
      c_4=\frac{\l_1}{4(\l_2^2-\l_1\l_3)},\qquad 
      c_5=\frac{\l_2^2}{4(\l_2^2-\l_1\l_3)^2}.$$
It is clear that the system $c_4=c_5=0$ has no solution that is the
function $f_0+Q_0$ cannot be of $A_5$-type.\QED

Combining two last Lemmas we complete the proof of the basic
relations Lemma formulated at the beginning of this Section.\QED



\subsection{Symmetries and Thom polynomials}

In this section we describe a method of computing Thom polynomials
which is based on Rim\'anyi's idea of using symmetries. Unlike the
direct method for computing Thom polynomials described in previous
sections this method uses an a priori Theorem~1 about the existence of these
polynomials. This method is less geometric but it uses less
computations. The idea is the following. We know that the class dual
to some singularity locus $\O$ is given by a certain polynomial
$P_\O$ in Chern classes so we need to compute the coefficients of
this polynomial.  Every example where both the cohomology class dual to
the singularity locus of $\O\subset M$ and  the classes $u,a_i$ can
be computed explicitly gives rise to a number of linear relations
on the coefficients of this polynomial $P_\O$. If the number of
examples is high enough then these relations could determine the
polynomial completely.  A number of examples are produced in the
following way. Consider some quasihomogeneous family of function
germs realizing a transversal to some singularity class, say
$$f(x,y,a_1,\dots,b_2)=y^5+x^2y+a_1y^4+a_2 y^3+a_3 y^2+b_1 x^2+b_2 xy.$$
Consider some line bundle $\xi\to B$ with the first Chern class
$t=c_1(\xi)\in H^2(B)$. With any variable
$x,y,a_1,\dots,b_2$ we associate a line bundle $\xi^{\otimes l}$ where
$l$ is the quasihomogeneous weight of the variable. Then the family
$f$ may be interpreted as a quasihomogeneous bundle map 
$$f:\xi^{\otimes2}\oplus\xi\oplus\xi\oplus\cdots
             \oplus\xi^{\otimes2}\to\xi^{\otimes5},\qquad
 x\oplus y\oplus a_1\oplus\cdots\oplus b_2\mapsto 
        f(x,y,a_1,\dots,b_2).$$

Now define $M$ to be the total space of the bundle
$\pi:\xi\oplus\cdots\oplus\xi^{\otimes2}\to B$ corresponding to the
parameters $a_1,\dots,b_2$ of the family $f$; $V\to M$ to be the rank
$2$ bundle $\pi^*\xi^{\otimes2}\oplus\pi^*\xi$ over $M$ corresponding to
variables $x,y$; and $I=\pi^*\xi^{\otimes 5}$. So we constructed vector
bundles $V,I\to M$ and a smooth bundle map $f:V\to I$ as in
Section~\ref{bmap}. The characteristic classes in this example are
$$u=c_1(I)=5t,\quad a=c(V^*\otimes I-V)=
    \frac{(1+(5-1)t)(1+(5-2)t)}{(1+t)(1+2t)}=
   1+4t-2t^2+\dots.$$
On the other hand we may compute the classes dual to the singularity
loci. This gives the following relations on the coefficients of the
Thom polynomials $P_\O(u,a_1,a_2,\dots)$:
\begin{itemize}
\item If $\O=D_6$ then $\O(f)$ is the zero section of the bundle
$M\to B$ and its dual cohomology class is the top Chern class 
$e=c_5(M\to B)$ of this bundle, so
$$P_{D_6}(5t,4t,-2t^2,\dots)=t\cdot 2t\cdot 3t\cdot t\cdot2t=
12t^5.$$

\item If $\O=A_6,E_6$ or $\O$ is any singularity class of greater
codimension  then $\O(f)=\emptyset$ and so $P_\O(5t,4t,-2t^2,\dots)=0$.

\end{itemize}

(We may compute the classes $P_{D_k}(5t,4t,-2t^2,\dots)$, $k<6$ in  a
similar way but these extra relations are redundant.) The 
characteristic classes for quasihomogeneous deformations of other
singularities of codimension $\le6$ are given in the following table.
$$\begin{array}{>{\rule{0pt}{5ex}}l|c|>{\displaystyle}c|c}
\multicolumn{1}{c|}{\O}&u&a=1+a_1+a_2+\dots&e\\ \noalign{\hrule}
A_{k}&(k{+}1)\,t&\frac{1+k t}{1+t}&(k{-}1)!\,t^{k-1}\\
D_{k}&2(k{-}1)\,t&\frac{(1+2(k{-}2)t)(1+kt)}
{(1+2t)(1+(k{-}2)t)}&(k{-}2)!!\,t^{k-1}\\
E_6&12\,t&\frac{(1+9t)(1+8t)}{(1+3t)(1+4t)}&6!\,t^5\\
E_7&9\,t&\frac{(1+7t)(1+6t)}{(1+2t)(1+3t)}&3\cdot5!\,t^6\\
P_8&3\,t&\frac{(1+2t)^3}{(1+t)^3}&t^6 \end{array} $$

Relations arising from these examples are sufficient to compute
all Thom polynomials of Theorem~1.\QED







%\newpage

\section{Characteristic classes of Lagrange\\ 
and Legendre singularities}

The correspondence between Lagrange (Legendre) singularities and 
critical point singularities described in this section is valid 
for both real and complex cases. We assume some familiarity of the
reader with the theory of Lagrange and Legendre singularities, see,
eg.~\cite{AVG,AG}. Our definitions should not be considered for the
introduction to symplectic or contact geometry.



\subsection{Lagrange singularities and characteristic
classes}


A Lagrange singularity is a projection singularity of a Lagrange
submanifold in the space of cotangent bundle to the base of the
bundle.

\ex Definition. A submanifold $M\subset T^*B$, $\dim M=\dim B$ is
called {\em Lagrangian}, if the {\em standard symplectic $2$-form\/}
$\o=\sum dp_i\wedge dq_i$ vanishes on $L$, where $q_i$ are some local 
coordinates on the base $B$, and $p_i$ a the corresponding coordinates
on the fibres of the cotangent bundle $T^*B\to B$. 

With any germ of Lagrange submanifold in the cotangent bundle one can
associate a critical point function singularity.  Namely, any Lagrange 
germ $M\subset T^*\C^n$ may be given by a germ of its
generating family of functions $F(x,q)$ according to the rule
(cf.~\cite{AVG,AG})
 $$L=\Bigl\{(p,q)\in T^*\C^n\Bigm|
   \exists x,\;\D F/\D x=0,\;p=\D F/\D q\Bigr\}.$$  
Here $x$ is the coordinate on the fibres of an auxiliary bundle
$(\C^{n+m},0)\to (\C^n,0)$. We associate with the Lagrange germ $M$
the the initial function germ of its generating family $f(x)=F(x,0)$.
The generating family is not unique but another choice of the family
or of the coordinates on the base lead to $R_{\rm st}$-equivalent
function germs. (Recall that two function germs in spaces of possibly
different dimensions are called {\em $R_{\rm st}$-equivalent}
(stably right equivalent) if after adding suitable
non-degenerate quadratic forms in new variables each of these
functions can be brought into another by a change of variables.)

\ex Example. Let $V$ be a vector space. With any function germ 
$f:V,0\to\C,0$ we associate a Lagrange germ $L(f)\subset T^*V$, the
graph of the differential $-df$ (it is convenient to put the sign
`$-$'
here). The natural isomorphisms $T^*V\cong V\oplus V^*\cong
TV^*$ allows to consider $L_w(f)$ also as a submanifold in  $T^*V^*$.
The symplectic structures induced on $V\oplus V^*$ by this
isomorphisms differ  by sign. Hence, $L(f)$ is also Lagrange as a
submanifold in $T^*V^*$. The critical point function singularity
corresponding to the germ $L(f)\subset T^*V^*$ is the singularity of
the function $f$ itself. Indeed, $L(f)\subset T^*V^*$ may be given by
the following generating family
  $$F(x,q)=\<x,q\>+f(x),\qquad q\in V^*,\quad x\in V.$$
  Here $x$ is considered as an additional variable and
$\<\cdot,\cdot\>:V\times V^*\to\C$ is the natural coupling.



\ex Definition. The {\em classifying space of Lagrange
singularities\/} $\Lc=\Lc_N^\C$ is the space of all $K$-jets of
Lagrange germs $M\subset (T^*\C^N,0)=(\C^{2N},0)$, where $K,N\gg0$ are
some large integers. This space is
homotopy equivalent to the {\em Lagrange Grassmannian\/} $\L=\L_N^\C$
consisting of Lagrange planes in $\C^{2N}$
since the natural projection $\Lc\to\L$ sending a Lagrange germ to
its tangent plane have contractible fibres. The  ring of {\em
Lagrange characteristic classes} is the limit cohomology ring
$\lim\limits_{N\to\infty}H^*(\L_N^\C)$.

The topology of real Lagrange Grassmannian is well studied
(see~\cite{F}). Its $\Z_2$-cohomology ring $H^*(\L_N^\R)$ is generated
by Stiefel-Whitney classes $\a_i$ of the tautological bundle, the
generators $\a_i$ satisfy relations $\a_i^2$=0. Similar description
exists for the (integer) cohomology ring of the complex Lagrange
Grassmannian.

\th Theorem \rm(\cite{P1,K3}). The ring of Lagrange characteristic
classes is isomorphic to the quotient of polynomial ring in variables
$a_1,a_2,\dots$ of degrees $2,4,\dots$ over the ideal
generated by elements
\begin{equation}\label{rellag}
a_i^2-2a_{i+1}a_{i-1}+2a_{i+2}a_{i-2}-2a_{i+3}a_{i-3}+\dots.
\end{equation}
The group of Lagrange characteristic classes is torsion free and 
the monomials $a_1^{i_1}\dots a_n^{i_n}$, $i_k\in\{0,1\}$ 
form a free additive basis.

For the generators we choose the classes
$a_i=c_i(\C^{2N}/L)=c_i(L^*)=(-1)^ic_i(L)\in H^{2i}(\L^\C_N)$, where
$L$ is  the tautological bundle $L\to\L^\C_N$. To express an element
of this ring in terms of the additive basis one should apply
repeatedly relation~(\ref{rellag}) to every monomial which contains
squares of generators. This will require a finite number of steps since
every newly appeared monomial has the degree strictly less than the
original one if one uses the `strange' filtration with the degree of
$a_i$ equal $i^2$.

The meaning of the relations is the following. The symplectic
form induces the canonical isomorphism $\C^{2N}/L\cong L^*$. By
Whitney formula we have $c(L)c(L^*)=1$ or
  $$(1+a_1+a_2+\dots)(1-a_1+a_2-\dots)=1$$
which is equivalent to~(\ref{rellag}).


\mdskip
By construction above the points of $\Lc$ are classified according to
the $R_{\rm st}$-classification of function germs. With any singularity
class $\O$ (given as an $R$-invariant
algebraic subset in some jet space of function germs) we associate the
corresponding subvariety $\O(\Lc)\subset\Lc$.


\ex Definition. The {\em Thom polynomial} associated with an
$R_{\rm st}$-class $\O$ of critical point singularity 
is the universal Lagrange
characteristic class $P_\O\in H^*(\L)$ (expressed in terms of
multiplicative generators $a_i$) represented by the intersection with
the variety $\O(\Lc)\subset\Lc_N$.

This definition is independent on $K,N$ provided these numbers are
large enough ($N$ must be larger than the codimension of the
singularity and $K$ is chosen so that the $K$-jet of the singularity is
sufficient, see~\cite{AVG,AGLV}).


\spa{Proof~of~Theorem~2~for~the~trivial~bundle~$I$}.
Consider vector bundles $V,I\to M$ and a fibre bundle map
$f:V\to I$ as in Section~\ref{bmap}. Assume that the bundle $I$ is 
{\em trivial}, $I=\C$.

For each point $w\in W$ we define the Lagrange germ $L_w(f)\subset
T^*V_x$ as above. We would like to extend
this correspondence between the critical point singularities and the Lagrange
singularities and to construct a classifying map $M\to\Lc$ which
preserves the Lagrange (or critical point) singularity type at
considered points.

The construction is as follows. Consider a bundle $U\to M$ such that
$V\oplus U$ is the trivial bundle $\C^N$. Then $V{\oplus} V^*\times
U{\oplus} U^*\cong \C^{2n}$ is also trivial. Hence all Lagrange germs
$\k(w)=L_w(f)\times 0{\oplus} U_w^* \subset V_w{\oplus} V_w^*\times
U_w{\oplus} U^*_w=T^*(V_w^*\times U_w^*)=T^*\C^N$ belong to the same
symplectic space $\C^{2N}$. One can see that the critical point
singularity corresponding to $\k(w)$ is the same as for $L_w(f)$,
i.e.\ $f_w$. Thus constructed map 
 $$\k:M\to\Lc_N,\qquad w\mapsto L_w(f)\times 0{\oplus} U^*,$$
induces both the classes dual to the loci of Lagrange singularities
and the characteristic classes $a_i\in H^*(\Lc_N)$. By definition,
$\k^*a_i=c_i((V\oplus U^*)^*)=c_i(V^*-V)$. 

We have proved, therefore, that {\em the cohomology class on $M$
Poincar\'e dual to the locus $\O(f)\subset M$ is equal to the defined
above polynomial $P_\O$ evaluated on the classes $a_i=c_i(V^*-V)$.}
This proves Theorem~2 of Section~\ref{bmap} in case when the bundle
$I$ is trivial.\QED

\th Theorem. The Thom polynomials of $R_{\rm st}$-singularities of
function of codimension $\le6$ are obtained from the polynomials of the
list of Table~{\rm\ref{thompol}} by setting $u=0$.

\spa{Proof}. There are two possible proofs of this theorem. First we
observe simply that the homomorphism
$\Z[a_1,a_2,\dots]\to\Z[c_1,c_2,\dots]$ which sends the generator
$a_i$ to the $i$th homogeneous term of the expansion
$(1-c_1+c_2-\dots)(1+c_1+c_2+\dots)^{-1}$ induces an injective
homomorphism of the ring of Lagrange characteristic classes to the
polynomial ring $\Z[c_1,c_2,\dots]$ (see~\cite{K3}). It follows
that the formulas for the characteristic classes found for the case of
fiber singularities can be applied to the case of Lagrange
singularities.

In another proof we oserve that all steps of our computations made in
Sections~\ref{gysin}--\ref{adjacency}  (including resolutions, the
formula for the Gysin homomorphism, markings, adjacensy exponents and
basic relations) can be carried out directly  for the case of
Lagrange singularities. For instance, the kernel of the second
differential $f_{(2)}$ of a function germ corresponds to the
intersection of the thangent plane of a Lagrange germ
$L\subset\C^{2N}$ with the fixed Lagrange plane $\C^N\subset\C^{2N}$,
zeroes of the third-order terms $f_{(3)}$ on $\ker f_{(2)}$ correspond
to the lines of higher order of tangency of Lagrange submanifolds ect.
In fact, our original computation of Thom polynomials of Theorem~1
was performed on the languaue of Lagrange (or Legendre) singularities
and only later we translated it to the language of fiber singularities
of functions.\QED 


\ex Example (characteristic classes of Lagrange submanifolds in
$T^*B$).  Let $M\to T^*B$ be a Lagrange immersion. A similar
construction exists for the map $\k:M\to\L_N$ which preserves
critical point singularity types associated with Lagrange germs (see
also~\cite{Au}). The characteristic classes induced by this map are
$a_i=c_i(T_MB-TM)$. Again, the classes dual to different singularity
loci of Lagrange projection $M\to T^*B\to B$ are given by universal
Thom polynomials evaluated on the classes $c_i(T_MB-TM)$.

\ex Remark. {The most general situation where Lagrange characteristic
classes appear is the following. Let $E\to M$ be a vector bundle of
even rank equipped with symplectic bilinear forms on its fibers
(given as a nowhere degenerating section of $\L^2E^*$). Let
$L_1,L_2\subset E$ be two Lagrange subbundles (in a sense that the
fibers of $L_1,L_2$ are Lagrange planes in the fibers of $E$. Then
the {\em relative Chern classes\/} (cf.~\cite{MN}) of the triple
$(E,L_1,L_2)$ are defined as $a_i=c_i(L_2^*-L_1)$. The equalities
$c(L_1+L_1^*)=c(L_2+L_2^*)=c(E)$ imply the identity~\ref{rellag}
for these classes.

The situations considered above fit into this pattern. In case of fiber bundle 
map $f:V\to \C$ we take $E=V\oplus V^*$, $L_1=V\oplus 0$, and
$L_2\cong L_1$ is the bundle of Lagrange planes tangent to the germs
of $L_w(f)\subset V_w\oplus V_w^*$.

In case of Lagrange immersion $M\to T^*B$ we set $E=T_M(T^*B)$,
$L_1=TM$, and $L_2\cong (T^*B)|_M$ is the bundle of `vertical' tangent
vectors to $T^*B$, corresponding to the kernel of the differential of
the projection $T^*B\to B$.\QED}



\subsection{Legendre singularities and characteristic
classes}\label{LegCl}

The theory of Legendre characteristic classes is a twisted version of
the theory of Lagrange ones. Consider vector spaces $V,I$ such that
$\dim I=1$. The space $V\oplus V^*{\otimes} I$ has the natural
nondegenerate skew-symmetric bilinear form with values in $I$. After
any isomorphism $I\cong\C$ this form turns into the standard
symplectic form on $V\oplus V^*\cong T^*V$. The Grassmannian
of Lagrange planes in $V\oplus V^*{\otimes}I$
with respect to this form is isomorphic to the usual Lagrange
Grassmannian $\L_n$, where $n=\dim V$.

\ex Definition. The {\em Legendre Grassmannian} $\Lt=\Lt_N^\C$ is the
total space over $\BU(1)$ whose fibers are formed by the Grassmannians of
Lagrange subspaces in the twisted symplectic fibers of the bundle
$\C^N\oplus \C^N{\otimes }\xi$, where $\BU(1)\cong\C P^\infty$ is the
classifying space of one-dimensional vector bundles (or some its
finite-dimensional approximation $\C P^{N'}$, $N'\gg0$), and $\xi\to
\BU(1)$ is the canonical line bundle.   The ring of {\em Legendre
characteristic classes\/} is the cohomology ring of the stable
Legendre Grassmannian $H^*(\Lt)=\lim\limits_{n\to\infty}
H^*(\Lt_N)$.


\th Theorem {\rm(cf.\ \cite{K3})}. 
The ring of Legendre characteristic classes is given by
generators $u,a_1,a_2,\dots$, and relations which
are homogeneous components of the equality
 \begin{equation}
 (1+a_1+a_2+a_3+\dots)\(1-\frac{a_1}{1+u}+
     \frac{a_2}{(1+u)^2}-\frac{a_3}{(1+u)^3}+\dots\)=1.
\label{leg}
\end{equation}

The class $u=c_1(\xi)$ is the standard generator of $H^*(\BU(1))$ and
the classes $a_i$ are defined as $a_i=c_i(-L)$, where
$L$ is the tautological bundle over the Grassmannian. The relation
above comes from the isomorphism $(\C^N\oplus \C^N{\otimes}\xi)/L\cong
L^*{\otimes}\xi$, or, formally, $L+(L^*-\C^N){\otimes }\xi=0$.

\ex Remark. The description above is valid for both complex case and
integer coefficients (with $\deg a_i=2i$) and real case and
$\Z_2$-coefficients (with $\deg a_i=i$). Note also that the monomials
$u^{i_0}a_1^{i_1}a_2^{i_2}$, $i_0\ge0$, $i_k\in\{0,1\}$ for $k>0$,
form a free additive basis. Nevertheless even for the case of
$\Z_2$-coefficients it is not isomorphic to
$\Z_2[u]{\otimes}\L_{\Z_2}(a_1,a_2,\dots)$ (The multiplicative
structure in the ring of Legendre characteristic $\Z_2$-classes is
wrongly computed in~\cite{K2}.) Indeed, the relation of degree $2$ is
$a_1u+a_1^2=0\pmod2$ and so the square of none element of degree $1$
vanishes. On the other hand one can show that for the cohomology with
coefficients in any field $K$ of characteristic different from $2$
there is an isomorphism $H^*(\Lt,K)\cong K[u]{\otimes} H^*(\L,K)$.

\spa{Proof}. The class $u$ generates the cohomology of the base and
the classes $a_1,a_2,\dots$ generate the cohomology of each fibre. It
follows that the spectral sequence of the bundle $\Lt\to \BU(1)$ 
degenerate at the second term and the classes $u,a_1,a_2,\dots$
generate the whole cohomology ring of $\Lt$. We know already
some set of relations and comparing the dimensions we see that there are
no other relations between the generators.\QED


Now we explain the relationship between the definition above
and the theory of Legendre singularities. Let $V,I$ be smooth
manifolds (not necessary vector spaces) of dimensions $n,1$
respectively. {\em The space $J^1(V,I)$ of $1$-jets of maps $V\to I$}
is the total space of the bundle $T^*V\otimes TI$ over $V\times I$.
This space carries the natural contact structure (a codimension $1$
subbundle in the tangent bundle). If $z:I\to\C$ is a local coordinate
on $I$ then we get  $J^1(V,\C)\cong T^*V\times\C$. The contact
structure on  $J^1(V,\C)=T^*V\times\C$  is given by the field of
kernels of the $1$-form
  $$\a=d z-\l,$$
where $\l$ is the Liouville form on $T^*V$ (written as $\l=\sum
p_i\,dq_i$ in canonical coordinates). Another choice of the coordinate
$z$ leads to a multiplication of $\a$ by a nonzero function so the
field of kernels of $\a$ is invariantly defined.

\ex Definition. A submanifold $M\subset J^1(V,I)$, is
called {\em Legendrean}, if it is tangent to the contact field at
every point.

With any germ of Legendre submanifold $L\subset J^1(V,I)$ one can
associate a critical point function singularity. To do that, we
choose a local coordinate $z$ on $I$. Observe that the image of $L$
under the natural projection $J^1(V,\C)=T^*V\times\C\to T^*V$ is
Lagrangian. Then we apply the construction of previous section. 
Another choice of the coordinate on $\C$ may lead to another function but
{\em the class of $V_{\rm st}$-equivalence of the critical point
function singularity is well defined} (see~\cite{AVG,AG}). Recall
that two function germs in spaces of possibly different dimensions
are called {\em $V_{\rm st}$-equivalent}\footnote{Sometimes in the
literature this equivalence is called contact equivalence. We prefer
following~\cite{AVG,AG} to keep the notion of contact equivalence for
contact diffeomorphisms of the ambient space.} if after adding suitable 
non-degenerate quadratic forms in new variables and multiplication by
non-vanishing functions they can be brought on into another by a 
change of variables.

Remark that the correspondence between Lagrange germs in $(T^*\C^n,0)$ and
Legendre germs in $(J^1(\C^n,\C)=T^*\C^n\times\C,0)$ is bijective.
Indeed, the $z$-coordinate is uniquely determined by the condition
$dz=p\,dq$ since the restriction of the form $p\,dq$ to a Lagrange
germ is closed (and hence, exact).



\ex Example. {Let $V$ and $I$ be vector spaces. Then we have the
following bijections
$$\begin{array}{ccccccccccccccc}
\hbox{Legendre}&&\hbox{Lagrange}&&\hbox{Lagrange}&&\hbox{Legendre}\\
\hbox{germs in}&\stackrel1\longleftrightarrow&
\hbox{germs in}&\stackrel2\longleftrightarrow&
\hbox{germs in}&\stackrel3\longleftrightarrow&\hbox{germs in}\\
(J^1(V,I),0)&&(T^*V,0)&&(T^*V^*,0)&&(J^1(V^*{\otimes} I,I),0)
\end{array}$$
where $2$ is induced by the isomorphism $T^*V=V\oplus V^*=T^*V^*$.
The correspondences $1$ and $3$ depend on the choice of coordinate on
$I\cong\C$. Nevertheless the resulting correspondence between Legendre
submanifolds in $J^1(V,I)$ and $J^1(V^*{\otimes} I,I)$ is invariantly
defined. Moreover, this correspondence is given by the global
(nonlinear) contactomorphism of these spaces. This contactomorphism
  $$h:J^1(V,I)\longrightarrow J^1(V^*{\otimes} I,I)$$
is called the {\em hodograph transform}. It is given by
  $$h:(v,u,z)\longmapsto  (v,u,\<v,u\>-z),
     \quad v\in V,\quad u\in V^*{\otimes}I,\quad z\in I$$
where we identify $ J^1(V,I)=V\times V^*{\otimes}I\times I= 
J^1(V^*{\otimes} I,I)$. (Remark that the two contact structures
induced on $V\times V^*{{\otimes}}I\times I$ are different.)

It is easy to verify that if $L(f)\subset J^1(V,I)$ is a germ of Legendre
submanifold given as the $1$-jet extension of the function $f:V,0\to
I,0$ then the class of $V$-equivalence of function singularities
associated with $h(L(f))\subset  J^1(V^*{\otimes} I,I)$ is represented
by the function germ $f$ itself.}



\ex Definition. The {\em classifying space of Legendre
singularities\/} $\Lct=\Lct_N^\C$ is the total space of the bundle over (a
finite dimensional approximation of) $\BU(1)$ with
the fibre over $x\in \BU(1)$ consisting of all $K$-jets of Legendre germs in 
$(J^1(\C^N,\xi_x),0)\cong (J^1(\C^N{\otimes}\xi_x,\xi_x),0)$, where
$\xi$ is the canonical line bundle over $\BU(1)$.
This space is homotopy equivalent to the Legendre Grassmannian
$\Lt=\Lt_N^\C$

As it is explained above, the points of $\Lct$ are classified according to
the $V_{\rm st}$-classification of function germs. With any singularity
class $\O$ (given as a $V$-invariant
algebraic subset in some jet space of function germs) we associate the
corresponding subvariety $\O(\Lct)\subset \Lct$.


\ex Definition. The {\em Thom polynomial} associated with a $V$-class $\O$ 
of critical point singularity is the universal Legendre
characteristic class $P_\O\in H^*(\Lct)$ (expressed in terms of the
multiplicative generators $u,a_i$) represented by the intersection with
the variety $\O(\Lct)\subset\Lct_N$.

Similar to the Lagrange case, this definition is independent on $K,N$ 
provided these numbers are large enough.

\mdskip
\spa{Proof~of~Theorem~2}. Consider vector bundles $V,I\to M$ and a 
fibre bundle map $f:V\to I$ as in Section~\ref{bmap}. 

With each point $w\in W$ we associate the Legendre germ
$L_w(f)\subset J^1(V,I)$ given as the $1$-graph of $f_w:V_w\to I_w$.
Using the hodograph transform we may consider this germ as a germ
$L_w(f)\subset J^1(V_w^*{\otimes}I_w,I_w)$. The $V$-singularity class
associated with this germ is the class of the germ $f_w$. We would
like to extend this correspondence between the critical point
singularities and the Legendre singularities and to construct a
classifying map $M\to\Lct$ which preserves the Legendre singularity 
type at considered points.

The construction is similar to that considered for Lagrange case in
previous section. Consider a vector bundle $U\to M$ such that
$V\oplus U$ is the trivial bundle $\C^N$. Then for each $w\in M$ 
the germ $L_w(f)$ defines the germ 
$L_w(f)\times 0\oplus U_w^*{\otimes}I_w\subset J^1(V_w^*{\otimes}I_w\oplus
U_w^*{\otimes}I_w,I_w)=J^1(\C^N{\otimes}I_w,I_w)$. It remains to
observe that the spaces   $J^1(\C^N{\otimes}I_w,I_w)$ form a
$U(1)$-bundle that can be induced from the universal one. The
universal bundle is, by definition, the space $\Lct_N$. The
correspondence used in this construction preserves the $V$-singularity
class associated to Legendre germs. The characteristic classes induced
by this construction are $u=c_1(I)$ and $a_i=c_i(-(V\oplus
U^*{\otimes}I))=c_i(V^*{\otimes}I-V)$.\QED




\th Theorem. The Thom polynomials of $V_{\rm st}$-singularities of
functions of codimension $\le6$ are those from the
list of Theorem~{\rm1}.

\spa{Proof} repeats the arguments used for the proof(s) of similar
theorem in Lagrange case of previous section.\QED 


\ex Example (characteristic classes of Legendre submanifolds in $PT^*B$). 
The space $PT^*B$ of projectivised cotangent bundle is formed by pairs
(a point $b\in B$, a hyperplane $h\subset T_bB$). Such pairs are
called {\em contact elements}. The space $PT^*B$ carries the
natural contact structure that can be defined as follows. Represent
(locally) the base $B$ as $B=M\times I$. Denote by $P_0\subset PT^*B$
the open set formed by contact elements that are transversal to the
lines $\{\pt\}\times I\subset M\times I$. Every such contact element
$h\in T_{w,z}(M{\times} I)=T_w M\oplus T_z I$ can be considered as a
linear map $h: T_w M\to T_z I$. This allows to identify $P_0\cong
J^1(M,I)$. The contact structure on $PT^*B$ is independent on the
presentation $B=M\times I$.

With every germ of Legendre submanifold $L\subset J^1(M,I)=P_0\subset 
PT^*B$ we can associate the class of $V_{\rm st}$-equivalence of function
singularities. This class is also independent of the local
representation $B=M\times I$. It is not difficult to construct a map
$\k:L\to\L_N$ which preserves
$V_{\rm st}$-singularity types associated with Legendre germs. 
The characteristic classes induced by this map are the following:
$u=c_1(I)$, where $I$ is the conjugate of the tautological line bundle
on $PT^*B$ (it can be also defined as the normal line bundle of the
contact structure); for the classes $a_i$ we have $a_i=c_i(T_MB-TM-I)$
(see below). Similar to the Lagrange case, the classes dual to different 
singularity loci of Legendre projection $M\to T^*B\to B$ are given by 
universal Thom polynomials evaluated on the classes $c_1(I)$, 
$c_i(T_MB-TM-I)$.

\ex Remark. {The most general situation where Legendre characteristic classes
appear is the following. Let $E\to M$ be a vector bundle of even rank.
Assume that the fibres of this bundle are 
equipped with symplectic bilinear forms that take values in the fibres
of some line bundle $I$ (i.e.\ we are given a
nowhere degenerating section of $\L^2E^*\otimes I$). Let $L_1,L_2\subset E$ 
be two Lagrange subbundles with respect to this twisted symplectic
form. Then the {\em relative Chern
classes\/} of the quadruple $(E,I,L_1,L_2)$ are defined as $u=c_1(I)$,
$a_i=c_i(L_2^*{\otimes}I-L_1)$. The equalities $c(L_1+L_1^*{\otimes}
I)=c(L_2+L_2^*{\otimes}I)=c(E)$
imply the identity~(\ref{leg}) for these classes.

The situations considered above fit into this pattern. In case of fibre bundle 
map $f:V\to I$ we take $E=V\oplus V^*{\otimes}I$, $L_1=V\oplus 0$, and
$L_2\cong L_1$ is the bundle of the tangent planes to the graphs of
the differentials of the maps $f_w:V_w\to I_w$, $w\in M$.

In case of Legendre immersion $M\to PT^*B$ we set $E\subset
T_M(PT^*B)$ to be the bundle of contact planes, $I=T_M(PT^*B)/E$
to be the normal bundle of the contact structure. The symplectic form
is given by $(\xi,\eta)\mapsto [\xi,\eta]\pmod E$, where $\xi,\eta$
are vector fields tangent to $E$ and $[\cdot,\cdot]$ is the commutator
of vector fields. (This is the invariant definition of the linear symplectic
structure on contact planes given by $d\a|_E$, where the contact
structure $E$ is the field of kernels of the $1$-form $\a$.)

The subbundles $L_1,L_2$ are $TM$, and the bundle of `vertical' tangent
vectors to $PT^*B$, corresponding to the kernel of the differential of
the projection $PT^*B\to B$. Since $E_w/L_{2w}$ is the hyperplane of
the contact element $w\in M\subset PT^*B$, we have 
$L_2^*{\otimes I}=E-L_2=T_MB-I$, i.e.\ 
 $$a_i=c_i(L_2^*{\otimes}I-L_2)=c_i(T_MB-TM-I).\QED$$}


\subsection{Real Lagrange and Legendre singularities and
characteristic classes} 

The theorems on complex characteristic classes have usually a real
analogue where the complex manifolds, maps and bundles are replaced by
the real
ones, integer cohomology by $\Z_2$-cohomology, Chern classes by
Stiefel-Whitney classes etc. This principle can be applied to the
problems studied in this paper as well. The notions of Lagrange and
Legendre characteristic classes, the correspondence between
singularities of functions and Lagrange (Legendre) singularities, the
definitions of classifying spaces and Thom polynomials repeat
word-by-word the corresponding notions defined in this paper for
complex case. The main difference is that the
homomorphism~(\ref{atoc}) of the ring of Legendre characteristic
classes to the polynomial ring in variables $u,c_1,c_2,\dots$ (which
is injective over $\Z$) has a big kernel over $\Z_2$. For instance,
it is trivial in Lagrange case when $u=0$ (indeed, the total
Stiefel-Whitney class $\o(V^*-V)$ is trivial for any real bundle
$V$). It follows that the characteristic classes of fibre
singularities of  real-valued functions are trivial, see~\cite{K1}. 
Nevertheless these characteristic classes are not trivial if they are
applied directly to the cycles of Lagrange singularities on Lagrange
submanifolds in $T^*B$ (respectively, Legendre singularities of
Legendre submanifolds in $PT^*B$).

\th Theorem. The Thom polynomials of real Legendre singularities of
real codimension $\le6$ are obtained from the list of Theorem~{\rm1} by
replacing the Chern classes $a_i$ by the corresponding Stiefel-Whitney
classes $\a_i$ and reducing the coefficients modulo $2$,
$$\begin{array}{@{}ccl@{}}
%\begin{eqnarray*}
A_2&=&\a_1\\
A_3&=& \a_2\quad+u\a_1\\
A_4&=& \a_1\a_2\quad+u^2\a_1\\
D_4&=& \a_1\a_2\quad+u\a_2\\
A_5&=& \a_1\a_3\quad+u^3\a_1\\
D_5&=& 0\\
A_6&=& \a_2\a_3\quad+ u(\a_1\a_3+\a_4)+
   u^2\a_1\a_2+u^3\a_2+u^4\a_1\\
D_6&=& 0\\
E_6&=& \a_2\a_3\quad+u\a_4+u^2\a_1\a_2+u^3\a_2\\
A_7&=& \a_1\a_2\a_3+\a_2\a_4\quad+
   u\a_5+
   u^2\a_1\a_3+u^4\a_2+u^5\a_1\\
D_7&=& 0\\
E_7&=& \a_1\a_2\a_3\quad+
   u(\a_2\a_3+\a_1\a_4)+
   u^2\a_4+u^3\a_1\a_2+u^4\a_2\\
P_8&=&\a_1\a_2\a_3\quad+   u\a_1\a_4
\end{array}$$
%\end{eqnarray*}}

In particular, let $M\subset PT^*B$ be a real Legendre immersion. Then the
$\Z_2$-cohomology class dual to some cycle of Legendre singularities of
$\codim\le6$ is equal to the corresponding Thom polynomial evaluated
on the classes $u=\o_1(I)$ and $\a_i=\o_i(T_MB-TM-I)$, where $I$ is
the normal line bundle of contact structure on $PT^*B$.

\th Theorem~{\rm(cf.~\cite{V})}. The Thom polynomials of real Lagrange
singularities of real codimension $\le6$ are obtained from the list
of previous theorem by setting $u=0$.

In particular, let $M\subset T^*B$ be a real Lagrange immersion. Then the
$\Z_2$-cohomology class dual to some cycle of Legendre singularities of
$\codim\le6$ is equal to the corresponding Thom polynomial evaluated
on the classes $\a_i=\o_i(T_MB-TM)$.

These theorems can be proved applying step by step the real versions of
all constructions used in Section~2 in the proof of the corresponding
formulas for complex case.\QED

\ex Remark. The formulas for classes of real Lagrange singularities of
$\codim\le6$ (except $A_7$) were obtained in~\cite{V} by a different
method. The expression $A_7=\a_1\a_2\a_3+\a_2\a_4$ as well as all
classes of real Legendre singularities are new.






%\newpage
\frenchspacing
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\end{document}

