%From: "Andrei A. Agrachev" <andrei@agrachev.mian.su>
%Date: Fri, 16 Oct 1998 19:00:12 +0300
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\title{ On Morse Theory for Piecewise Smooth Functions}
\author{Andrei A. Agrachev, Diethard Pallaschke, and Stefan Scholtes}

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\begin{document}
\maketitle

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\vspace{0.2cm}
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%                    A B S T R A C T
%
%----------------------------------------------------------------
\begin{abstract}
%----------------------------------------------------------------
Lower level sets of continuous selections of \(C^2\)-functions defined
on a smooth manifold in the vicinity of a nondegenerate critical
point in the sense of \cite{jon3} are studied. It is shown that the
lower level set is homotopy equivalent to the join of the lower level sets
of the smooth and the nonsmooth part, respectively,
of the corresponding normal form.
Some generalized Morse inequalities are deduced from this result.
%
\end{abstract}
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%----------------------------------------------------------------
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{~}\\
{\bf Key Words: }Continuous selection, critical point,
nonsmooth Morse theory\\
\\
{\bf AMS(MOS) Subject Classification: }26A27, 58C27, 90C30\\
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\vspace{0.2cm}
{~}\\
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%                I n t r o d u c t i o n
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%----------------------------------------------------------------
\section{Introduction}
%----------------------------------------------------------------
%
For a smooth function  $f:M \longrightarrow \R$ defined on a
smooth $n$-dimensional manifold $M,$ Morse Theory studies
the topological types  of the lower level sets
\[M^{\alpha}=\{~x~|~f(x)\leq\alpha\}\]
for \(\alpha\in\R\) (cf. \cite{milnor}). The first Morse Lemma states
that for a \(C^1\)-function \(f\) and \(\alpha <\beta\) the
lower level set \(M^{\alpha}\) is a strong deformation retract
of \(M^{\beta}\) provided that \(M^{\beta}\) is compact and
\(f^{-1}([\alpha,\beta])\) does not contain a critical point.
The second Morse Lemma is concerned with the local behaviour
of \(C^2\)-functions in a neighborhood of a nondegenerated critical
point \(x_0 \in M \). It states that whenever the Hessian of
\(f\) at \(x_0\) is regular, the function \(f\) is locally
topologically equivalent in local coordinates to a function
\[q(y)=f(x_0)-\sum_{j=1}^{l} y_j^2+\sum_{j=l+1}^n y_j^2\]
where \(l \in\{0,\ldots,n\}\) is the index of the critical point
\(x_0 \in M.\){~} A version of the
second Morse Lemma for piecewise smooth functions can be
found in \cite{jon3}. The latter result is the starting point for
our study.\\
\par
A version of Morse theory for smooth functions
restricted to nonsmooth subvarieties of a smooth manifold
is developed in \cite{goresky} (see also \cite{agrachev}).
In contrast to this work we shall be
concerned with extentions of Morse theory to nonsmooth functions
defined on smooth manifolds. Some preliminary work in this
direction has been done in \cite{Bonnisseau,Bougeard}.
Following \cite{jon3}, we shall use a critical point concept for locally
Lipschitz continuous functions based on Clarke's
generalized derivative \cite{cla} which is defined by
\[f^o(x_0,y)=\limsup_{x\to x_0\atop {\alpha\to 0\atop\alpha>0}}
\frac{f(x+\alpha y)-f(x)}{\alpha}.\]
The function \(f^o(x_0,.)\) is sublinear; its support set is called
Clarke's subdifferential of \(f\) at \(x_0\) and denoted by
\(\partial f(x_0)\); i.e.
\[f^o(x_0,y)=\max_{v\in\partial f(x_0)}\langle v,y\rangle.\]
The important chain rule [5] implies that the Clarke subdifferential of
a locally Lipschitz continuous function $f:M\longrightarrow\R$ defined on a
smooth manifold \(M\) has an
invariant meaning, i.e. does not depend on the choice of local coordinates
in $M$. More precisely, $\partial f(x)$ is a compact convex subset in
$T^*_xM.$
%
%----------------------------------------------------------------
\begin{definition} \label{critical}
Let \(M\) be a smooth $n$-dimensional manifold,
\(f:M \longrightarrow \R\) be a locally Lipschitz-continuous function.
A point \(x_0\in M\) is called a critical point of \(f\) if
\(0\in\partial f(x_0)\).
\end{definition}
%----------------------------------------------------------------
%
The following proposition extends the first Morse Lemma to
locally Lipschitzian functions.
\medskip\noindent
\begin{proposition}\label{MorseI}
Let $f:M\to R$ be a locally Lipschitz continuous
function, $M_t=f^{-1}((-\infty,t]).$ Suppose $M_\beta$ is compact and
$f^{-1}([\alpha,\beta])$  does not contain critical points of $f$.
Then there exists a Lipschitz continuous mapping \linebreak
$F:M_\beta\times [0,1]\to M_\beta$ such that
$$
F(M_\beta,t)\subset M_{\beta+t(\alpha-\beta)},\quad
F\mid_{M_\alpha\times[0,1]}=id.
$$
\end{proposition}
\medskip\noindent
{\bf Proof.} Take $x\in f^{-1}([\alpha,\beta])$.
Since $x$ is not a critical point for $f$, there
exists $v(x)\in T_xM$ such that
$\sup\limits_{\xi\in\partial f(x)}\langle\xi,v(x)\rangle<0.$
Choose a vector field \(v\) on \(M\) with value $v(x)$ at \(x\).
The upper semi-continuity of the Clarke subdifferential implies that
$\sup\limits_{\xi\in\partial f(y)}\langle\xi,v(y)\rangle<-\varepsilon_x$
for any $y\in O_x$, where $O_x$ is a neighborhood of $x$ in $M$ and
$\varepsilon_x >0$. The neighborhoods $O_x,\ x\in f^{-1}([\alpha,\beta])$,
form a covering of $f^{-1}([\alpha,\beta])$, let $O_{x_i},\
i=1,\ldots,m$, be a finite subcovering and let $v_i$ be the
corresponding smooth vector fields
on $M$, i.e.
$\sup\limits_{\xi\in\partial f(y)}\langle\xi,v_i(y)\rangle<-\varepsilon$,
$\forall y\in O_{x_i}$, where
\(\varepsilon=\min\limits_{1\leq i\leq m}\varepsilon_{x_i}\).

Suppose $\varphi_i,\ i=1,\ldots,m$, is a smooth partition of unity
subordinated to the covering $O_{x_i},\ i=1,\ldots,m$, of
$f^{-1}([\alpha,\beta])$. This means that $\varphi_i(x)\ge 0\ \forall x\in M$,
$\varphi_i(x)=0\ \forall x\in M\setminus O_{x_i},$ and
$\sum\limits_{i=1}^m\varphi_i(x)=1\ \forall x\in f^{-1}([\alpha,\beta])$.
We set $v(x)=\sum\limits_{i=1}^m\varphi_i(x)v_i(x)$. Then $v$ is a smooth
vector field on $M$;
$\sup\limits_{\xi\in\partial f(x)}\langle\xi,v(x)\rangle<-\varepsilon$
$\forall x\in f^{-1}([\alpha,\beta])$; $v(x)=0$
$\forall x\in M\setminus\bigcup\limits_{i=1}^m O_{x_i}.$
We may assume without lack of generality that
$\bigcup\limits_{i=1}^m O_{x_i}$ has a compact closure.
Then the vector field $v$ has a compact support
and is thus a complete field.

Let us denote by $t\mapsto y(t;x)$,
$x\in f^{-1}([\alpha,\beta])$, the solution of the differential equation
$\dot y=v(y)$ with the initial condition $y(0;x)=x$. Then
$t\mapsto f(y(t;x))$ is a scalar Lipschitz continuous function and the
derivative of this function, which is defined almost everywhere, satisfies
the inequalities $\frac d{dt}f(y(t;x))\le 0\ \forall t$ and
$\frac d{dt}f(y(t;x))<-\varepsilon$, if $f(y(t;x))\in [\alpha,\beta]$.
Hence there exists the Lipschitz continuous function
$\tau\mapsto a(\tau;x)$, $\tau\in [\alpha,f(x)]$, which is inverse to the
function $t\mapsto f(y(t;x))$. Moreover, it follows from the Lipschitzian
version of the implicit function theorem \cite{cla} that $a$ is Lipschitz
continuous as a function of the pair
$(\tau;x)$, where $x\in f^{-1}([\alpha,\beta])$, $\tau\in [\alpha,f(x)]$.
We have $f(y(a(\tau;x);x))=\tau$; thus
\[F(x,t)=\left\{\begin{array}{ll}
y(a((\alpha-f(x))t+f(x);x);x),&\mbox{if}~\alpha\le f(x)\le\beta;\\
x,&\mbox{if}~f(x)\le\alpha
\end{array}\right.\]
satisfies the assertion.\hfill\(\Box\)
%
%----------------------------------------------------------------
\section{The second Morse Lemma for piecewise smooth functions}
\begin{definition} \label{selection}
Let $M$ be an $n$-dimensional topological
manifold, and let \(f:M \longrightarrow \R\) and
\(f_1,\ldots,f_m:M \longrightarrow \R \) be continuous functions.
If \(I(x)=\{i \in \{1,...,m\} |f_i(x)=f(x)\}\) is nonempty at
every point \(x\in M \), then \(f\) is called a continuous selection
of the functions \(f_1,\ldots,f_m\). We denote by \(CS(f_1,\ldots,f_m)\)
the set of all continuous selections of \(f_1,\ldots,f_m\). The set
\(I(x)\) is called the active index set of \(f\) at the point \(x\).
\end{definition}
%----------------------------------------------------------------
%
Typical examples of continuous selection of the functions
\(f_1,\ldots,f_m\) are functions of the form
\[f(x)=\max_{i\in I}\min_{j\in J_i} f_j(x),\]
where \(I\) is a finite index set and \(J_i\subseteq \{1,\ldots,m\}\).
We shall confine ourselves to continuous selections of smooth functions.
A continuous selection \(f\) of \(C^1\)-functions
\(f_1,\ldots,f_m:\R^{n} \longrightarrow \R\) is locally Lipschitz
continuous and Clarke's subdifferential of \(f\) is given by
\[\partial f(x)=\conv\{\nabla f_i(x)|i\in\hat{I}(x)\},\]
where
\[\hat{I}(x)=\{i|x\in\closure\interior\{x|f(x)=f_i(x)\}\},\]
and "\(\conv\)", "\(\closure\)", and "\(\interior\)" denote the convex hull,
the closure, and the interior operators on the respective sets, \cite{scho}.
%
%
%           M o r s e    L e m m a   f o r   CS
%
%----------------------------------------------------------------
%-------------------------------------------------------
%
In the sequel we use the following notion of
a nondegeneracy critical point for continuous selections of
\(C^2\)-functions (cf. \cite{jon3}).
%
%----------------------------------------------------------------
\begin{definition} \label{nondegenerated}
Let \(M \) be a smooth $n$-dimensional manifold,
\(f_1,\ldots,f_m: M \longrightarrow \R ~~  C^2\)-functions, and
\(f\in CS(f_1,\ldots,f_m).\)
A critical point \(x_0 \in M \) of \(f\)
is called nondegenerate if the following two conditions hold:\\
\par
(ND\(1\))~~\it For each \(i\in\hat{I}(x_{0})\), the set
of differentials \(\{df_{j}(x_{0})|j\in\hat{I}(x_{0})\backslash\{i\}\}\)
is linearly independent; \\
\par
(ND\(2\))~~\it The second differential \(d^2 L(.,\hat{\lambda})(x_0)\)
of \(x \mapsto L(x,\hat{\lambda}) \) is regular on
\[\hat{T}(x_{0})=\bigcap_{i \in \hat{I}(x_{0})} \mbox{kern~}df_{i}(x_{0}),\]
where
\[L(x,\lambda)=\sum_{i\in\hat{I}(x_0)}\lambda_i f_i (x),\]
and the reals \(\hat{\lambda}_i\) are such that
\[dL(.,\hat{\lambda})(x_0)=0,~
\sum_{i\in\hat{I}(x_{0})}\hat{\lambda}_i =1,~
\hat{\lambda}_i\geq 0 \mbox{ for every \(i\in\hat{I}(x_{0})\)}.\]
The dimension of a maximal linear subspace of \(\hat{T}(x_{0})\)
on which \(d^2 L(.,\hat{\lambda})(x_0)\)
is negative definite is called the quadratic
index of the critical point $x_{0}.$
\end{definition}
%----------------------------------------------------------------
{~}\\
Note that the reals \(\hat{\lambda}_i,\) \(i\in\hat{I}(x_0)\),
in (ND2) exist and are unique in the presence of (ND1).
If \(f\) is twice continuously differentiable at a critical point,
then (ND2) corresponds to the classical nondegeneracy condition.
Geometrically speaking, a point \(x_0\) is a nondegenerate
critical point of the continuous selection \(f\) if
the graphs of the functions \(f_j,\) \(j\in\hat{I}(x_0)\), have
a transversal intersection at the point \((x_0,f(x_0))\) and
\(x_0\) is a nondegenerate critical point of the restriction
of the mapping \((x,f(x))\mapsto f(x)\) to the intersection of
these graphs. Note that the
intersection of the graphs is a smooth submanifold and that
\((x,f(x))\) is a smooth function on this submanifold.\\

The following result of \cite{jon3} is the basis for our study
(cf. also \cite{kuscho} for an alternative proof).\\
%
%----------------------------------------------------------------
\begin{theorem} \label{critical point} Let \(U \subseteq \R^{n}\)
be an open subset, \(x_{0} \in U, \)
 \(f_1,\ldots,f_m:U \longrightarrow \R\)
be twice continuously differentiable
functions, and let \(f\in CS(f_1,\ldots,f_m)\).
Then the
following two statements hold:\\
i) If \(x_0\) is a non-critical point of \(f\), then \(f\) and
\(f(x_0)+y_1\) are locally topologically equivalent at the
point \((x_0,0)\).\\
ii) If \(x_0\) is a nondegenerate critical point of \(f\), then
\(f\) is locally
topologically equivalent at \((x_0,0)\) to a function of the form
\[f(x_0)+l(y_1,\ldots,y_k)-\sum_{i=k+1}^{k+\mu}
y_{i}^{2}+\sum_{j=k+\mu+1}^n y_j^2,\]
where \(k=|\hat{I}(x_0)|-1,~l\in CS(y_1,\ldots,y_k,-\sum_{i=1}^k y_i),\)
and \(\mu\) is the quadratic index of \(f\) at \(x_0\).\\
Moreover, the set of all \(m\)-tuples \((f_1\ldots,f_m)\) of
\(C^2\)-functions
such that all critical points of every continuous selection of the
functions \(f_1\ldots,f_m\) are nondegenerate
is an open and dense subset in the strong \(C^2\)-topology {\rm (see
\cite{jon2}, Chapter 7).}
\end{theorem}
%----------------------------------------------------------------
%
We call a collection of linear
mappings \(l_1,....,l_m :\R^{n} \longrightarrow \R\) given by
\(l_i(x)= \langle a_i, x \rangle\) affinely independent
if the vectors
\[\\
\klamauf a_1\\1\\ \klamzu,\ldots,\klamauf a_m\\1\\ \klamzu \in\ \R ^{n+1}\\
\]
are linearly independent. The following representation result is
proved in \cite{barku}.
%
%----------------------------------------------------------------
\begin{theorem} \label{max-min}
Let \(l_1,\ldots,l_m\ \in \R^{n^{\ast}} \). If
\(l\in CS(l_1,\ldots,l_m),\) then
there exists a finite collection of index sets \(\{M_1,\ldots,M_r\}\)
with \(M_i\subseteq\{1,\ldots,m\}\) and
\(M_i\not\subseteq{M_j}\) for \(i\neq j\) such that
\[l(x)=~\max_{i\in{\{1,\ldots,r\}}} \min_{j\in{M_i}} l_{j}(x).\]
Moreover, if \(l_1,\ldots,l_m\) are affinely independent, then
the collection of index sets is unique.
\end{theorem}
%----------------------------------------------------------------
%
{~}\\
If \(l_1,\ldots,l_m\) are affinely independent, then the unique
max-min representation
$$l(x)=~\max_{i\in{\{1,\ldots,r\}}} \min_{j\in{M_i}} l_{j}(x)$$
of a function \(l\in CS(l_1,\ldots,l_m)\)
is called the max-min normal form of \(l\). Using the laws for
distributive lattices, this normal form
can be transformed into a min-max representation
\[l(x)=~\min_{k\in{\{1,\ldots,s\}}} \max_{j\in{N_k}} l_{j}(x)\]
where the index sets \(N_i\subseteq \{1,\ldots,m\} \)
are such that  \( N_i\neq\subseteq N_{j} \) for \(i\neq j\).
If \(l_1,\ldots,l_m\) are affinely independent, then the
collection of index sets \(\{N_1,\ldots,N_s\}\)
is again unique and the above representation is called the
min-max normal form of \(l\), \cite{barku}.\\
\par
For \(l\in CS(l_1,\ldots,l_{m})\) we denote by
\[ K_{+}(l) =\{x \in   \R^{n}   ~|~ l(x)~>~0 ~~\} \mbox{~~and~~}
K_{-}(l) =\{x\in   \R^{n}   ~|~l(x)~<~0 ~~\}, \]
the ascent- and descent cones of \(l\).
The max-min and min-max representations
\[ l(x) = \max_{i\in\{1,\ldots,r\}}\min_{j\in M_i} l_j(x)~
      =~ \min_{k\in\{1,\ldots,s\}}\max_{j\in N_k} l_j(x)\]
provide the following
decomposition of \(K_+(l)\) and \(K_-(l)\) into convex cones:
\[K_+(l)=\bigcup_{i=1}^r K_+^i, \mbox{~~~and~~~~~~~}
K_{-}(l)=\bigcup_{k=1}^s K_{-}^k,\]
with  \(K_+^i=\{ x\in \R^{n}~|~ l_j(x)>0,~j\in M_i\}\)
and \(K_{-}^k=\{ x\in \R^{n}~|~ l_j(x)<0,~j\in N_k\}.\)
{~}\\
%
%----------------------------------------------------------------
If (ND1) holds at the origin for \(l\in CS(l_1,\ldots,l_m),\)
then the combinatorial structure of
\(K_+(l)\) and \(K_-(l)\) is further characterized by the following
result from \cite{barku}.\\
\begin{proposition} \label{prop1}
Let \(l_1,\ldots,l_m \in \R^{n^{\ast}} \) and \(l\in CS(l_1,\ldots,l_m)\).
If \(l\) satisfies (ND1) at the origin, then
\begin{itemize}
\item[ a)] \(\dim\closure K^i_+\cap\closure K^j_+\in\{0,n\}\) and
\(\dim\closure K^i_+\cap\closure K^j_+=n\) if and only if
\(|M_i\cup M_j|\leq n\).
\item[ a)] \(\dim\closure K^k_-\cap\closure K^l_-\in\{0,n\}\) and
\(\dim\closure K^k_-\cap\closure K^l_-=n\) if and only if
\(|N_k\cup N_l|\leq n\).
\end{itemize}
\end{proposition}
%----------------------------------------------------------------
%
%
%           H o m o t o p y   T y p e   of the Lower-Level S e t
%
%----------------------------------------------------------------
\section{The Homotopy Type of the Lower-Level Set }
%----------------------------------------------------------------
%
%---------------------------------------------------------
%
\begin{proposition} \label{prop2}
Let \(l\in CS(l_1,\ldots,l_k)\) with \(l_i\in\R^{n^\ast}\)
such that \((ND1)\) holds at the origin,
and let \(q: \R^{m} \longrightarrow \R\) be a quadratic function of the form
\[q(v)=-\sum_{j=1}^{\mu} v_j^2+\sum_{j=\mu+1}^m v_j^2.\]
Then the sets
\[ {\cal L}=\{(u,v) \in \R^{n} \times \R^{m}~|~ l(u)+q(v) ~<~0 ~\}\]
and
\[ {\cal L}_{0} =\{(u,v) \in \R^{n} \times \R^{m}~|~ l(u)~\leq~0 ~,
q(v)~\leq ~0 \mbox{~and~} (l(u),q(v))\not=(0,0)~\} \]
are homotopy equivalent.
\end{proposition}
%---------------------------------------------------------
{\bf Proof:}
Let \( i: {\cal L}_{0}  \hookrightarrow  {\cal L}\)
denote the inclusion map. We shall construct a
retraction \(R: {\cal L}  \longrightarrow  {\cal L}_{0}\)
such that \( R \circ i = \mbox{Id}_{{\cal L}_{0}} \)
and that \( i \circ R \simeq \mbox{Id}_{{\cal L}}\)
(\(i\circ R\) is homotopic to \( \mbox{Id}_{{\cal L}}).\)\\
\par
Note first that, by Proposition \ref{prop1}, the sets
\[K_{l} =\{u \in \R^{n}~|~ l(u)~\leq~0 \} \subset \R^{n} \mbox {~and~}
K_{q}= \{v \in \R^{m}~|~ q(u)~\leq~0 \} \subset \R^{m} \]
are locally finitely triangulable and contractible and are thus absolute
retracts in view of \cite{hu} Chapter III, Corollary 6.4 and Proposition 7.2.
Hence there exist retractions
\(\rho: \R^{n} \longrightarrow K_{l}\) and
\(\sigma: \R^{m} \longrightarrow K_{q}\) and thus the mapping
\(R: {\cal L} \longrightarrow  K_{l} \times K_{q}\) defined by
\[R(u,v)=(\rho(u),\sigma(v)) \]
is a retraction as well.
%
%Note that the canonical projection of
%\({\cal L}\) into \(u\)-space is surjective if
%\(\mu\geq 1\) and that the canonical projection into
%\(v\)-space is surjective, provided there exists a vector
%\(u\) with \(l(u)<0\).\\
%
By definition, \( R \circ i = \mbox{Id}_{{\cal L}_{0}} \) since for every
\( u \in K_{l} \)  and  every  \( v \in K_{q} \) we have
\( \rho(u) =u \) and \( \sigma(v)=v .\)
Moreover, \(R({\cal L}) \subset   {\cal L}_{0}\). In fact,
if \((u,v)\in {\cal L}\) then either \(l(u)<0\) and hence \(\rho(u)=u\) or
\(q(v)<0\) and hence \(\sigma(v)=v\). In both cases we obtain
\((l(\rho(u)),q(\sigma(v)))\neq (0,0)\).\\

Hence, it remains to be shown that \(i \circ R\) is homotopic
to the identity \(\mbox{Id}_{ {\cal L} }\).
To this end we consider the function
\(H: {\cal L} \times [0,1] \longrightarrow  \R^n\times\R^m\)
defined by
\[ H((u,v),t)= \left\{ \begin{array}{l@{\quad:
\quad}l}
(u~,~v) & l(u) \leq 0,~ q(v) \leq 0    \\
((1-t) \cdot \rho(u)+t \cdot u~,~\eta(u,v,t) \cdot v) & l(u) > 0,~ q(v) < 0  \\
(\zeta(u,v,t)u~,~(1-t) \cdot \sigma(v)+t \cdot v) & l(u) < 0,~ q(v) > 0  \\
\end{array} \right.\]
with
\[\eta(u,v,t)= \left\{ \begin{array}{l@{\quad {~}
\quad}l}
\sqrt{ 1- \displaystyle\frac{\max \{ l((1-t) \rho(u)+tu)-l(u),0\}}{q(v)}}
&,~~ l(u) > 0 \\
&~~~ q(v) < 0   \\
{~} & {~} \\
 1    &,~~ l(u) \cdot q(v) = 0 \\
 &~~~ l(u)+q(v) < 0
\end{array} \right.\]
\[\zeta(u,v,t)= \left\{ \begin{array}{l@{ {~}
\quad}l}
 1-\displaystyle\frac{(1-t^2) q(v) + 2t(t-1)
 (\sum\limits_{j=\mu+1}^m \sigma(v)_{j} v_{j})-
 \sum\limits_{j=1}^{\mu}  \sigma(v)_{j} v_{j}) }
 {l(u)} &,~~ l(u) < 0 \\
 &~~~ q(v) > 0   \\
{~} & {~} \\
 1    &,~~ l(u) \cdot q(v) = 0\\
 &~~~ l(u)+q(v) < 0.
\end{array} \right.\]
By definition \(H\) is continuous on the open set
\[\{(u,v)\in {\cal L}\times [0,1]~|~l(u)>0, q(v)<0~~\mbox{or}~~
l(u)<0, q(v)>0\}.\]
Moreover, if \((u_n,v_n,t_n)_{n \in \N} \) is a sequence in
\({\cal L}\times [0,1]\) which converges to a point
\((u_{0},v_{0},t_0) \in {\cal L}\times[0,1]\) not
contained in the above set, then
\(H(u_n,v_n,t_n)\to (u_0,v_0)\). In fact, if
\(l(u_0)=0\) and \(q(v_0)<0\), then  \(\rho(u_n)\to\rho(u_0)=u_0\) and
thus \(\eta(u_n,v_n,t_n)\to 1\), and if
\(q(v_0)=0\) and \(l(u_0)<0\), then \(\sigma(v_n)\to\sigma(v_0)=v_0\) and
thus \(\zeta(u_n,v_n,t_n)\to 1\). Note that \((l(u_0),q(v_0))\neq (0,0)\)
since \((u_0,v_0)\in {\cal L}\). Hence we conclude that
\(H\) is continuous on \({\cal L}\times [0,1]\).\\
Next we show that \(H(u,v,t)\in {\cal L}\) for every
\(t\in [0,1]\), provided \((u,v)\in {\cal L}\).
If \(l(u)\leq 0\) and \(q(v)\leq 0\), then \(H(u,v,t)=(u,v)\), so that
it suffices to consider the case that at least one of the values
\(l(u)\) and \(q(v)\) is positive. If, on the one hand,
\(l(u)>0\), then \(q(v)<0\) since \((u,v)\in {\cal L}\) and thus
\(H(u,v,t)=((1-t)\rho(u)+tu,\eta(u,v,t) v)\). Note that
\(\eta=\eta(u,v,t)\) solves the equation
\[ \max \{ l((1-t) \rho(u)+tu)-l(u),0\} = (1-\eta^{2}(t)) \cdot q(v). \]
Hence
\begin{eqnarray*}
l((1-t) \rho(u)+tu)+ q(\eta(u,v,t)v)
&=& l(u) + (l((1-t) \rho(u)+tu)-l(u)) + q(v)\\
&~&  \qquad + (q(\eta(u,v,t)v) - q(v))\\
&\leq & l(u) +q(v) + \max \{ l((1-t) \rho(u)+tu)-l(u),0\} \\
&{~}& \qquad + (1-\eta(u,v,t)^{2}) \cdot q(v)\\
&=& l(u)+q(v) <0 ,
\end{eqnarray*}
and thus \(H(u,v,t)\in {\cal L}\). If, on the other hand,
\(l(u)<0\) and \(q(v) >0 \), then
\(H(u,v,t)=(\zeta(u,v,t)u,(1-t)\sigma(v)+tv)\) and
\begin{eqnarray*}
l(\zeta(u,v,t) u)+ q((1-t) \sigma(v)+tv)
&=& \zeta(u,v,t)l(u) -\sum_{j=1}^{\mu} ((1-t) \sigma(v)_{j}+tv_{j})^2\\
&~& \qquad \qquad + \sum_{j=\mu+1}^m ((1-t) \sigma(v)_{j}+tv_{j})^2 \\
&=& \zeta(u,v,t)l(u) -(1-t)^{2} \sum_{j=1}^{\mu} ( \sigma(v)_{j})^2+
(1-t)^{2} \sum_{j=\mu+1}^m (\sigma(v)_{j})^2\\
&{~}& \qquad -{~~}t^{2} \sum_{j=1}^{\mu} ( \sigma(v)_{j})^2+
t^{2} \sum_{j=\mu+1}^m (\sigma(v)_{j})^2 \\
&~& \qquad -2t(1-t) \sum_{j=1}^{\mu}  \sigma(v)_{j} v_{j}+
2t(1-t) \sum_{j=\mu+1}^m \sigma(v)_{j} v_{j}\\
&\leq & \zeta(u,v,t)l(u) + t^{2} q(v)
-2t(1-t) (\sum_{j=1}^{\mu}  \sigma(v)_{j} v_{j}+
\sum_{j=\mu+1}^m \sigma(v)_{j} v_{j}),
\end{eqnarray*}
since
\[q(\sigma(v))=-\sum_{j=1}^\mu (\sigma(v)_j)^2+
\sum_{j=\mu+1}^{m} ( \sigma(v)_{j})^2 \leq 0. \]
Hence we obtain
\begin{eqnarray*}
l(\zeta(u,v,t) u)+ q((1-t) \sigma(v)+tv)
&\leq & \zeta(u,v,t)l(u) + t^{2} q(v)
-2t(1-t) (\sum_{j=1}^{\mu}  \sigma(v)_{j} v_{j}+
\sum_{j=\mu+1}^m \sigma(v)_{j} v_{j})\\
&=& (\zeta(u,v,t)-1) l(u) +l(u) + (t^{2}-1) q(v)\\
&~& \qquad \qquad +q(v)
-2t(1-t) (\sum_{j=1}^{\mu}  \sigma(v)_{j} v_{j}+
\sum_{j=\mu+1}^m \sigma(v)_{j} v_{j})\\
&=& l(u) + q(v) < 0 ,
\end{eqnarray*}
in view of the fact that \(\zeta=\zeta(u,v,t)\) solves
\[ (\zeta-1)\cdot l(u)= (1-t^2)q(v)+ 2t(t-1)
 (-\sum_{j=1}^{\mu}  \sigma(v)_{j} v_{j}+
 \sum_{j=\mu+1}^m \sigma(v)_{j} v_{j}).\]
\par
We conclude that the path \( t \mapsto H((u,v),t) \) does not leave
the set \({\cal L}.\) We have thus proved the proposition since
\(H(u,v,0)=R(u,v)\) and \(H(.,.,1)= Id_{\cal L}\).\hfill{$\Box$}
%
%---------------------------------------------------------
%
{~}\\
{~}\\
%
%---------------------------------------------------------
%
%
\begin{proposition} \label{prop3}
Let \(l\in CS(l_1,\ldots,l_m)\) with
\(l_1,\ldots,l_{n+1}\ \in \R^{n^{\ast}}\) and suppose (ND1) holds
at the origin. Let \(q: \R^{m} \longrightarrow \R\) be a quadratic function
of the form
\[q(v)=-\sum_{j=1}^{\mu} v_j^2+\sum_{j=\mu+1}^m v_j^2\]
Then the sets
\[ {\cal L}_{0} =\{(u,v) \in \R^{n} \times \R^{m}~|~ l(u)~\leq~0 ~,
q(v)~\leq ~0 \mbox{~and~} (l(u),q(v))\not=(0,0)~\} \]
and
\[ {\cal L}_{0}^{\ast} =\{(u,v) \in {\cal L}_{0}~|~ l(u)~<~0 ~,
\mbox{~for~} u \not= 0 \mbox{~ and~} q(v)~< ~0 \mbox{~for~} v\not= 0 ~\} \]
are homotopy equivalent.
\end{proposition}
%
%---------------------------------------------------------
%
{\bf Proof:}
%For the ascent cones
%\[ K_{-}(l) = \{u \in \R^{n}~|~ l(u) < 0 \} \mbox{~~~and~~~}
% K_{-}(q) = \{v \in \R^{m}~|~ q(v) < 0 \} \]
%of the function \(l\) and \(q\)
%we construct and increasing family of cones
%\[ K_{-}(l_{\varepsilon})
%           = \{u \in \R^{n}~|~ l_{\varepsilon}(u) < 0 \}
%\mbox{~~~and~~~}
% K_{-}(q_{\varepsilon})
%           = \{v \in \R^{m}~|~ q_{\varepsilon}(u) < 0 \} \]
%with
%\[ K_{-}(l) =
%\bigcup_{0 < \varepsilon < {\varepsilon}_{0}} K_{-}(l_{\varepsilon})
%\mbox{~~~and~~~}
%K_{-}(q) =
%\bigcup_{0 < \varepsilon < {\varepsilon}_{0}} K_{-}(q_{\varepsilon})\]
%and such that for every \( u \in  K_{-}(l_{\varepsilon})\setminus\{0\} \)
%and for every \( v \in  K_{-}(q_{\varepsilon})\setminus\{0\} \)
%the strict inequalities \(l(u) <0 \) and \(q(v) <0 \) hold.\\
%\par
Let \(l_S:S^{n-1}\to\R\) be defined by \(l_S(u)=l(u)\) for every
\(u\in S^{n-1}\). Clearly \(l_S\) is piecewise smooth and
\(u\in S^{n-1}\) is a critical point of \(l_S\) if and only if
there exists a scalar \(\lambda\) such that
\(\lambda u\in\partial l(u)\). Let \(I\subseteq\{1,\ldots,n+1\}\)
with \(|I|\leq n\). Note that, in view of (ND1),
\[\alpha_I=\min\{\|\sum_{i\in I}\lambda_i a_i\|_2~|~\lambda_i\geq 0,
\sum_{i\in I}\lambda_i=1\}>0.\]
Let \(\alpha<\alpha_I\) for every \(I\subseteq\{1,\ldots,n+1\}\)
with \(|I|\leq n\). We
show that \(l_S^{-1}([-\alpha,\alpha])\) does not contain a critical point.
Let \(u\in S^{n-1}\) and \(I(u)=\{i~|~l_i(u)=l(u)\}\). Note that
\(|I(u)|\leq n\) since \(l_i(u)=l_j(u)\) for every \(i,j\in\{1,\ldots,n+1\}\)
implies \(u=0\), in view of (ND1). If \(u\) is a critical point of
\(l_S\), then there exists a scalar \(\lambda\) and convex
multipliers \(\lambda_i\) such that
\(\lambda u=\sum_{i\in I(u)}\lambda_i \nabla l_i(u)\). Note that
\(\lambda=l(u)\) in view of the definition of \(I(u)\) and
that \(\|\sum_{i\in I(u)}\lambda_i\nabla l_i(u)\|\geq \alpha\).
Thus \(l(u)\geq\alpha\).
We conclude from Proposition \ref{MorseI} and the homogeneity of the
function \(l\) that there exists a continuous mapping
\[ R:\mbox{cl}K_{-}(l) \times [0,1] \longrightarrow \mbox{cl}K_{-}(l)\]
with \(R(u,t)<0\) for every \(t\in (0,1]\) and every
\(u\in\mbox{cl}K_-(l)\backslash\{0\}\).\\
\par
%
By assumption, the function \(q\) is given by:
\[q(v)=- \sum_{j=1}^{\mu} v_{j}^2+\sum_{j=\mu+1}^m v_{j}^2.\]
Now let \(0 < \varepsilon < {\varepsilon}_{0} \) be sufficiently small
and define
\[ {q}_{\varepsilon}:\R^{m} \longrightarrow \R \mbox{~~by~~}
{q}_{\varepsilon}(v)=-(1+\varepsilon)\sum_{j=1}^{\mu} v_j^2
+\sum_{j=\mu+1}^m v_j^2.\]
Then
\[ K_{-}(q_\varepsilon) = \{x \in \R^{m}~|~ q_\varepsilon(x) < 0 \}
\subset_{\atop{\!\!\!\!\! \not=}}
   K_{-}(q) = \{x \in \R^{m}~|~ q(x) < 0 \}.\]
In fact, the definition of \({q}_{\varepsilon}\) implies that \(q(u)<0\)
for every \(u \in\mbox{cl}K_{-}({q}_{\varepsilon})\setminus\{0\}\).\\
\par
By integrating the vectorfield of inner normals on the level sets
one obtains a family of homeomorphisms
\[ d_{\varepsilon}: K_{-}(q) \longrightarrow K_{-}(q_{\varepsilon})\]
which depends continuously on the parameter \(\varepsilon \)
and gives a homotopy
\[D: \mbox{cl}(K_{-}(q)) \times [0,1] \longrightarrow \mbox{cl}(K_{-}(q),
~~~~~D(v,t)= d_{t \cdot \varepsilon_{0}}(v) \]
between \(\mbox{Id}_{K_{-}(q)}\) and \(d_{\varepsilon_{0}}.\)\\
\par
%
In order to see that the cones
\[ {\cal L}_{0} =\{(u,v) \in \R^{n} \times \R^{m}~|~ l(u)~\leq~0 ~,
q(v)~\leq ~0 \mbox{~and~} (l(u),q(v))\not=(0,0)~\} \]
and
\[ {\cal L}_{0}^{\ast} =\{(u,v) \in \R^{n} \times \R^{m}~|~ l(u)~<~0 ~, \mbox{~~for~~}
u \not= 0 \mbox{~~and~~}
q(v)~<~0 \mbox{~for~~} v \not= 0~\} \]
are homotopy equivalent, not first that
\[ {\cal L}_{0}^{\ast} \subset {\cal L}_{0} \]
and that
\[W:  {\cal L}_{0} \longrightarrow {\cal L}_{0}^{\ast} \]
with
\[ W(u,v) = (R(u,q(v)), D(v,l(u))) \]
maps \( {\cal L}_{0} \) onto \( {\cal L}_{0}^{\ast} \)
since for every \((u,v) \in {\cal L}_{0}\) we have \((l(u),q(v)) \not=
(0,0). \)  Moreover \(W\) is homotopic to the identity by the above
construction.  \hfill{$\Box$}
%
{~}\\
%
%----------------------------------------------------------------
\begin{proposition} \label{prop4}
Let \(l_1,\ldots,l_{n+1}\ \in \R^{n^{\ast}} \) be
the following linear functions:
\[ l_{i}(u)=u_{i} \mbox{~~for ~~} i \in \{1,...,n\} \mbox{~~and~~}
l_{n+1}(u) = -\sum_{j=1}^{n} u_j \]
and let  \(l\in CS(l_1,\ldots,l_m)\) be a
continuous selections.
Furthermore let \(q: \R^{m} \longrightarrow \R\) be given by:
\[q(v)=-\sum_{j=1}^{\mu} v_j^2+\sum_{j=\mu+1}^m v_j^2\]
Then the sets
\[ {\cal L}_{0}^{\ast} =\{(u,v) \in {\cal L}_{0}~|~ l(u)~<~0 ~,
\mbox{~for~} u \not= 0 \mbox{~ and~} q(v)~< ~0 \mbox{~for~} v\not= 0 ~\} \]
and
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1} =
\{(u,v) \in {\cal L}_{0}^{\ast}~|~
 \langle u,u\rangle ~+~\langle v,v \rangle =1 ~\} \]
are homotopy equivalent.
\end{proposition}
%
%----------------------------------------------------------------
%
{\bf Proof:} Since \((0,0) \in \R^{n+m} \) is not contained in
\( {\cal L}_{0}^{\ast},\)
the retraction \( \displaystyle{x \mapsto \frac{x}{\Vert x \Vert}}\)
of\\
\( \R^{n+m} \setminus \{(0,0)\} \) to the
\( (n+m-1)\)-sphere gives the homotopy equivalence between
\[ {\cal L}_{0}^{\ast} =\{(u,v) \in {\cal L}_{0}~|~ l(u)~<~0 ~,
\mbox{~for~} u \not= 0 \mbox{~ and~} q(v)~< ~0 \mbox{~for~} v\not= 0 ~\} \]
and
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1} =
\{(u,v) \in {\cal L}_{0}^{\ast}~|~
 \langle u,u\rangle ~+~\langle v,v \rangle =1 ~\} \]
 \hfill{$\Box$}
{~}\\
{~}\\
The join \(X\star Y\) of two topological spaces
\(X\) and \(Y\), \cite{bred,span}, is defined to be the quotient
space of the product \(X \times Y \times [0,1]\) by the decomposition
whose elements are the sets \(\{ \{x\} \times Y \times \{0\} ~|~
 x \in X\} \cong X  \) and \(\{X \times \{y\} \times \{1\}~|~
 y \in Y\} \cong Y \) and the individual points of the
set \(X \times Y \times [0,1] \setminus
(X \times Y \times \{0\}) \cup (X \times Y \times \{0\}).\)
With this definition we obtain the following homotopy equivalence
(cf. \cite{bred} p. 468 )
%
%----------------------------------------------------------------
%
\begin{proposition} \label{join}
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1} \cong
\left\{ \begin{array}{r@{\quad:
\quad}l}
\{u \in \R^{n}~|~ l(u) ~<~0 ~\} \star S^{\mu -1}  \mbox{~~for~~}
 &   \mu \geq 1\\
\{v \in \R^{m}~|~ l(u) ~<~0 ~\}
 &  \mu =0
\end{array}  \right.  \]
\end{proposition}
%
%----------------------------------------------------------------
%
{\bf Proof} Assign
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1}  \longrightarrow
\{u \in \R^{n}~|~ l(u) ~<~0 ~\} \star
\{v \in \R^{m}~|~ q(u) ~<~0 ~\} ,\]
with
\[ (u,v) \mapsto ((u,v)~,~\langle u,u \rangle ).\]
Since the lower level set of the function \(q\), i.e.
\(\{v \in \R^{m}~|~ q(u) ~<~0 ~\} \) is homotopy equivalent
to the \(\mu-1\)-sphere for \(\mu\geq 1\), we obtain
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1} \simeq
\{u \in \R^{n}~|~ l(u) ~<~0 ~\} \star S^{\mu-1}.\]
For \( \mu=0\) we have
\[ {\cal L}_{0}^{\ast} \cap S^{n+m-1} \simeq
\{u \in \R^{n}~|~ l(u) ~<~0 ~\} \]
since
\(\{v \in \R^{m}~|~ q(u) ~<~0 ~\} \) is contractible. \hfill{$\Box$}
{~}\\
{~}\\
%
%----------------------------------------------------------------
%
\begin{proposition} \label{prop5}
Let \(l_1,\ldots,l_{n+1}\ \in \R^{n^{\ast}} \) be
the following linear functions:
\[ l_{i}(u)=u_{i} \mbox{~~for ~~} i \in \{1,...,n\} \mbox{~~and~~}
l_{n+1}(u) = -\sum_{j=1}^{n} u_j \]
and let  \(l\in CS(l_1,\ldots,l_m)\) be a
continuous selections.
Furthermore let \(q: \R^{m} \longrightarrow \R\) be given by:
\[q(v)=-\sum_{j=1}^{\mu} v_j^2+\sum_{j=\mu+1}^m v_j^2\]
and let
\[ {\cal L} =\{(u,v) \in \R^{n} \times \R^{m} ~|~ l(u)~+q(v)<~0 ~\}.\]
Then
\[ H_{p}({\cal L})= \left\{ \begin{array}{lcl}
H_{p-\mu}(\{ u \in \R^{n}~|~ l(u)~<~0~\}) &:& p ~\geq~1,\mu \geq 0  \\
{~} &{~~}   & {~~~}\\
\Z &:& p~=~0, ~~\mu > 0 \\
H_{0}(\{ u \in \R^{n}~|~ l(u)~<~0~\})  &:& p~=~0, ~~\mu=0
\end{array} \right. \]
for the singular homology in $\Z$.
\end{proposition}
%
%----------------------------------------------------------------
%
{\bf Proof:} Since
\begin{eqnarray*}
{\cal L} &=& \{(u,v) \in \R^{n} \times \R^{m} ~|~ l(u)~+q(v)<~0 ~\} \\
 &\simeq& \{u \in \R^{n}~|~ l(u) ~<~0 ~\} \star S^{\mu-1},
\end{eqnarray*}
and a join with a sphere is a suspension, we obtain that, by an analogue of the
K\"unneth formula (see for instance \cite{bred}  Chapter IV,
Problem 6.1),
\[ H_{p}({\cal L})=
H_{p-\mu}(\{ u \in \R^{n}~|~ l(u)~<~0~\}) \]
for \( p \geq 1\) and \( \mu \geq 0 \).
For \(p=0\) and \(\mu > 0\) the join of
\(\{v \in \R^{m}~|~ q(u) ~<~0 ~\} \) with the
\(\mu-1\)-sphere is pathwise connected, i.e.
\( H_{0}({\cal L})= \Z.\)
For \(p=0\) and \(\mu = 0\) the assertion
follows from
\( {\cal L}_{0}^{\ast} \cap S^{n+m-1} \simeq
\{u \in \R^{n}~|~ l(u) ~<~0 ~\}.\) {~~~}
\hfill{$\Box$}
%
%           M o r s e    I n q u l i t i e s
%
%----------------------------------------------------------------
\section{The Morse Inequalities }
%----------------------------------------------------------------
%
Let \(l_1,\ldots,l_{n+1}\) be a collection of linear functions such
that (ND1) holds. Then a continuous selection
\(l\in CS(l_1,\ldots,l_{n+1})\) has a unique min-max normal form
\[ l(x) ~=~ \min_{k\in\{1,\ldots,s\}}\max_{j\in N_k} l_j(x),\]
\cite{barku}. The continuous selection \(l\) is thus described by the
system of sets
\(\displaystyle{\{(\bar{N}_{k})~|~k \in \{1,...,s\}\} } .\)
Proposition \ref{prop1} b) implies that
\( K_{-}(l) \) has the homotopy type of the simplicial complex
\({\cal N}_{l} \) which is the nerve (see \cite{span}) of
the system of sets
\( \displaystyle{\{(\bar{N}_{k})~|~k \in \{1,...,s\}\} } ,\)
i.e. the simplicial complex  \({\cal N}_{l} \)
which consists of the
vertices \(\bar{N}_1,...,\bar{N}_s \) together with all symplices
\( \Delta(\bar{N}_{j_{1}},...,\bar{N}_{j_{k+1}}) \) for which
 \(\bar{N}_{j_{1}}\cap,...,\cap \bar{N}_{j_{k+1}}\not= \emptyset. \)
Recall that \(\bar{N}_j\) is the set
\(\{1,\ldots,n+1\}\setminus N_j\). There is a simple simplicial
complex with the same nerve as \({\cal N}_l\) and only
\((n+1)\) vertices. Take the standard simplex with \((n+1)\) vertices.
Every proper subset of \(\{1,\ldots,n+1\}\) defines a face of the
simplex. Our complex is the union of the faces defined by the
subsets \(\bar{N}_j\). Note that the number of vertices of this
comples does not exceed the dimension of the manifold plus one.
Note that, by Proposition \ref{prop1},
for every  \(l\in CS(l_1,\ldots,l_{n+1})\)
with
\[l(x) ~=~ \min_{k\in\{1,\ldots,s\}}\max_{j\in N_k} l_j(x)\]
the collection
\( \displaystyle{\{(\bar{N}_{k})~|~k \in \{1,...,s\}\} } \)
satisfies
\[ \bigcup_{ k=1}^{s}  N_k =\{1,...,n+1\};\]
hence
\[ \bigcap_{ k=1}^{s} \bar{N}_k =\emptyset.\]
Thus all subcomplexes of the standard simplex appear as
representations of \({\cal N}_{l}, ~l\in CS(l_1,\ldots,l_{n+1})\),
except the homotopically trivial ones, where all simplices of the
complex have a common point.\\
\par
Following \cite{milnor}, the Morse inequalities are expressed
in terms of relative homology of lower level sets. To this end,
let \(M\) be a smooth manifold,
\(f_1,\ldots,f_m:M \longrightarrow \R\)
be twice continuously differentiable
functions, \(f\in CS(f_1,\ldots,f_m)\), and let
\(x_{0} \in M \) be a nondegenerate critical point of \(f\).
The Morse Lemma says that there is a system of coordinates
\((u,v)\) defined in a neighbourhood \(Q_{x_0}\) of \(x_0\) such that
in these coordinates the function \(f\) is of the form
%
$$ f(x_0)+ \min_{r\in\{1,\ldots,s\}}\max_{j\in N_r} l_j(u)
-\sum_{i=k+1}^{k+\mu} v_{i}^{2}+\sum_{j=k+\mu+1}^m v_j^2,  \eqno (*) $$
where \(k=|\hat{I}(x_0)|-1,\) every \(l_j\) is one of the functions
\(u\to u_i\) or \(u\to-\sum_{i=1}^k u_i\),
and \(\mu\) is the quadratic index of \(f\) at \(x_0\).\\
\par
Consider the lower level sets
\[ Mf_{\leq}(x_0) =  \{ x \in Q_{x_{0}}~|~ f(x)~\leq~f(x_{0})~\}
\mbox{~~and~~}
Mf_{<}(x_0) = \{ x \in Q_{x_{0}}~|~ f(x)~<~ f(x_{0})~\}.\]
We shall calculate the local homology groups
\[ H_{p}(f,x_{0})~=~H_{p}( Mf_{\leq}(x_0),Mf_{<}(x_0))\]
with coefficients in \(\Z.\)\\
%
%----------------------------------------------------------------
%
\begin{theorem} \label{homolog}
If \(x_0\) is a nondegenerate critical point of \(f\) with the normal
form $(*)$ and
\(Mf_{<}(x_0)\not= \emptyset,\)
then the singular homology groups can be computed in the following
way:
\begin{itemize}
\item[ a)] If \(k=0\) then:
%
\[ H_{p}(f,x_{0})~= \left\{ \begin{array}{lcl}
\Z &:& p = \mu \\
{~} &{~~}   & {~~~}\\
0 &:& \mbox{otherwise.}
\end{array} \right. \]
%
\item[ b)] If \(k \not= 0\) then:
\[ H_{p}(Mf_{<}(x_0))= \left\{ \begin{array}{lcl}
H_{p-\mu}({\cal N}_{l}) &:& p ~\geq~1, \\
{~} &{~~}   & {~~~}\\
\Z &:& p~=~0, ~~\mu \not= 0, \\
H_{0}({\cal N}_{l})  &:& p~=~0, ~~\mu=0,
\end{array} \right. \]
where \({\cal N}_l\) is the simplicial complex with $k+1$ vertices
and the maximal simplices $\bar N_1,\ldots,\bar N_s$.
\end{itemize}
\end{theorem}
%
%----------------------------------------------------------------
%
{\bf Proof:} Part $a)$ can be shown as in the book of
 J.Milnor (\cite{milnor}, p. 29) and part $ b)$  follows immediately
from Proposition \ref{prop5}
and Theorem \ref{critical point}.
\hfill{$\Box$}
{~}\\
{~}\\
%
%
%----------------------------------------------------------------
\begin{theorem} \label{local homology}
Let \(x_0\) be a nondegenerate critical point of \(f\).
\begin{itemize}
\item[ a)]
If \(f\) has a local minimum in \(x_0 \in M, \) i.e.
\( Mf_{<}(x_0) = \emptyset \), then
\[ H_{p}(f,x_{0})~=
   H_{p}( Mf_{\leq}(x_0),\emptyset)~=
   H_{p}( Mf_{\leq}(x_0))
=~\left\{ \begin{array}{lcl}
\Z &:& p =0 , \\
{~} &{~~}   & {~~~}\\
0 &:& p~\not=~0.
\end{array} \right. \]
\item[ b)]
If \( Mf_{<}(x_0) \not= \emptyset \), then
\[ H_{p}(f,x_{0})~=
   H_{p}( Mf_{\leq}(x_0), Mf_{<}(x_0))~=
~\left\{ \begin{array}{lcl}
   H_{p-1}(Mf_{<}(x_0))  &:& p \geq 2 , \\
{~} &{~~}   & {~~~}\\
\Z^{\rho-1} &:& p~=1,\\
 0          &:& p~=0,
\end{array} \right. \]
\end{itemize}
where \(\rho\) is the number of connected components of the
set \( Mf_{<}(x_0)\) (with \(\Z^0=0\)).
\end{theorem}
%
%----------------------------------------------------------------
%
{\bf Proof:} Since a nondegenerate local minimizer of
\(f\) is strict, we obtain
\[ H_{p}(f,x_{0})~=
   H_{p}( Mf_{\leq}(x_0),\emptyset))~=
   H_{p}( Mf_{\leq}(x_0))
=~\left\{ \begin{array}{lcl}
\Z &:& p =0 , \\
{~} &{~~}   & {~~~}\\
0 &:& p~\not=~0,
\end{array} \right. \]
{~}\\
Now suppose \( Mf_{<}(x_0) \not= \emptyset. \)
Since, by Theorem \ref{critical point}, the set \(Mf_{\leq}(x_0)\)
is contractible, the long exact homology sequence
\[ \begin{array}{ccccccccc}
\rightarrow & H_{p}( Mf_{\leq}(x_0)) & \rightarrow
H_{p}( Mf_{\leq}(x_0),Mf_{<}(x_0))& \rightarrow &
H_{p-1}(Mf_{<}(x_0)) & \rightarrow & H_{p-1}( Mf_{\leq}(x_0))
& \rightarrow \\
{~} & \parallel & {~} & {~} & {~} & {~} & \parallel & {~}\\
\rightarrow & 0 & \rightarrow
H_{p}( Mf_{\leq}(x_0),Mf_{<}(x_0))& \rightarrow &
H_{p-1}(Mf_{<}(x_0)) & \rightarrow & 0 & \rightarrow
\end{array} \]
implies the isomorphism
\[ H_{p}( Mf_{\leq}(x_0),Mf_{<}(x_0)) \cong H_{p-1}(Mf_{<}(x_0)) \]
for \(p\geq 2\).
For \(p = 1\), the long exact sequence of the
reduced homology (see \cite{span}) implies that
\[ H_{1}( Mf_{\leq}(x_0),Mf_{<}(x_0)) \cong
\tilde{H}_{0}(Mf_{<}(x_0)). \]
Hence, if the set \((Mf_{<}(x_0))\) has \(\rho\) connected
components, then
\[ H_{1}( Mf_{\leq}(x_0),Mf_{<}(x_0))
=\Z^{\rho -1}\]
It remains to consider the case \(p=0.\) Here we use again the long
exact homology sequence:
\[0~\rightarrow~H_{1}( Mf_{\leq}(x_0),Mf_{<}(x_0))
~\stackrel{\partial_{*}}{\rightarrow}~ \] \[ H_{0}
( Mf_{<}(x_0)) ~\stackrel{i_{*}}{\rightarrow}
~H_{0}( Mf_{\leq}(x_0)) ~\stackrel{j_{*}}{\rightarrow}
~H_{0}( Mf_{\leq}(x_0),Mf_{<}(x_0))~\rightarrow~0 \]
%
Since the homomorphism \(i_{*}: ~H_{0}( Mf_{<}(x_0))~\longrightarrow
~H_{0}( Mf_{\leq}(x_0)) \) is always surjective, the homomorphism
\(j_{*}\) maps
\(H_{0}( Mf_{\leq}(x_0))= \Z\) into \(0\); hence
\[H_{0}( Mf_{\leq}(x_0),Mf_{<}(x_0))~=~0.\]
\hfill{$\Box$}
%
{~}\\
{~}\\
Let us consider a continuous selection \(f\) of \(C^{2}\)-functions
on a smooth \(n\)-dimensional manifold with compact lower level sets
and suppose all critical points of \(f\) are nondegenerate.
By definition, we have \(H_p(f,x)=0\) if \(x \in M \) is not a critical
point. Following J.Milnor \cite{milnor} we consider the following
inequality
\[ \sum_{j=0}^{p} (-1)^{2p-j} \mbox{rank~}H_{p-j}(M) \leq
\sum_{j=0}^{p} (-1)^{2p-j}(\sum_{x \in M} \mbox{rank~}H_{p-j}(f,x)), \mbox{ ~~for all~~}
p \in \N \cup \{0\}. \]
This results in the following theorem.
\begin{theorem}
\begin{itemize}
\item[ a)] (Inequalities of Betti-numbers)\\
\[ \mbox{rank~} H_{p}(M)~\leq~\sum_{x \in M} \mbox{rank~}H_{p}(f,x),~~
p \in \N \cup \{0\} \]
\item[ b)] (Equality of Euler-Characteristics)\\
\[ \chi(M) ~=~ \sum_{x \in M} \chi(f,x) \]
with
\[ \chi(f,x)~=~ \sum_{p=0}^{n} (-1)^{p} \mbox{rank~}H_{p}(f,x).\]
\end{itemize}
\end{theorem}
\par
One may ask whether the use of upper level sets instead of lower
level sets would lead to different
Morse inequalities in the case of piecewise smooth functions.
This is not the case as can be seen by applying Alexander duality.
In fact, the upper level set is given by
\[K=\{(u,v)\in\R^{k} \times \R^{n-k} ~|~l(u)+q(v)>0\}=
 \R^{k}\times \R^{n-k} \setminus  \{(u,v)~|~l(u)+q(v)\leq0\},\]
and, analogously to Proposition \ref{prop3}, one can show that
\(\{(u,v) \in \R^{k} \times \R^{n-k}~|~l(u)+q(v)\leq0\} \) is homotopy
equivalent to the
set \(\tilde{K}=\{(u,v)\R^{k}\times \R^{n-k}~|~l(u)+q(v) ~<~0\}.\)
Hence we have for \(\R^n \setminus K \) and \(\tilde{K} \) the same
homology groups. Now consider the upper level set \(K\)
as a subset of the one-point compactification of \(\R^n,\) i.e.
\( K \subset \R^{n} \cup \{\infty\} = S^{n+1}. \)
Alexander duality (see \cite{bred} Chapter VI, Corollary 8.6)
gives \( \mbox{rank~} \tilde{H}_{i}(K)= \mbox{rank~}\tilde{H}_{n-i}(
S^{n+1}\setminus K) \) for the reduced homology
for all \(i \in \Z.\) Hence, we obtain the
same Morse inequalities as for lower level sets.\\

Classical Morse inequalities are often used to obtain lower bounds
for the number of nondegenerate critical points of a function
on a manifold. In order to use our inequalities for the same
purpose, one has to estimate the maximal possible contribution
to the Betti numbers of one nondegenerate critical point. We come to
the interesting problem: Given a simplicial complex \(C\) with
\((k+1)\) vertices, what are the maximal possible values of
Betti numbers \(b_i={\rm rank}~H_i(C)\) and of the total Betti
number \(b=b_0+\ldots+b_k\)? Let us denote these maximal possible
values by \(B_i(k)\) and \(B(k),\) respectively. We have
\(B_0(k)=k+1\), \(B_{k-1}(k)=1,\) \(B_j(k)=0\) for \(j>k-1\).
Direct caculations give also \(B(2)=3\), \(B(3)=4\), \(B(4)=7\),
\(B_1(3)=3\), \(B_1(4)=6\), \(B_2(4)=4\). A general formula for
these numbers is presently not known.
%
%           E x a m p l e s
%
%----------------------------------------------------------------
\section{Examples }
%----------------------------------------------------------------
%
We consider the two-dimensional case. Here we have the following
types of nonsmooth normal-forms:
%
\begin{center}
\begin{tabular}{lcl}
  {~~}      &      {~~~~~~~}               \\
$f_{1}(x_1,x_2)$ &= & $ \max \{ x_1,x_2,-x_1-x_2\}  $  \\
  {~~}& &     {~~~~~~~}                    \\
$f_{2}(x_1,x_2)$ &= & $ \min \{ x_1,x_2,-x_1-x_2\}  $  \\
  {~~}&  &    {~~~~~~~}                  \\
$f_3(x_1,x_2) $ &= & $    \max \{ x_1,\min\{x_2,-x_1-x_2\}\}  $  \\
  {~~}& &     {~~~~~~~}                    \\
$f_4(x_1,x_2) $ &= & $  \max \{\min \{x_1,x_2\},\min\{x_1,-x_1-x_2\}\}  $  \\
  {~~}&  &    {~~~~~~~}                   \\
$f_5(x_1,x_2) $ &= & $ \max \{\min \{x_1,x_2\},\min\{x_2,-x_1-x_2\}\}  $  \\
  {~~}&  &    {~~~~~~~}                    \\
$f_6(x_1,x_2) $ &=& $  \max \{ x_2,\min\{x_1,-x_1-x_2\}\}  $  \\
  {~~}&  &    {~~~~~~~}                   \\
$f_7(x_1,x_2) $ &=& $ \max \{-x_1-x_2,\min\{x_1,x_2\}\}       $  \\
  {~~}&  &    {~~~~~~~}                   \\
$f_8(x_1,x_2) $ &= & $ \max \{\min \{-x_1-x_2,x_1\},\min\{x_2,-x_1-x_2\}\} $  \\
  {~~}&  &    {~~~~~~~}                     \\
$f_9(x_1,x_2)  $ &= & $ \max \{\min \{x_1,-x_1-x_2\},\min\{x_2,-x_1-x_2\},
 \min\{x_1,x_2\}\}  $  \\
\end{tabular}
\end{center}
and the smooth normal formes:
\[ g_1(x_1,x_2)= x_{1}^{2} + x_{2}^{2},{~~~}
   g_2(x_1,x_2)= -x_{1}^{2} - x_{2}^{2}
\mbox{~~and~~} g_3(x_1,x_2)= -x_{1}^{2} + x_{2}^{2} \]
{~}\\
For a function \(f \in \{g_1,f_1\} \) we have the following
Betti-numbers for the local homology:
\[\mbox{(Max)~~~~~~~~~~~~~} \beta_{i}= \mbox{rank~}H_{i}(f,0)= \left\{ \begin{array}{r@{\quad:
\quad}l}
0 & 0 \leq 1 \\
1 & i=2 \\
0 & i \geq 3
\end{array}  \right.  \]
and for  \(f \in \{g_2,f_2\} \)
\[\mbox{(Min)~~~~~~~~~~~~~} \beta_{i}= \mbox{rank~}H_{i}(f,0)= \left\{ \begin{array}{r@{\quad:
\quad}l}
0 & i<0 \\
1 & i=0 \\
0 & i \geq 1
\end{array} \right. \]
%-------------------------------------------------------
For a function \(f \in \{g_3,f_3,f_4,f_5,f_6,f_7,f_8\} \) we have the
Betti-numbers:
\[\mbox{(SP1)~~~~~~~~~~~~~} \beta_{i}= \mbox{rank~}H_{i}(f,0)= \left\{ \begin{array}{r@{\quad:
\quad}l}
0 & i \leq 0 \\
1 & i=1 \\
0 & i \geq 2
\end{array} \right. \]
%-------------------------------------------------------
Finally, for the function \(f_9\) we have:
\[\mbox{(SP2)~~~~~~~~~~~~~}  \beta_{i}= \mbox{rank~}H_{i}(f_9,0)= \left\{ \begin{array}{r@{\quad:
\quad}l}
0 & i\leq 0 \\
2 & i=1 \\
0 & i \geq 1
\end{array} \right..\]
Hence by classifying the normal-forms with respect to Betti-numbers,
we have the above mentioned 4 types.\\
\par
For the types (Max), (Min) and (SP1) we have the same situation
as in the smooth case. First note that, by \cite{barku} Theorem 3.1,
every function
\(h \in \{f_3,f_4,f_5,f_6,f_7,f_8\}\) is topologically equivalent to
the function
\[ f:[-1,+1]\times [-1,+1] \longrightarrow \R \mbox{~~with~~}
f(x_1,x_2)= \vert x_1 \vert - \vert x_2 \vert ,\]
since all these functions have two ascent- and descent cones.
This function \(f\) induces a function on the torus with a nonsmooth
maximum and minimum and two nonsmooth saddle points of type (SP1).
The saddle points are  \((0,0) \) and the point which results
from identifying \( (1,1),(1,-1),(-1,-1),(-1,1),\)
the maximum is the point which results from the identification
of \( (1,0)\) with \( (-1,0) \) and the minimum is achieved at
the point which results from the identification of
\((0,1)\) with \((0,-1).\) The smooth analogon to this example is
given by the function \(\tilde{f}(x_1,x_2)=x_1^2-x_2^2.\)\\
\par
The function \(f_9\)
has three ascend- and descent cones. In new coordinates
\(u_1,u_2\) defined by  \(x_1=u_1-u_2,~x_2=u_1+u_2 \) we have the
representation
\[f_9(u_1,u_2)  = \max \{\min \{u_1-u_2,-2u_1\},\min\{u_1+u_2,-2u_1\},
\min\{u_1-u_2,u_1+x_2\}\}.\]
Note that
\begin{enumerate}
\item[    i)] $f_9(0,u_2)=0$ for all $v \in \R$
\item[   ii)] $f_9(u_1,u_2)=f(u_1,-u_2)$ for all $u_1,u_2 \in \R$
\item[  iii)] $f_9(-u_1,u_2)=-f(u_1,u_2)$ for all $u_1,u_2 \in \R,$
\end{enumerate}
with the last identity resulting from the fact that
\begin{eqnarray*}
&~&\max \{\min \{u_1-u_2,-2u_1\},\min\{u_1+u_2,-2u_1\},
\min\{u_1-u_2,u_1+x_2\}\} \\
&=& \min \{\max \{u_1-u_2,-2u_1\},\max\{u_1+u_2,-2u_1\},
\max\{u_1-u_2,u_1+x_2\}\}.
\end{eqnarray*}
Thus the function \(f_9\) induces a function on the cylinder
\(\R \times [-1,+1]  \) where the points \( (u,1)\) and \((u,-1) \)
are identified for every \(u \in \R.\) Moreover \(f_9\) is antisymmetric
with respect to the first variable and vanishes on the circle \(\{0\}
\times [-1,1].\)  If we contract this circle to a point, then we can
consider \(f_9\) as function defined on a quadratic cone  in \(\R^3\)
given by \(u_1u_2=u_3^2\) which is antisymmetric with respect
to the reflection at the origin.\\
\par
Let us consider a smooth function  \(p:\R^2 \longrightarrow \R \)
which is identical one
on a disk with center \( (0,0) \in \R^2 \) and radius \( r>0 \)
and which vanishes outside the disk around \( (0,0) \in \R^2 \)
with radius \(r+\varepsilon \) for some \(\varepsilon > 0\) and which
satisfies \( p(x_1,x_2) = p(-x_1,-x_2)=p(x_1,-x_2)=p(-x_1,x_2),\)
e.g. the function
\[ p_{r,\varepsilon}: \R^{2} \longrightarrow \R \mbox{~~with~~}
p_{r,\varepsilon}(x_1,x_2)= 1-\frac{ \eta (\Vert(x_1,x_2) \Vert -r)}{\eta ( \Vert(x_1,x_2)
\Vert - r)+ \eta (\varepsilon +r - \Vert (x_1,x_2) \Vert) }  \]
where \( \Vert(x_1,x_2)\Vert \) denotes the Euclidean norm of the vector
\( (x_1,x_2) \) and
\[\eta(t)= \left\{ \begin{array}{l@{\quad
\quad}l}
e^{-\frac{1}{t}} & t > 0 \\
{~} & {~} \\
0 & t \leq 0.
\end{array} \right.\]
{~}\\
Now the support of the function \( f= f_9\cdot p_{r,\varepsilon} \) is the disk
\( D=\{(x_1,x_2)~|~ \Vert (x_1,x_2) \Vert \leq r+\varepsilon \} \)
and \(f\) vanishes on the boundary \( \partial D \) of \(D.\)
Since \(f(-x_1,-x_2) = f_9(-x_1,-x_2) \cdot p_{r,\varepsilon}(x_1,x_2) =
-f_9(x_1,x_2) \cdot p_{r,\varepsilon}(x_1,x_2) \) and \(f_9\) has three ascend- and descent
cones, we get a function on the two-sphere \( S^2 \) with three maxima,
three minima and two saddle-points of type (SP2), if we identify
\( \partial D \) to a point.\\
\par
As a further example consider for \(r+\varepsilon <1 \) the function
\[ f:[-1,1] \times [-1,1] \longrightarrow \R
\mbox{~~with~~}
f(x_1,x_2)=f_9(x_1,x_2)\cdot p_{r,\varepsilon}(x_1,x_2) + (1-p_{r,\varepsilon}
(x_1,x_2))\cdot(\vert x_1 \vert - \vert x_2 \vert ) \]
as a function on the torus. This is possible, since
\( p(x_1,x_2) = p(x_1,-x_2)=p(-x_1,x_2). \)
This function has a maximum, a minimum and a saddle-point of type (SP1)
which results from the second summand, a saddle-point of type (SP2),
three maxima and three minima which result from the first summand,
and five saddle-points of type (SP1) which result from the combination
of both functions by the chosen partition of unity.
%
%-------------------------------------------------------
%       R e f e r e n c e s
%-------------------------------------------------------
%
\begin{thebibliography}{99}
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Theory and Optimal Control Problems, in:
\it Nonlinear Synthesis \rm Progress in System and Control Theory 9,
Birkh\"auser Verlag 1991, pp 1-11.
%
\bibitem{barku} Bartels, S. G., Kuntz, L., and Scholtes, S. (1995).
Continuous Selections of Linear Functions and Nonsmooth Critical
Point Theory.
\it Nonlinear Analysis, Theory, Meth. \& Appl. \rm {\bf 24} 385-407.
%
\bibitem{bred} Bredon, G. E.(1993).
Topology and Geometry. \rm
Springer Verlag, Berlin, Heidelberg, New York.
%
\bibitem{Bonnisseau} Bonnisseau, J.-M. and Cornet, B. (1990).
Fixed-Point Theorems and Morse's Lemma for Lipschitzian Functions.
\it J. Math. Anal. Appl. \rm {\bf 146} 318-332.
%
\bibitem{Bougeard} Bougeard, M.L. (1988).
Morse Theory for Some Lower \(C^2\)-Functions in Finite Dimension.
\it Math. Prog. \rm {\bf 41} 141-159.
%
\bibitem{cla} Clarke, F. H. (1989). \it Optimization and Nonsmooth
Analysis.\rm~ Les publications CRM, Universit\'{e} de Montr\'{e}al.
%
\bibitem{goresky} Goresky, M and MacPherson, R.
\it Stratified Morse Theory. \rm
Springer Verlag, Berlin 1988.
%
\bibitem{hu} Hu, Sze-Tsen (1965). \it Theory of Retracts.\rm~
Wayne State University Press, Detroit.
%
\bibitem{jon1} Jongen, H. Th., Jonker, P., and Twilt, F. (1983)
\it Nonlinear Optimization in $\R^{n},$ I. \it Morse Theory,
Chebyshev Approximation. \rm
Peter Lang Verlag, Frankfurt, New York.
%
\bibitem{jon2} Jongen, H. Th., Jonker, P., and Twilt, F. (1986)
\it Nonlinear Optimization in $\R^{n},$ II. \it Transversality,
Flows, Parametric Aspects. \rm
Peter Lang Verlag, Frankfurt, Bern, New York.
%
\bibitem{jon3} Jongen, H. Th. and Pallaschke, D. (1988). On Linearization
and Continuous Selections of Functions.
\it Optimization \rm\(\bf 19\)~3 343-353.
%
\bibitem{kuscho} Kuntz, L. and Scholtes, S. (1995). Qualitative Aspects
of the Local Approximation of a Piecewise Smooth Function.
\it Nonlinear Analysis, Theory, Meth. \& Appl. \rm {\bf 25} 197-215.
%
\bibitem{milnor} Milnor, J. (1963) \it Morse Theory. \rm
Princeton University Press, Princeton, New Jersey.
%
\bibitem{scho} Scholtes, S. (1994) \it Introduction to piecewise
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%
\bibitem{span} Spanier, E. H. (1966) \it Algebraic Topology. \rm
McGraw-Hill Book Company, New York, San Francisco, St. Louis, Toronto,
London, Sydney.
%
\end{thebibliography}
%
%
%
%   A d r e s s e s
%
%
%------------------------------------------------
%
\begin{tabular}{ll}
\\
Andrei A. Agrachev  {~~~~~~~~~~~~~~~~~~~}&     Diethard Pallaschke \\
Steklov Mathematical Institute           &     Institute for Statistics and\\
Russian Academy of Sciences              &     Mathematical Economics  \\
ul. Vavilova 42                          &     University of Karlsruhe \\
117966 Moscow, Russia                    &     Kaiserstr. 12  \\
{~~~~~~~~~~~~~~~~~~~}                    &     D-76128 Karlsruhe, Germany \\
{~~~~~~~~~~~~~~~~~~}                     &            {~~~~~~~~~~~~~}\\
Stefan Scholtes                     & {~~~~~~~~~~~~~}\\
Department of Engineering           & {~~~~~~~~~~~~~}\\
University of Cambridge             & {~~~~~~~~~~~~~}\\
Trumpington Street                  & {~~~~~~~~~~~~~}\\
Cambridge CB 2 1AG, Great Britain   & {~~~~~~~~~~~~~}\\
\end{tabular}
%------------------------------------------------
\end{document}
%
%-------
%
