Vertical Cohomologies and Their Application to Completely Integrable Hamiltonian Systems
Z. Tevdoradze
Some functorial and topological properties of vertical cohomologies and their application to completely integrable Hamiltonian systems are studied.
math0202088.tex
\documentstyle[righttag,amssymb,intlim,amscd,bezier]{amsart}

%\def\currentvolume{5}
%\def\curryr{98}
%\def\currmonth{0900}
%\def\currentnumber{5}
%\def\currstpg{0483}
%\def\currentstartpage{483}
%\def\currentendpage{500}

\setcounter{page}{1}

\subjclass{53C12, 70H05}
\keywords{Foliation, vertical cohomology, Hamiltonian systems,
    isoenergetic surfaces, Liouville torus}

\newcommand{\nc}{\newcommand}
\nc{\al}{\alpha}
\nc{\bt}{\beta}
\nc{\gm}{\gamma}
\nc{\dl}{\delta}
\nc{\Lb}{\Lambda}
\nc{\vf}{\varphi}
\nc{\om}{\omega}
\nc{\tht}{\theta}

\nc{\bR}{\Bbb{R}}
\nc{\bT}{\Bbb{T}}
\nc{\bZ}{\Bbb{Z}}
\nc{\cF}{\cal{F}}
\nc{\cP}{\cal{P}}
\nc{\cT}{\cal{T}}

\nc{\pa}{\partial}
\nc{\sbs}{\subset}
\nc{\wh}{\widehat}
\nc{\wt}{\widetilde}
\nc{\ol}{\overline}
\nc{\os}{\overset}
\nc{\us}{\underset}
\nc{\str}{\stackrel}
\nc{\map}{\longmapsto}
\nc{\hook}{\hookrightarrow}

\nc{\sgrad}{\operatorname{sgrad}}
\nc{\grad}{\operatorname{grad}}
\nc{\Inv}{\operatorname{Inv}}
\nc{\End}{\operatorname{End}}

\newtheorem{th}{\ \ \ Theorem}[section]
\newtheorem{prop}[th]{\ \ \ Proposition}
\newtheorem{cor}[th]{\ \ \ Corollary}
\newtheorem{lem}[th]{\ \ \ Lemma}

\theoremstyle{definition}
\newtheorem{defi}[th]{\ \ \ Definition}

\theoremstyle{remark}
\newtheorem{rem}[th]{\ \ \ Remark}

\numberwithin{equation}{section}

\hyphenation{ho-mo-to-pic}

\begin{document}

\title[Vertical Cohomologies and Their Application]
    {Vertical Cohomologies and Their Application to
        Completely  Integrable Hamiltonian  Systems}

\author{Z. Tevdoradze}

\maketitle
\bigskip

\hfill {\it Dedicated to the memory of Roin Nadiradze}
\vskip-2cm

\begin{abstract}
    Some functorial and topological properties of vertical
    cohomologies and their application to completely integrable
    Hamiltonian systems are studied.
\end{abstract}


%   \S 1.
\section{Introduction }

    If on a smooth Riemannian manifold $M^n$ we have a distribution $V$ of
dimension $k$, which is actually a smooth section of the Grassman
fiber bundle $G_k(TM^n)\to M^n$ adjoint to the tangent fibration
$TM^n$ to the manifold $M^n$, then by means of the Riemannian
metric we obtain $TM^n=V\oplus N$, where $N$ is a normal fiber
bundle to $V$. Let $P:TM^n\to V\sbs TM^n$ be a natural projection.
The operator $P$ defines the mapping $P^*:\Lb^*(M^n)\to
\Lb^*(M^n)$ ($(\Lb^*(M^n),d^*)$ is the de Rham differential
complex) by the formula
$(P^*\al)(X_1,\dots,X_q)=\al(PX_1,\dots,PX_q)$, where $\al\in
\Lb^q(M^n)$ and $X_1,\dots,X_q\in S(M^n)$ are the smooth vector
fields on $M^n$.

    Denote by $\Lb^*_V(M^n)$ all fixed points of the operator $P^*$.
In what follows we shall consider the case, where $V$ is
integrable, i. e., where $M^n$ is partitioned into leaves and the
tangent space to the leaf that passes through the point  $x\in
M^n$ is $V_x$. Then the pair $(\Lb^*_V(M^n),d_V^*)$ forms a
differential complex with the differential $d_V^*=P^*\circ d^*$.

    The cohomologies  of the complex $(\Lb_V^*(M^n),d_V^*)$ are
called vertical and denoted by $H_V^*(M^n)$ ([1]). It has turned
out that these cohomologies coincide with those of the classical
BRST operator ([2], [3]).

    In \S 2 the vertical cohomologies are defined without fixing
the metric on $M^n$, and some of their functorial properties are
studied. The FOL category  of smooth foliations and leaf-to-leaf
transforming mappings is introduced, and a natural transformation
of the de Rham functor $\Lb^*$ to the functor $\Lb^*_{\cF}$ is
construced (Proposition 2.2). The notion of leaf-to-leaf
transforming homotopic mappings is introduced, and the homotopy
axiom for vertical cohomologies is proved (Theorem 2.5). The
notion of  a relative group of vertical cohomologies is introduced
by analogy with de Rham's theory, and the long exact cohomologic
sequences (2.6) and (2.7) are derived. Moreover, for a
leaf-to-leaf transforming mapping $f:(M^n,{\cF}_1)\to
(N^m,{\cF}_2)$, the cohomology groups $H^*(f)$ are constructed and
proved (Theorem 2.8) to be isomorphic for leaf-to-leaf
transforming homotopic mappings. Finally, a double complex
$(K^{**},D^*)$ is constructed for the countable covering
$U=\{u_\al\}_{\al\in A}$ of the manifold $M^n$. It is shown that
the cohomologies of $(K^{**},D^*)$ are isomorphic to the vertical
cohomologies. A combinatorial definition of vertical comologies in
the \v{C}ech sense (Theorems 2.10 and 2.12) is also given.

    In \S 3 some of the main facts from the topological theory of
integrable  Hamiltonian systems  ([5], [6]) are presented. Using
the notion of vertical cohomologies, the groups corresponding to
nonresonance Hamiltonian systems are constructed (Theorems 3.3 and
3.4).

    In \S 4 the case of a spherical pendulum is considered as an example.


%   \S 2.
\section{Vertical Cohomologies}

%   2.1.
\subsection{Definition of Vertical Cohomologies.}
 Let $M^n$ be
a smooth $n$-dimensional manifold, and $V$ be a $k$-dimensional
involutive distribution on $M^n$ whose foliation is denoted by
$\cF$.  The bundle of exterior $p$-forms on $V$ is denoted by
$A^p(V)$, and the  set of  smooth sections of the bundle $A^p(V)$
by $\Lb^p_{\cF}(M^n)$. Then $\Lb_{\cF}^p(M^n)$ is a module over
the algebra of infinitely differentiable functions $C^\infty(M^n)$
on $M^n$.

    Let $\al\in \Lb_{\cF}^p(M^n)$, and let $X_1,\dots,X_p$ be the smooth
vector fields on $M^n$ which are tangent to the leaves of the
foliation $\cF$, i.e., they are the smooth sections of the bundle
$V\str{p_{{}_{\cF}}}{\to} M^n$. Then the mapping  defined by the
formula
%   (2.1)
\begin{equation}
    \al(X_1,\dots,X_p):x\str{j}{\map} \al(X_1(x),\dots,X_p(x)),
\end{equation}
which on the module of  sections $S(V)$ of $V$ assigns an exterior
$p$-form to an element $\al\in \Lb_{\cF}^p(M^n)$, is an
isomorphism.

    Let us now define the operator
$$  d_{\cF}^p:\Lb_{\cF}^p(M^n)\to \Lb_{\cF}^{p+1}(M^n)  $$
by the relation
\begin{gather*}
    (d_{\cF}^p\al)(X_1,\dots,X_{p+1}) =\sum_{i=1}^{p+1}
        (-1)^{i-1} X_i\al(X_1,\dots,\wh{X}_i,X_{i+1},
        \dots,X_{p+1})+  \\
    +\sum_{i<j} (-1)^{i+j} \al([X_i,X_j],X_1,\dots,\wh{X}_i,\dots,
        \wh{X}_j,\dots,X_{p+1}),
\end{gather*}
where $\al\in \Lb_{\cF}^p(M^n)$, $X_i\in S(V)$, $i=\ol{1,p+1}$.

        It is easy to verify that the embedding
$i:S(V)\hook S(TM^n)$ induces the projection $i_q^*:A^q(S(TM))\to
A^q(S(V))$. Indeed, let $\al\in A^q(S(V))$. Define $\ol{\al}\in
A^q(S(TM))$ so that $i_q^*\ol{\al}=\al$. Any Riemannian metric
defines a smooth section $P:M^n\to \End (TM^n)$ of the bundle of
homomorphisms $\End(TM^n)$ of the tangent bundle $TM^n$, where
$P(x):T_xM^n\to V_x\sbs T_xM^n$ is the orthogonal  projection
onto  a leaf of the foliation $\cF$ which passes through the point
$x\in M^n$. The element $\ol{\al}$ can be defined by the formula
$$  \ol{\al}(X_1,\dots,X_q)=\al(PX_1,\dots,PX_q),\quad
        X_i\in S(TM^n),\;\;\;i=\ol{1,q}.    $$

        The following diagram is commutative:
$$  \begin{CD}
        A^q(S(TM^n)) @>{i_q^*}>> A_{\cF}^q(M^n) \\
        @VV{d^q}V @VV{d_{\cF}^q}V \\
        A^{q+1}(S(TM^n)) @>{i_{q+1}^*}>> A_{\cF}^{q+1}(M^n)
    \end{CD}\;,     $$
where by $A_{\cF}^*(M^n)$ is denoted $A^*(S(V))$. Indeed,
\begin{gather*}
    (d_{\cF}^qi_q^*\al)(X_1,\dots,X_{q+1}) =\sum_{j=1}^{q+1}
        (-1)^{j-1} (X_j)(i_q^*\al)(X_1,\dots,\wh{X}_j,
        \dots,X_{q+1})+  \\
    +\sum_{j<t} (-1)^{j+t} (i_q^*\al)([X_j,X_t],X_1,\dots,\wh{X}_j,\dots,
        \wh{X}_t,\dots,X_{q+1})= \\
    =(d^q\al)(X_1,\dots,X_{q+1})=(i_{q+1}^* d^q\al)(X_1,\dots,X_{q+1}).
\end{gather*}

      The commutativity $d_{\cF}^q\circ i_q^*=i_{q+1}^*\circ d^q$
implies $d_{\cF}^q\circ d_{\cF}^{q-1}=0$. Thus $i^*$ is a cochain
mapping between the differential complexes $(A^*(S(TM^n),d^*)$ and
$(A_{\cF}^*(M^n),d_{\cF}^*)$.

%        DEFINITION 2.1.
\begin{defi}
Cohomology groups of the complex $(A_{\cF}^*(M),d_{\cF}^*)$ are
called vertical cohomologies of the foliation $(M^n,\cF)$, and we
denote them by $H_{\cF}^*(M^n)$.
\end{defi}

        It is easy to verify that $d_{\cF}^*$ is an antidifferentiation
of order 1, i.e., if $\al\in A_{\cF}^q(M^n)$, $\bt\in
A_{\cF}^l(M^n)$, we have
$$  d_{\cF}^{q+l}(\al\wedge \bt)=(d_{\cF}^q\al)\wedge \bt+
        (-1)^q\al\wedge (d_{\cF}^l\bt),     $$
where $\wedge$ is an exterior product. This implies that the
homomorphism $i^*$ induces a homomorphism  between the de Rham
cohomology algebra and the cohomology algebra $H_{\cF}^*(M^n)$. We
denote this homomorphism by the same symbol $i^*$. Since we
already know that the cochain mapping $i^*$ is an epimorphism, we
have a short exact sequence of cochain differential complexes
%   (2.2)
\begin{equation}
    0 @>>> Z_{\cF} @>{j^*}>> A^* (S(TM^n)) @>{i^*}>>
        A_{\cF}^*(M^n) @>>> 0,
\end{equation}where by $(Z_{\cF}^*,d^*)$ is denoted the kernel of the mapping
$i^*$. Sequence (2.2) induces a long exact sequence of cohomology
groups
%   (2.3)
\begin{equation}
    0 @>>> \wt{H}_{\cF}^0(M^n) @>{j^*}>> H^0(M^n) @>{i^*}>>
        H_{\cF}^0(M^n) @>{\dl}>> \wt{H}_{\cF}^1(M^n)
        @>{j^*}>> \cdots \;.
\end{equation}
Here by $\wt{H}_{\cF}^*(M^n\!)$ are denoted the cohomology groups
of the complex $(Z_{\cF}^*,d)$. Since $H_{\cF}^{k+m}(M^n)=0$,
$m>0$, from (2.3) we get $\wt{H}_{\cF}^{m+k}(M^n) \approx
H^{m+k}(M^n)$, $m>1$.

        Denote by FOL a category whose objects are smooth foliations
$(M^n,\cF)$, and morphisms from $(M_1^n,{\cF}_1)$ to
$(M_2^m,{\cF}_2)$ are leaf-to-leaf transforming mappings, i.e.,
smooth mappings $h$ from $M_1^n$ to $M_2^n$ which preserve the
foliation structure by perform a leaf-to-leaf transformation.

        If $h$ is the leaf-to-leaf transforming  mapping between the foliations
\linebreak  $(M_1^n,{\cF}_1)$ and $(M_2^n,{\cF}_2)$, then $h$
defines the morphism $h^*$ between the    \linebreak
$C^\infty(M_2^m)$-module $\Lb_{{\cF}_2}^*(M_2^m)$ and the
 $C^\infty(M_1^n)$-module $\Lb_{\cF_1}^*(M_1^n)$
by the formula
%   (2.4)
\begin{equation}
    (h^*\al)(X_1,\dots,X_p)(x)=\al(h^\top X_1(x),\dots,
        h^\top X_p(x)),
\end{equation}
where $\al\in \Lb_{{\cF}_2}^p(M_2)$ and $X_1,\dots,X_p\in S(V_1)$,
$V_1$ is the distribution associated with ${\cF}_1$, and $h^\top$
is the tangent mapping to $h$. We have the commutative diagram
$$  \begin{CD}
        \Lb^*(M_2^m) @>{h^*}>> \Lb^*(M_1^n) \\
        @VV{i_2^*}V @VV{i_1^*}V \\
        \Lb_{{\cF}_2}^*(M_2^m) @>{h^*}>> \Lb_{{\cF}_1}^*(M_1^n)
    \end{CD}\;.     $$

        A direct calculation shows that $h^*$ is a cochain mapping, i.e.,
$d_{{\cF}_1}^*\circ h^*=h^{*+1}\circ d_{{\cF}_2}^*$. Hence we have

%        PROPOSITION 2.2.
\begin{prop}
The mapping $i^*$ is a natural transformation of the de Rham
functor $\Lb^*$ to the functor $\Lb_{\cF}^*$, where $\Lb^*$ and
$\Lb_{\cF}^*$ are the contravariant functors from the \rom{FOL}
category to the category of differential graded algebras and their
homomorphisms.
\end{prop}

%        2.2.
\subsection{A Homotopy Axiom for Vertical Cohomologies.}

        Let $(M^n,\cF)$ be a foliation of dimension $k$. On the manifold
$M^n\times \bR$ we define naturally a foliation $\wh{\cF}$ of
dimension $k+1$ whose leaves are manifolds $L_\al\times {\bR}$,
$\al\in A$, where $L_\al$, $\al\in A$, are the leaves of the
foliation $\cF$.

%        LEMMA 2.3.
\begin{lem}
The projection $\pi:M^n\times {\bR}\to M^n$ defines an isomorphism
in vertical cohomologies.
\end{lem}

%        PROOF.
\begin{pf}
Consider the zero section $s$ of the trivial bundle $M^n\times
{\bR} @>{\pi}>> M^n$, i.e., $s(x)=(x,0)$, $x\in M^n$. Then the
mappings $\pi$ and $s$ are the leaf-to-leaf transforming mappings
which define the cochain mappings
$\pi^*:(\Lb_{\cF}^*(M^n),d_{\cF}^*) \to
    (\Lb_{\wh{\cF}}^*(M^n\times {\bR}),d_{\wh{\cF}}^*)$ and
$s^*:(\Lb_{\wh{\cF}}^*(M^n\times {\bR}),d_{\wh{\cF}}^*) \to
    (\Lb_{\cF}^*(M^n),d_{\cF}^*)$. Since $s^*\circ \pi^*=1$,
$\pi^*$ is an monomorphism. We shall show that $\pi^*$ induces an
isomorphism at the cohomology level. To this end, we shall
construct a cochain equivalence of the mappings 1 and $\pi^*\circ
s^*$.

        Note that each form from $\Lb_{\cF}^*(M^n\times {\bR})$
can be uniquely represented by linear combinations of the
following two types of forms:
\begin{align*}
        \text{(I)} \;\; & (\pi^*\vf)\cdot f, \;\;\; \vf\in \Lb_{\cF}^*(M^n),
        \;\;\;f\in C^\infty(M^n\times {\bR});  \\
        \text{(II)} \;\; & (\pi^*\vf)\wedge dt\cdot f, \;\;\;
        \vf\in \Lb_{\cF}^*(M^n),
        \;\;\;f\in C^\infty(M^n\times {\bR}),
\end{align*}
where $t$ is the coordinate on the straight line ${\bR}$. Define
the operator $K^*:\Lb_{\wh{\cF}}^*(M^n\times {\bR})\to
{\Lb_{\wh{\cF}}^*}^{-1}(M^n\times {\bR})$ as follows:
\begin{align*}
    & K^*((\pi^*\vf)\cdot f)=0,  \\
    & K^*((\pi^*\vf)\wedge dt\cdot f)=\pi^*\vf\cdot \int_0^t f\,dt.
\end{align*}

        A direct calculation shows that the relation
$$  1-\pi^*\circ s^*=(-1)^{q-1} (d_{\wh{\cF}}^{q-1}K^q-K^{q+1}d_{\wh{\cF}}^q)   $$
is fulfilled on the forms of types (I) and (II). %Thus the lemma is complete.
\end{pf}

%        DEFINITION 2.4.
\begin{defi}
Two leaf-to-leaf transforming mappings $f,g:({M_1}^n,{\cF}_1)
\linebreak  \to ({M_2}^m,{\cF}_2)$ between the foliations
$({M_1}^n,{\cF}_1)$ and $({M_2}^m,{\cF}_2)$ are called
leaf-to-leaf transforming homotopic if there exists a leaf-to-leaf
transforming mapping
$$  F:(M_1^n\times {\bR},\wh{\cF}_1)\to (M_2^m,{\cF}_2)     $$
such that
$$  \begin{cases}
        F(x,t)=f(x), & t\geq 1, \\
        F(x,t)=g(x), & t\leq 0,
    \end{cases} \quad x\in M,\;\;\;t\in {\bR}.        $$
\end{defi}

%        THEOREM 2.5
\begin{th}[A Homotopy Axiom]
Leaf-to-leaf transforming  ho- \linebreak motopic mappings induce
identical map\-pings in vertical cohomologies.
\end{th}

%        PROOF.
\begin{pf}
Let $f,g:(M_1^n,{\cF}_1)\to ({M_2}^m,{\cF}_2)$ be the leaf-to-leaf
transforming homotopic mappings and $F:(M_1^n\times
{\bR},\wh{\cF}_1)\to (M_2^m,{\cF}_2)$ be the homotopy between $f$
and $g$. Denote by $s_0$ and $s_1$ the sections
$s_0,s_1:(M_1^n,{\cF}_1)\to (M_1^n\times {\bR},\wh{\cF}_1)$,
$s_0(x)=(x,0)$, $s_1(x)=(x,1)$, $x\in M_1^n$. Then $f=F\circ s_1$
and $g=F\circ s_0$. Hence we have $f^*=s_1^*\circ F^*$ and
$g^*=s_0^*\circ F^*$. From the proof of Lemma 2.3 it follows that
$s_1^*=s_0^*=(\pi_1^*)^{-1}$, where $\pi_1:M_1^n\times {\bR}\to
M_1^n$
is the projection. Therefore $f^*=g^*$. %Thus the theorem is complete.
\end{pf}

        The foliations $(M_1^n,{\cF}_1)$ and
$(M_2^m,{\cF}_2)$ will be said to be  of the same homotopy type if
there are  leaf-to-leaf transforming smooth mappings $f:M_1^n\to
M_2^m$ and $g:M_2^m\to M_1^n$ such that $g\circ f$ and $f\circ g$
are leaf-to-leaf transforming homotopic to the identical mappings
of the foliations $(M_1^n,{\cF}_1)$ and $(M_2^m,{\cF}_2)$,
respectively.

%        COROLLARY 2.6.
\begin{cor}
If two foliations $(M_1^n,{\cF}_1)$ and $(M_2^m,{\cF}_2)$ are of
the same homotopy type, then their vertical cohomologies are
isomorphic.
\end{cor}

%        2.3.
\subsection{Relative Vertical Cohomologies}

        Let $(M^n,{\cF}_1)$ and $(N^m,{\cF}_2)$ be two foliations, and let
$f$ be a leaf-to-leaf transforming smooth mapping $f:M^n\to N^m$.
Define the differential complex
$$  (\Lb^*(f),\ol{d}{}^*), \quad \Lb^*(f)=
        \us{q\geq 0}{\oplus} \Lb^q(f),      $$
where
$$  \Lb^q(f)=\Lb_{{\cF}_2}^q(N^m)\oplus \Lb_{{\cF}_1}^{q-1}(M^n), \quad
        \ol{d}{}^*(\om,\tht)=(-d_{{\cF}_2}^*\om,
            f^*\om+d_{{\cF}_1}^*\tht).    $$

        We easily verify that $\ol{d}{}^2=0$ and denote the cohomology
groups of this complex by $H^*(f)$. Note that the complex
$(\Lb^*(f),\ol{d}{}^*)$ is the cone of the cochain mapping
$f^*:\Lb_{{\cF}_2}^*(N^m)\to \Lb_{{\cF}_1}^*(M^n)$. If we
regraduate the complex $\Lb_{{\cF}_1}^*(M^n)$ as
$\wt{\Lb}_{{\cF}_1}^p(M_1^n)\equiv \Lb_{{\cF}_1}^{p-1}(M^n)$, then
we obtain an exact sequence of differential complexes
%   (2.5)
\begin{equation}
    0 @>>> \wt{\Lb}_{{\cF}_1}^*(M^n) @>{\al}>> \Lb^*(f) @>{\bt}>>
        \Lb_{{\cF}_2}^*(N^m) @>>> 0
\end{equation}
with the obvious mappings $\al$ and $\bt$: $\al(\tht)=(0,\tht)$,
$\bt(\om,\tht)=\om$. From (2.5) we have an exact sequence in
cohomologies
$$  \cdots @>>> H_{{\cF}_1}^{q-1}(M^n) @>{\al^*}>> H^q(f) @>{\bt^*}>>
        H_{{\cF}_2}^q(N^m) @>{\dl^*}>> H_{{\cF}_1}^q(M^n)
        @>>> \cdots\;.      $$

        It is easily seen that $\dl^*=f^*$. Let $\om\in \Lb_{{\cF}_2}^q(N^m)$
be the closed form, and $(\om,\tht)\in \Lb^q(f)$. Then
$\ol{d}(\om,\tht)=(0,f^*\om+d_{{\cF}_1}^{q-1}\tht)$, and by the
definition of the operator $\dl^*$ we have
$\dl^*[\om]=[f^*\om+d_{{\cF}_1}^{q-1}\tht]=f^*[\om]$. Hence we
finally get a long exact sequence
%   (2.6)
\begin{equation}
\ \hskip-1cm    \cdots @>>> H_{{\cF}_1}^{q-1}(M^n) @>{\al^*}>>
H^q(f) @>{\bt^*}>>
        H_{{\cF}_2}^q(N^m) @>{f^*}>> H_{{\cF}_1}^q(M^n)
        @>{\al^*}>> \cdots \;.
\end{equation}

%        COROLLARY 2.7.
\begin{cor}
If the foliations $(M^n,{\cF}_1)$ and $(N^m,{\cF}_2)$ are of the
$p$-th and $q$-th dimension, respectively, then
\begin{itemize}
\item[(i)] $\bt^*:H^{p+1}(f)\to H_{{\cF}_2}^{p+1}(N^m)$
        is an epimorphism, \\
            $\al^*:H_{{\cF}_1}^q(M^n)\to H^{q+1}(f)$ is an epimorphism, \\
            $\bt^*:H^i(f)\to H_{{\cF}_2}^i(N^m)$ is an isomorphism for $i>p+1$, \\
            $\al^*:H_{{\cF}_1}^i(M^n)\to H^{i+1}(f)$ is an isomorphism for $i>q$;

\item[(ii)] $H^i(f)=0$ for $i>\max\{p+1,q\}$.
\end{itemize}
\end{cor}

 %       THEOREM 2.8.
\begin{th}
If $f,g:(M^n,{\cF}_1)\to (N^m,{\cF}_2)$ are leaf-to-leaf
transforming homotopic mappings, then $H^*(f)=H^*(g)$.
\end{th}

%        PROOF.
\begin{pf}
Let $F:(M^n\times {\bR},\wh{{\cF}}_1)\to (N^m,{\cF}_2)$ be the
homotopy mapping between $f$ and $g$. Let $s_0$ and $s_1$ be the
zero and the unit section, respectively, of the trivial bundle
$M^n\times {\bR}@>{\pi}>> M^n$. Then $F\circ s_0=g$ and $F\circ
s_1=f$. Hence we have a homomorphism between the short exact
sequences
$$  \begin{CD}
        0 @>>> \wt{\Lb}_{\wh{{\cF}}_1}^*(M^n\times {\bR})
            @>{\al'}>> \Lb^*(F) @>{\bt'}>>
            \Lb_{{\cF}_2}^*(N^m) @>>> 0 \\
        @. @VV{s_1^*}V @VV{id\oplus s_1^*}V @VV{id}V \\
        0 @>>> \wt{\Lb}_{{\cF}_1}^*(M^n) @>>> \Lb^*(f) @>>>
            \Lb_{{\cF}_2}^*(N^m) @>>> 0
    \end{CD}\;.     $$

        This homomorphism defines a homomorphism  between the corresponding
long cohomologic sequences
$$  \begin{smallmatrix}%array}{ccccccccccccc}
        \cdots & \to & H_{{\cF}_2}^q(N^m) & \to
            & H_{\wh{{\cF}}_1}^q(M^n\times {\bR}) & \to
            & H^{q+1}(F) & \to
            & H_{{\cF}_2}^{q+1}(N^m) & \to
            & H_{\wh{{\cF}}_1}^{q+1}(M^n\times {\bR}) & \to
            & \cdots \\
        & & \downarrow{\vcenter{\rlap{$\scriptstyle id$}}}
            & & \downarrow{\vcenter{\rlap{$\scriptstyle s_1^*$}}}
            & & \downarrow{\vcenter{\rlap{$\scriptstyle \gm$}}}
            & & \downarrow{\vcenter{\rlap{$\scriptstyle id$}}}
            & & \downarrow{\vcenter{\rlap{$\scriptstyle s_1^*$}}}&&\\
        \cdots & \to & H_{{\cF}_2}^q(N^m) & \to
            & H_{{\cF}_1}^q(M^n) & \to
            & H^{q+1}(f) & \to
            & H_{{\cF}_2}^{q+1}(N^m) & \to
            & H_{{\cF}_1}^{q+1}(M^n) & \to
            & \cdots
    \end{smallmatrix} \;, $$%array}\;,          $$
where $\gm$ is the mapping induced by $id\oplus s_1^*$. Since
$s_1^*$ is an isomorphism (Lemma 2.3), by virtue of the lemma on
five homomorphisms we conclude that $\gm$ is also an isomorphism,
i.e., $H^*(f)\approx H^*(F)$. By a similar reasoning we can
conclude that $H^*(g)\approx H^*(F)$.
\end{pf}

        If $(M^n,{\cF}_1)$ is a subfoliation of  the  foliation
$(W^m,{\cF}_2)$, i. e.,  the embedding $M^n \str{j}{\hook} W^m$ is
simultaneously a leaf-to-leaf transforming mapping, then the
cohomology algebra $H^*(j)$ will be said to be the algebra of
relative vertical cohomologies. Denote it by
$H_{{\cF}_2,{\cF}_1}^*(W;M)$.

        Now sequence (2.6) can be rewritten as
%   (2.7)
\begin{gather}
    \cdots @>>> H_{{\cF}_1}^{q-1}(M) @>{\al^*}>> H_{{\cF}_2,{\cF}_1}^q(W;M)
        @>{\bt^*}>> H_{{\cF}_2}^q(W) @>{j^*}>> \notag  \\
    @>{j^*}>> H_{{\cF}_1}^q(M)
        @>{\al^*}>> \cdots \;.
\end{gather}

        Note that if we forget the structure of the foliation,
then, as is known, the embedding $M^n \str{j}{\hook} W^m$ defines
a long exact cohomological sequence of the pair $(W^m,M^n)$ in de
Rham's theory. One can easily verify that the homomorphism $i^*$
from Proposition 2.2 defines a morphism between the long exact
cohomological sequence of the pair $(W^m,M^n)$ in de Rham's theory
and sequence (2.7).


%        2.4.
\subsection{The Generalized Mayer--Vietoris Principle for Vertical Cohomologies.
            A Combinatorial Definition of Vertical Cohomologies.}
        Let  $(M^n,{\cF})$ be a smooth foliation of dimension
$k$; let $U=\{u_\al\}_{\al\in A}$ be an open countable covering of
the manifold $M^n$. Similarly to \v{C}ech-de Rham's theory, we
define a double complex which will be used to calculate vertical
cohomologies of the foliation $(M^n,{\cF})$.

        Denote by $u_{\al_0\cdots\al_p}$ the intersection of open sets
$u_0,\dots,u_p$ and by $\coprod$ the disjunctive union. Then we
have a sequence of open sets
$$  M^n \;\longleftarrow \;\coprod_{\al_0} u_{\al_0}\;
            \begin{matrix}
                {\scriptstyle \pa_0} \\[-2.5mm]
                \longleftarrow \\[-3mm]
                \longleftarrow \\[-2.5mm]
                {\scriptstyle \pa_1}
            \end{matrix} \;
        \coprod_{\al_0<\al_1} u_{\al_0\al_1} \;
            \begin{matrix}
                {\scriptstyle \pa_0} \\[-2.5mm]
                \longleftarrow \\[-3mm]
                {\scriptstyle \pa_1} \\[-3mm]
                \longleftarrow \\[-3mm]
                \longleftarrow \\[-2.5mm]
                {\scriptstyle \pa_2}
            \end{matrix} \;
        \coprod_{\al_0<\al_1<\al_2} u_{\al_0\al_1\al_2} \;
            \begin{matrix}
                \longleftarrow \\[-3mm]
                \longleftarrow \\[-3mm]
                \longleftarrow \\[-3mm]
                \longleftarrow
            \end{matrix} \; \cdots\;,           $$
where $\pa_i$ is the embedding $\pa_i(u_{\al_0\cdots
\al_p})=u_{\al_0\cdots \wh{\al}_i\cdots\al_p}$. This sequence of
embeddings induces a sequence of restriction mappings of vertical
forms
\begin{gather*}
    \Lb_{{\cF}}^*(M^n) \; \str{r^*}{\longrightarrow} \;
        \prod_{\al_0} \Lb_{{\cF}}^*(u_{\al_0})\;
            \begin{matrix}
                {\scriptstyle \dl_0} \\[-2.5mm]
                \longrightarrow \\[-3mm]
                \longrightarrow \\[-2.5mm]
                {\scriptstyle \dl_1}
            \end{matrix} \;
        \prod_{\al_0<\al_1} \Lb_{{\cF}}^*(u_{\al_0\al_1}) \;
            \begin{matrix}
                {\scriptstyle \dl_0} \\[-2.5mm]
                \longrightarrow \\[-3mm]
                {\scriptstyle \dl_1} \\[-3mm]
                \longrightarrow \\[-3mm]
                \longrightarrow \\[-2.5mm]
                {\scriptstyle \dl_2}
            \end{matrix} \;  \\
        \begin{matrix}
            {\scriptstyle \dl_0} \\[-2.5mm]
            \longrightarrow \\[-3mm]
            {\scriptstyle \dl_1} \\[-3mm]
            \longrightarrow \\[-3mm]
            \longrightarrow \\[-2.5mm]
            {\scriptstyle \dl_2}
        \end{matrix} \;
        \prod_{\al_0<\al_1<\al_2} \Lb_{{\cF}}^*(u_{\al_0\al_1\al_2}) \;
            \begin{matrix}
                \longrightarrow \\[-3mm]
                \longrightarrow \\[-3mm]
                \longrightarrow \\[-3mm]
                \longrightarrow
            \end{matrix} \; \cdots\;,
\end{gather*}
where $\dl_i$ is induced by the imbedding $\pa_i$, i.e.,
$\dl_i=(\pa_i)^*$.

        Let us define the difference operator $\dl$ by the following
rule: if $\om_{\al_0\cdots\al_p}\in
\Lb_{{\cF}}^q(u_{\al_0\cdots\al_p})$ denotes the components of the
element $\om\!\in\!\! \prod\limits_{\al_0<\cdots<\al_p}\!\!\!\!
\Lb_{{\cF}}^q(u_{\al_0\cdots\al_p})$, then
%   (2.8)
\begin{equation}
    (\dl\om)_{\al_0\cdots\al_{p+1}}=\sum_{i=0}^{p+1}(-1)^i
        \om_{\al_0\cdots\wh{\al}_i\cdots\al_{p+1}},
\end{equation}
where $\om_{\al_0\cdots\wh{\al}_i\cdots\al_{p+1}}\equiv \dl_i
        (\om_{\al_0\cdots\al_{i-1}\al_{i+1}\cdots\al_{p+1}})$.

        By a standard reasoning one can verify that $\dl^2=0$.

        Consider now the double complex
%   (2.9)
\begin{equation}
    K^{p,q}\equiv C^p(U,\Lb_{{\cF}}^q)=\prod_{\al_0<\cdots<\al_p}
        \Lb_{{\cF}}^q(u_{\al_0\cdots\al_p}).
\end{equation}

whose horizontal mappings are the operators $\dl^*$ and vertical
mappings are the operators $d_{{\cF}}^*$. As is known, this double
complex can be reduced to an ordinary differential complex
$(K^*,D^*)$:
$$  K^n=\us{p+q=n}{\oplus} K^{p,q}, \quad D^n=\dl^p+(-1)^pd_{{\cF}}^q
        \quad \text{on} \quad K^{p,q}.      $$

%        LEMMA 2.9.
\begin{lem}
The sequence
$$  \cdots @>>> \Lb_{{\cF}}^*(M^n) @>{r^*}>> K^{0,*} @>{\dl^0}>> K^{1,*}
        @>{\dl^1}>> K^{2,*} @>{\dl^2}>> \cdots      $$
is exact.
\end{lem}

%        PROOF.
\begin{pf}
Let $q\geq 0$ be an integer number. Clearly, $\Lb_{{\cF}}^q(M^n)$
is the kernel of the first operator $\dl^0$, since an element from
$\prod\limits_{\al_0}\Lb^q(u_{\al_0})$ is a global form on $M^n$
if and only if its components consistent at the intersections.

        Let $\{\tht_\al\}_{\al\in A}$ be the partitioning of unity
subordinate to the covering $U=\{u_\al\}_{\al\in A}$. If $\om\in
K^{p,q}$ is the cocycle of the operator $\dl^p$, then we can
assign to it $p-1$ cochains $K\om$ by the formula
$(K\om)_{\al_0\cdots\al_{p-1}}=\sum\limits_\al \tht_\al
    \om_{\al\al_0\cdots\al_{p-1}}$ (it is assumed here that
$\om_{\dots,\al,\dots,\bt,\dots}=-\om_{\dots,\bt,\dots,\al,\dots}$,
if $\al>\bt$; clearly, this is consistent with the operation
$\dl$, i.e., $(\dl\om)_{\dots,\bt,\dots,\al,\dots}=
    -(\dl\om)_{\dots,\al,\dots,\bt,\dots}$). In that case
\begin{gather*}
    (\dl^{p-1}K\om)_{\al_0\cdots\al_p}=\sum_i (-1)^i
        (K\om)_{\al_0\cdots\wh{\al}_i\cdots\al_p}=
        \sum_{i,\al} (-1)^i\tht_\al
        \om_{\al\al_0\cdots\wh{\al}_i\cdots\al_p}= \\
    =\sum_\al\tht_\al[\om_{\al_0\cdots,\al_p}-
        (\dl\om)_{\al\al_0\cdots\al_p}]=
        \sum_\al\tht_\al \om_{\al_0\cdots\al_p}=\om_{\al_0\cdots\al_p}.
\end{gather*}

        Hence $\dl^{p-1}(K\om)=\om$.
\end{pf}

%        THEOREM 2.10.
\begin{th}
The cohomologies of the double complex $K^{p,q}$, i.e., the
cohomologies of the complex $(K^*,D^*)$ are isomorphic to the
vertical cohomologies $H_{{\cF}}^*(M^n)$. This isomorphism is
obtained  by the mapping of the restriction $r^*$.
\end{th}

%        PROOF.
\begin{pf}
Since
$D^*r^*=(\dl^0+d_{\cF}^*)r^*=\dl^0r^*+d_{\cF}^*r^*=d_{\cF}^*r^*=
    r^{*+1} d_{\cF}^*$, $r^*$ is a cochain mapping, it induces the
mapping in cohomologies $r_*$.

        Let $\vf\in K^m$, be the cocycle, i.e., $D^m\vf=0$. We can represent
$\vf$ as a sum $\vf=\vf^{0,m}+\vf^{1,m-1}+\cdots+\vf^{p,m-p}$,
where $\vf^{i,j}\in K^{ij}$ and $\vf^{p,m-p}\neq 0$. Then
$\dl^p\vf^{p,m-p}=0$. Hence by Lemma 2.9 we find that there exists
an element $\vf'\in K^{p-1,m-p}$ such that
$\dl^{p-1}\vf'=\vf^{p,m-p}$. Then the element $\vf-D^{m-1}\vf'$ is
obviously cohomologic to $\vf$ and has no component in
$K^{p,m-p}$. After repeating this procedure several times, we
obtain an element $\wt{\vf}\in K^{0,m}$ which is cohomologic to
$\vf$. For this element we have $d_{\cF}^m\wt{\vf}=0$,
$\dl^0\wt{\vf}=0$, i.e., $\wt{\vf}$ defines the global vertical
closed form on $M^n$ which by means of $r^m$ transforms to
$\wt{\vf}$. This shows that the mapping $r_*^m$ is epimorphic. Let
us now show that the mapping $r_*^m$ is monomorphic.

        Let $\vf\in \Lb_{\cF}^m(M^n)$ and $d_{\cF}^m\vf=0$. Then if
$r^m(\vf)=D^{m-1}\vf'$, $\vf'\in K^{m-1}$,
$\vf'=\vf'{}^{0,m-1}+\cdots+\vf'{}^{p,m-p-1}$, where
$\vf'{}^{p,m-p-1}\neq 0$, then $\dl^p\vf'{}^{p,m-p-1}=0$ (because
$D^{m-1}\vf'$ has only one component in $K^{0,m}$).

        Proceeding as above, we can find an element $\vf''$ such that
$\vf''\in K^{0,m-1}$ and $\vf'-\vf''\in D^{m-2}(K^{m-2})$. Then
$r^m(\vf)=d_{\cF}^{m-1}\vf''$ and $\dl^0\vf''=0$. Therefore
$\vf''$ defines a global form $\wt{\vf}''$ on $M^n$ such that
$d_{\cF}^{m-1}\wt{\vf}''=\vf$.
\end{pf}  %        Thus the proof is complete.

        By analogy with de Rham's theory Theorem 2.10 can be called
the generalized Mayer--Vietoris principle.

        From the lower row of the double complex $K^{*,*}$ let us
choose a subcomplex, namely, a kernel of the differential
$d_{\cF}^0$. We get a sequence
%   (2.10)
\begin{equation}
    C_{\cF}^0(U) @>{\dl^0}>> C_{\cF}^1(U) @>{\dl^1}>>
        C_{\cF}^2(U) @>{\dl^2}>> \cdots \;,
\end{equation}
where $C_{\cF}^i(U)\equiv \ker d_{\cF}^0\sbs
\prod\limits_{\al_0<\cdots<\al_i}
    \Lb_{\cF}^0(u_{\al_0\cdots \al_i})$, and $\dl^*$
is the difference operator defined above. The cohomologies of
complex (2.10) will be called the \v{C}ech cohomologies of the
foliation $(M^n,\cF)$ for the covering $U$. They are a purely
combinatorial object and will be denoted by $H_{\cF}^*(U)$.  Let
the covering $U=\{u_\al\}_{\al\in A}$ consist of foliated open
sets such that all finite nonempty intersections are contractible.
Any manifold is known to have such a covering. Then, in view of
the fact that Poincare's lemma ([I]) is valid for vertical
cohomologies, we obtain

%        LEMMA 2.11.
\begin{lem}
The sequence
$$  0 @>>> C_{\cF}^p(U) @>{j^p}>> \prod_{\al_0<\cdots<\al_p}
        \Lb_{\cF}^0 (u_{\al_0\cdots\al_p})
        @>{d_{\cF}^0}>> \prod_{\al_0<\cdots<\al_p}
        \Lb^1 (u_{\al_0\cdots\al_p}) @>{d_{\cF}^1}>> \cdots     $$
is exact, $p\geq 0$, where $j^p$ is the embedding.
\end{lem}

By Lemma 2.11 and the same reasoning as we used in proving Theorem
2.10  we can prove

%        THEOREM 2.12.
\begin{th}
The cochain mapping $j^*$ defines an isomorphism between the
\v{C}ech cohomologies $H_{\cF}^*(U)$ and cohomologies of the
double complex $K^{*,*}$. Hence the \v{C}ech cohomologies and
vertical ones are isomorphic:
$$  H_{\cF}^*(U)\approx H_{\cF}^*(M^n). $$
\end{th}

%        COROLLARY 2.13.
\begin{cor}
A zero-dimensional vertical cohomology $H_{\cF}^0(M^n)$ is
\linebreak isomorphic to a group of smooth functions on the
foliated manifold  $M^n$, which are constant on the leaves.
\end{cor}

%        \S 3.
\section{Completely Integrable Hamiltonian Systems and Vertical
        Cohomologies}

%        3.1.
\subsection{Topology of Constant Energy Surfaces of Completely Integrable
                Hamiltonian Systems}

        In this subsection we shall briefly recall the basic facts from the
topological theory of integrable Hamiltonian systems ([5]).

        Let $v=\sgrad H$ be a Hamiltonian system on the symplectic manifold
$(M^{2n},\om)$, where $\om$ is a symplectic 2-form on $M^n$, and
let $v$ be integrable. Thus there exist $n$ independent (almost
everywhere) smooth integrals $f_1=H$, $f_2,\dots,f_n$ in
involution, i.e., $\{f_i,f_j\}=0$, $i,j=1,n$, where $\{\ ,\ \}$ is
the Poisson bracket. Let $F:M^{2n}\to {\bR}^n$ be the moment
mapping which corresponds to these integrals, i.e.,
$F(x)=(f_1(x),\dots,f_n(x))$, $x\in M^{2n}$. Let $N$ be the set of
critical points of the moment mapping,
 and $\Sigma=F(N)$ be the set of all critical values which is
called the bifurcation diagram.

        Clearly, we have two cases: (a) $\dim \Sigma<n-1$ and
(b)~$\dim\Sigma=n-1$. In the case (a) the set $\Sigma$ does not
separate the space ${\bR}^n$ and therefore  all nonsingular leaves
$B_a=F^{-1}(a)$ are diffeomorphic to one another (it is well known
that if they are compact, then they are diffeomorphic to the tori
$T^n$, and if they are noncompact, then they are diffeomorphic to
the cylinders $T^k\times {\bR}^{n-k}$). The case (b) is more
difficult. Below we shall consider a theorem from [5].

        Suppose that the restriction $f=f_1|_{X^{n+1}}$ to a joint compact
nonsingular surface of the level of the rest of $n-1$ integrals
$X^{n+1}=\{x\in M^{2n}|f_i(x)=c_i$, $i=\ol{2,n}\}$ is a Bott
function, i.e., all critical points of this restriction are
organized into nondegenerate critical submanifolds (a critical
submanifold $L^k\sbs X^{n+1}$ is nondegenerate if the restriction
of the function $f$ to every normal plane $P^{n+1-k}$ has a
nondegenerate Morse singularity at the point $P^{n+1-k}\cap L^k$).

        Let $c_1$ be a critical value of the function $f$ on
the surface $X^{n+1}$, and let $c=(c_1,\dots,c_n)$, i.e.,
$c=(c_1,c_2,\dots,c_n)\in \Sigma$. Let $B_c=F^{-1}(c)$ be a
critical fiber of the moment mapping. Thus $B_c=\{f_1=c_1\}$ is a
critical level surface of $f_1$ on $X^{n+1}$.

%        THEOREM 3.1
\begin{th}[A. T. Fomenko]
Each connected compact component $B_c^0$ of the critical fiber
$B_c$ is homeomorphic to a set which is one of the following four
types: $(1)$ a torus $T^n$; $(2)$ the nonorientable manifolds
$K_0^n$ and $K_1^n$; $(3)$ a torus $T^{n-1}$; or $(4)$ a cell
complex $T^n_1\cup T^n_2$ obtained by removing  $n-1$-dimensional
tori $T^{n-1}_1$ from $T_1^n$ and $T^{n-1}_2$ from $T_2^n$, which
realize nonzero generators of the homology groups
$H_{n-1}(T_1^n,{\bZ})$ and $H_{n-1}(T_2^n,{\bZ})$,  and glueing
$T_1^n$ and $T_2^n$ together by identifying  only the tori
$T_1^{n-1}$ and $T_2^{n-1}$ $($by means of a diffeomorphism$)$. In
cases $(1)$--$(3)$  the critical fibers  consist entirely of
critical points of the function $f$ on which a maximum or a
minimum is attained. In case $(4)$ the critical points of $f$ in
the critical fiber $B^0_c$ forms a torus $T^{n-1}$ $($the result
of glueing together the tori $T^{n-1}_1$ and $T^{n-1}_2)$ which is
a ``saddle'' for the function $f$.
\end{th}

        The manifolds $K_1^n$ and $K_2^n$ from Theorem 3.1 are the factor-sets
of the torus $T^n$ generated by the action of some group on $T^n$
without fixed points ([5]).

    It is important to investigate a particular case of
the integrable Hamiltonian system $v=\sgrad H$ on the
four-dimensional symplectic manifold $(M^4,\om)$. For mechanical
and physical reasons, it is useful to study the integrability
effect on an individual isoenergetic surface $Q^3\sbs M^4$ given
by the equation $H(x)=h$, where $h$ is a regular value of the
Hamiltonian $H$. The restriction of the system $v$ on $Q^3$ will
be denoted by the same symbol $v$. In what follows $Q^3$ will be
assumed to be the closed manifold. We also assume that an
additional integral $f$ is the Bott function on $Q^3$. Then to the
critical fibers from Theorem 3.1 there correspond the following
manifolds:

(1)~the torus $T^2$;

(2)~Klein's bottle $K^2$;

(3)~the circle $S^1$; (4) a piecewise smooth two-dimensional
polyhedron with a singularity of the type of "a fourfold line" (a
transversal intersection of two planes).

        The Hamiltonian $H$ is called the nonresonance one on $Q^3$
if the set of Liouville tori with irrational windings is
everywhere dense in $Q^3$.

    We say that the Hamiltonian system $v$ is integrable in the Bott sense
if among Bott functions there is an additional first integral of
$v$.


%        DEFINITION 3.2
\begin{defi}[{[6]}]
Two  integrable, in the Bott sense, nonresonance Hamiltonian
systems $v_1,v_2$ on the oriented manifolds $Q_1^3,Q_2^3$ are said
to be topologically equivalent if there exists an orientation
preserving diffeomorphism $g:Q_1^3\to Q_2^3$ which transforms the
Liouville tori of the system $v_1$ to those of the system $v_2$
(critical tori are included into the number of Liouville tori),
and the isolated critical circles to the isolated critical circles
with the same orientation which is by the field $v$.
\end{defi}

The partitioning of the manifold $Q^3$ into Liouville tori and
critical level surfaces of the Bott integral $f$ is called the
Liouville foliation on $Q^3$ (for a given integrable nonresonance
Hamiltonian system $v$). Obviously, the diffeomorphism $g$,
appearing in Definition 3.2, is a leaf-to-leaf mapping between
Liouville foliated manifolds.

        Note that the Liouville foliation defined above does not depend on
a choice of the Bott integral $f$.

        An analogous definition can be introduced for the multidimensional
case as well.  Two completely integrable, in a Bott sense,
nonresonance Hamiltonian systems $v_1$ and $v_2$ on the
nonsingular level surfaces $X_1^{n+1}$ and $X_2^{n+1}$ are said to
be topologically equivalent if there exists a diffeomorphism
$h:X_1^{n+1}\to X_2^{n+1}$ which transforms the Liouville tori of
the system $v_1$ to those of the system $v_2$ (critical tori $T^n$
are included into the number of Liouville tori), and critical
fibers $B_c$ of the system $v_1$ to those of the system $v_2$.

Note that the Liouville foliation is not a foliation in the sense
of \S 2.

Let $v=\sgrad H$ be a completely integrable, in the Bott sense,
nonresonance Hamiltonian system on the isoenergetic surface $Q^3$.
As we already know, this system defines the Liouville foliation on
$Q^3$. Denote it by $\cF$.

        If from $Q^3$ we remove all critical
(i. e., maximal, minimal and "saddle"-type)  circles $S^1$, then
on the resulting open manifold $Q^{'3}$  the foliation $\cF$
induces a true foliation $\cF'$. Thus we have defined the vertical
cohomology groups $H_{\cF'}^i(Q^{'3})$, $i=0,1,2$. Similarly, one
can derive the vertical cohomology groups $H_{\cF'}^i(X^{'n+1})$
in the multidimensional case, $i=0,1,\dots,m$.

        If now  $v_1$ and $v_2$ are assumed to be the topologically
equivalent systems on the manifolds $Q_1^3$ and $Q_2^3$,
respectively, then the diffeomorphism $g:Q_1^3\to Q_2^3$ which
preserves the Liouville foliation obviously induces the
leaf-to-leaf transforming diffeomorphism $g':Q_1^{'3}\to
Q_2^{'3}$. Therefore the following theorem is valid.

%        THEOREM 3.3.
\begin{th}
The cohomology groups  $H_{\cF'}^i(Q^{'3})$ are a topological
invariant of completely integrable, in the Bott sense,
nonresonance Hamiltonian systems $v=\sgrad H$ on the isoenergetic
surface $Q^3$.
\end{th}

       Note that in the multi-dimensional case the groups
$H_{\cF'}^i(X^{'n+1})$, $i=\ol{0,n}$, are also  topological
invariants.

      Thus to determine a topological invariant we had to deal with
the open manifold $Q^{'3}$. To avoid this, we shall slightly
modify the definition of cohomology groups.

       The isoenergetic surface $Q^3$ can be considered  as the set of
Liouville tori, circles $S^1$ (which are maximal, minimal and
"saddle"-type)  and rings of the type $S^1\times (0,1)$ (which are
obtained from a hyperbolic critical fiber of type (4) (Theorem
3.1) by removing hyberbolic critical circles of the integral $f$).
We denote this partitioning of the manifold $Q^3$ by $\cP$. By
$S(Q^3)$ we denote a Lie algebra of smooth vector fields on $Q^3$
and by $\ol{S}(Q^3)$ a Lie subalgebra of the algebra $S(Q^3)$
consisting of smooth vector fields on $Q^3$ tangent to the
manifolds of the partitioning $\cP$. The algebra $C^\infty(Q^3)$
is, obviously, a module over $\ol{S}(Q^3)$ with respect to the
product $x\cdot g=xg$, $x\in \ol{S}(Q^3)$, $g\in C^\infty(Q^3)$.

        Under a $k$-dimensional cochain of the algebra $\ol{S}(Q^3)$
with coefficients in $C^\infty(Q^3)$ we shall mean an element of
the space $A^k(\ol{S}(Q^3))$ which is a skew-symmetric $k$-linear
functional on $\ol{S}(Q^3)$ with values in $C^\infty(Q^3)$. The
differential $d^k:A^k(\ol{S}(Q^3))\to A^{k+1}(\ol{S}(Q^3))$ is
defined by the formula
\begin{gather*}
    d^k\al(X_1,\dots,X_{k+1})=\sum_{i=1}^{k+1}(-1)^{i-1}X_i
        \al(X_1,\dots,\wh{X}_i,X_{i+1},\dots,X_{k+1})+  \\
    +\sum_{i<j} (-1)^{i+j} \al([X_i,X_j],X_1,\dots,\wh{X}_i,\dots,\wh{X}_j,
        \dots,X_{k+1}),
\end{gather*}
where $\al\in A^k(\ol{S}(Q^3))$; $X_1,\dots,X_{k+1}\in
\ol{S}(Q^3)$.

        It is easy to verify that $d^k\circ d^{k-1}=0$. Thus
$(A^*(\ol{S}(Q^3)),d^*)$ is a cochain complex. We denote the
cohomologies of the complex $(A^*(\ol{S}(Q^3)),d^*)$ by
$H_{\cP}^*(Q^3)$. Obviously, $H_{\cP}^i(Q^3)=0$, $i>2$.

        We have also defined the cohomology groups $H_{\cP}^*(X^{n+1})$
for the miltidimensional case. Note that here
$H_{\cP}^i(X^{n+1})=0$, $i>n$.

        Obviously, $A^*(\ol{S}(Q^3))$ is a graduated ring, and
the differential $d$ is the antidifferentiation. Hence the
cohomologies $H_{\cP}^*(Q^3)$ also acquire a ring structure.

        If now $v_1$ and $v_2$ are as above the topologically equivalent
systems on the manifolds $Q_1^3$ and $Q_2^3$, respectively, then
the diffeomorphism $g:Q_1^3\to Q_2^3$ defines an isomorphism
between the differential complexes $(A^*(\ol{S}(Q_1^3)),d^*)$ and
$(A^k(\ol{S}(Q_2^3)),d^*)$. Therefore we have

%        THEOREM 3.4.
\begin{th}
Let $v=\sgrad H$ be an integrable, in the Bott sense, nonresonance
Hamiltonian system on the isoenergetic surface $Q^3$, and let
$\cP$ be the partitioning of the manifold $Q^3$ into Liouville
tori, critical circles of the additional Bott integral $f$ and
rings of the type $S^1\times (0,1)$. Then the cohomology groups
$H_{\cP}^*(Q^3)$ are topological invariants of the system $v$,
i.e., to the topologically equivalent systems there correspond the
isomorphic groups.
\end{th}

       An analogous theorem holds in the multidimensional case as well.
Here the partitioning $\cP$ consists of Liouville tori $T^n$,
$(n-1)$-dimensional tori $T^{n-1}$ and rings of the type
$T^{n-1}\times (0,1)$.

        Since $A^0(\ol{S}(X^{n+1}))=C^\infty(X^{n+1})$, we have
\begin{gather*}
    H_{\cP}^0(X^{n+1})=\ker(d^0:C^\infty(X^{n+1})\to
        A^1(\ol{S}(X^{n+1}))= \\
    =\big\{g\in C^\infty(X^{n+1})\,|\,Xg=0,\;\;
        \forall X\in \ol{S}(X^{n+1}\big\}=
        \Inv_{\ol{S}}C^\infty(X^{n+1}),
\end{gather*}
where $\Inv_{\ol{S}}C^\infty(X^{n+1})$ denotes the set of smooth
functions on $X^{n+1}$ whose restrictions are constant functions
on the elements of the partitioning  $\cP$.

        Denote by $G$ the factor set of the space $X^{n+1}$ with respect
to Liouville tori $T^n$ and the connected components of critical
fibers of the integral $f$. If we introduce the factor topology on
$G$, then $H_{\cP}^0(X^{n+1})$ will coincide with the set of
continuous functions on $G$ whose liftings to $X^{n+1}$ by the
natural projection $X^{n+1}\to G$ are smooth functions.

%        REMARK 3.5.
\begin{rem}
For the four-dimensional case, a complete topological invariant
was introduced in [6]. This is the so-called labelled molecule
consisting of a graph with edges to which are attached rational
numbers from $[0,1)$ or $\infty$.  Note that in the
four-dimensional case the above-mentioned $G$ coincides with the
graph-molecule from [6]. Therefore the zero-dimensional groups
$H_{\cP}^0(X^{n+1})$ already pick up "nonzero" information on the
topological equivalence of integrable Hamiltonian systems.
\end{rem}

%        \S 4.
\section{The Case of a Spherical Pendulum}

        For a spherical pendulum the phase space is a cotangent bundle to
the two-dimensional sphere $T^*S^2$, where $S^2=\{x\in
{\bR}^3:x_1^2+x_2^2+x_3^2=1\}$. Using the Riemannian metric on
$S^2$, we can identify $T^*S^2$ with the tangent bundle $TS^2$ and
define the Hamiltonian of the system by the energy  function
%   (4.1)
\begin{equation}
    E(x,v)=\frac{1}{2}\langle v,v \rangle+x_3,\quad x\in S^2,
        \quad v\in T_xS^2.
\end{equation}

        A kinetic moment with respect to the $x_3$-axis has the form
%   (4.2)
\begin{equation}
    I(x,v)=x_1v_2-v_1x_2.
\end{equation}

        The critical points of the moment mapping $F=(E,I):TS^2\to {\bR}^2$
are the points $x=\pm(0,0,1)$, $v=0$ and $v=\al(-x_2,x_1,0)$,
$1+\al^2x_3=0$, $x_3\neq \pm 1$. The corresponding singular values
of the moment mapping are
%   (4.3)
\begin{equation}
    F=(\pm 1,0) \;\; \text{and} \;\;
        F=\Big(\frac{1}{2}\al^2-\frac{3}{2}\al^{-2},\al-\al^{-3}\Big),
        \;\; \text{where} \;\; |\al|>1.
\end{equation}
 When $\al$ tends to $\pm 1$, the point of the curve defined by
formula (4.3) tends to $(-1,0)$. For $I\neq0$, i.e., for $x\neq
\pm (0,0,1)$, we can introduce the polar coordinates
$$  x_1=\sin\vf\cos\tht, \quad x_2=\sin\vf\sin\tht,\quad x_3=\cos\vf, $$
where $\tht\in [0.2\pi]$, $\vf\in ]0,\pi[$.

        The functions $E$ and $I$ in terms of these coordinates are written as
%   (4.4)
\begin{equation}
    E=\frac{1}{2}\,\dot{\vf}^2+V_I(\vf), \quad I=(\sin^2\vf)\dot{\tht},
\end{equation}
where $V_I(\vf)=\frac{1}{2}(\sin^2\vf)\dot{\tht}^2+\cos\vf$ is the
effective potential, and $(\vf,\tht,\dot{\vf},\dot{\tht})$ define
the local coordinate system on $TS^2$.  The image $F$ is given  by
the relation $I=\al-\al^{-3}$, $E\geq
\frac{1}{2}\al^2-\frac{3}{2}\al^{-2}$, $|\al|\geq 1$. For regular
values of the mapping $F$ we have
$E>\frac{1}{2}\al^2-\frac{3}{2}\al^{-2}$; the point $(0,1)$ should
be discarded. %(see Fig. 1).
%\bigskip
%\input tevd-fig.pic

%\centerline{Fig. 1}
%\smallskip
        Let us consider the function $I$ on the isoenergetic surface
$Q^3=\{E=\frac{1}{2}\}$. If $(\vf,\tht,\dot{\vf},\dot{\tht})$ is a
critical point of the function $I$ on $Q^3$, then at that point
$\grad E$ and $\grad I$ are colinear. Hence we obtain
$\vf_0=\arccos(\frac{1-\sqrt{13}}{6})$,
$\dot{\tht}_1=\sqrt{\frac{\sqrt{13}+1}{2}}$,
$\dot{\tht}_2=-\sqrt{\frac{\sqrt{13}+1}{2}}$. Thus gives us two
isolated critical circles
$S_1=\{(\vf_0,\tht,\dot{\vf}=0,\dot{\tht}_1)| \tht\in [0,2\pi]\}$
and $S_2=\{(\vf_0,\tht,\dot{\vf}=0,\dot{\tht}_2)| \tht\in
[0,2\pi]\}$. Using the method of constant Lagrange multipliers, we
can conclude that $S_1$ is the maximal critical circle, and $S_2$
is the minimal one.

        By Morse's  lemma it follows that the compact isoenergetic surface
corresponding to the values of energy $E$ close to $-1$ is the
three-dimensional
 sphere $S^3$. Therefore $Q^3$ is also diffeomorphic to $S^3$, since
the points $\pm 1$ are critical values of the integral $E$. The
isoenergetic surface $Q^3$ can be represented as two solid tori
$S_1^1\times D^2$ and $S_2^1\times D^2$ glued together along the
boundary tori by means of a diffeomorphism which transforms the
parallel to the meridian, and vice versa (here $D^2$ is a
two-dimensional disk, and the central circles of the  solid tori
$S_1^1\times \{0\}$ and $S_2^1\times \{0\}$ coincide with the
critical circles of the integral $I$); the tori $S_i^1\times S_r$,
$r\in (0,1]$, $i=1,2$, where $S_r=\{x\in D^2\,|\,\|x\|=r\}$, are
the usual Liouville tori of the Hamiltonian system $v=s\grad E$.
In that case, using the notation from [6], the labelled molecule
has the form
$$  A \bullet\!\!\!\us{r=0}{\text{---\!\!---\!\!---\!\!---}}\!\!\!\bullet A, $$
and the factor set $G$ coincides with the segment.

        Now consider the function $I$ on the isoenergetic surface
$Q^3=\{E=\al>1\}$. A direct calculation shows that, as in the
preceding case, for the integral $I$ we have two isolated critical
circles $S_1=\{(\vf_0,\tht,0,\dot{\tht}_1)\,|\,\tht\in [0,2\pi]\}$
and $S_2=\{(\vf_0,\tht,0,\dot{\tht}_2)\,|\,\tht\in [0,2\pi]\}$,
where $\vf_0=\arccos\frac{\al-\sqrt{\al^2+12}}{3}$ and
$\dot{\tht}_1=-\sqrt{\frac{3}{\sqrt{\al^2+12}-\al}}$,
$\dot{\tht}_2=\sqrt{\frac{3}{\sqrt{\al^2+12}-\al}}$.

        In terms of the coordinates
$(\vf,\tht,\dot{\vf},\dot{\tht})$ the Euclidean metric is written
as $ds^2=d\vf^2+\sin^2\vf d\tht^2$ so that the the equation
$E=\al$ takes the form
$$  \|\xi\|^2+2\cos\vf=2\al,    $$
where $\xi=(\dot{\vf},\dot{\tht})$ is the tangent vector at the
point $(\vf,\tht)\in S^2$, and $\|\xi\|$ denotes the norm of
$\xi$. Hence it follows that the norm of the vector $\xi$ is a
function of $\vf$,and $\|\xi\|\neq 0$. Therefore $Q^3$ is
diffeomorphic to the space  $T_1S^2$, where
$T_1S^2=\{(a,\xi)\,|\,x\in S^2$, $\|\xi\|=1\}$. As is well-known,
$T_1S^2$ is diffeomorphic to the three-dimensional projective
space ${{\bR} P}^3$ so that $Q^3\approx {{\bR} P}^3$. As above,
$Q^3$ can be represented as two solid tori $S_1^1\times D^2$ and
$S_2^1\times D^2$ glued together along the boundary tori by means
of the diffeomorphism $h$ whose corresponding induced mapping
$h_*:H^1(T^2;{\bZ})\to H^1(T^2;{\bZ})$ is given by the matrix
$\begin{pmatrix} 1 & 2 \\ 1 & 1 \end{pmatrix}$. Thus for the
integrable Hamiltonian system $v=\sgrad E$ on $Q^3$ the labelled
molecule has the form
$$  A \bullet \!\!\!\us{r=\frac{1}{2}}{\text{---\!\!---\!\!---\!\!---}}\!\!\!\bullet A, $$
and the factor set again coincides with the segment. This example
shows that vertical cohomologies are not a complete topological
invariantof integrable Hamiltonian systems.


\section*{Acknowledgements}

The author expresses his gratitude to Dr. Z. Giunashvili for
helpful discussions.

%This research was partly carried out under Grant No.
%RJV200 from  the Intenational Science Foundation.

\vskip+0.8cm

\centerline{\sc References}
\vskip+0.4cm

1. Jos\'{e} M. Figueroa-o'Farrill, A topological characterization of classical
    BRST cohomology. {\em Commun. Math. Phys.}
    {\bf 127}(1990), 181--186.

2. M. Henneaux and C. Teitelboim,. BRST cohomology in classical mechanics.
    {\em Commun. Math. Phys.} {\bf 115}(1988), 213--230.

3. J. Fisch, M. Henneaux, J. Stasheff, and C. Teitelboim, Existence, uniqueness,
    and cohomology of the classical BRST charge with ghosts of ghosts.
    {\em Preprint,} 1988.

4. I. Tamura, Topology foliations. {\em Japan: Iwanami Shoten,} 1976.

5. A. T. Fomenko, The topology of surfaces of constant energy in integrable
    hamiltonian systems, and obstructions to integrability.
    {\em Math. USSR Izvestiya} {\bf  29}(1987), No. 3, 629--658.

6. A. V. Bolsinov, S. V. Matveev, and A. T. Fomenko, Topological classification of
    integrable Hamiltonian systems with two degrees of freedom.
    A list of systems of small complexity.
    (Translation from Russian) {\em  Math. Surveys} {\bf 45}(1990), No. 2,
    49--77.

\

\

%\centerline{(Received 1.03.1996)}

\

Author's address:

Faculty of Mechanics and Mathematics

I. Javakishvili Tbilisi State University

2, University St., Tbilisi 380043  Georgia

 E-mail address:
tevzaz@@ti.net.ge
\end{document}

