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\begin{document}
\date{Received July 19, 2001; received in final form May 23, 2003}
\title{Alexander-Spanier cohomology of foliated manifolds}
\author{Xos\'e M. Masa}
\address{Departamento de Xeometr\'{\i}a e Topolox\'{\i}a\\
Universidade de Santiago de Compostela\\
15782-Santiago de Compostela\\
Spain}
\email{xmasa@zmat.usc.es}
\thanks{Partially supported by grant DGICYT PB98-0618}
\subjclass{57R30, 55N30}
\begin{abstract}
For a smooth foliated manifold $(M,\mathcal F)$, the basic and
the foliated cohomologies are defined by using the de Rham complex of $M$.
These cohomologies are related with the cohomology of the manifold by
the de Rham spectral
sequence of $\mathcal F$.
A foliated manifold is an example of a space with two topologies, one coarser
than the other. For these spaces one can define a continuous cohomology
that, for a foliated manifold, corresponds to the continuous
foliated (or leafwise)
cohomology.
In this paper we introduce a construction
for spaces with two topologies based upon the Alexander-Spanier
continuous cochains. It allows us
to define a spectral sequence, similar to the de Rham spectral
sequence for a
foliation. In particular, continuous basic and foliated cohomologies
are defined and related
with the cohomology of the space.
For a smooth foliated manifold, we also consider Alexander-Spanier
differentiable cochains. We compare the continuous and differentiable
cohomologies, and the latter with the de Rham cohomology. We prove that all
three spectral sequences are isomorphic from $E_2\/$ onwards
if $\mathcal F\/$
is a Riemannian foliation. As a consequence, we conclude that this spectral
sequence is a
topological invariant of the Riemannian foliation.
We also
compute some examples. In particular, we give an isomorphism between the
$E_2\/$ term for a $G$-Lie foliation and the \emph{reduced
cohomology\/} of $G\/$
(in the
sense of S.-T. Hu) with coefficients in the reduced foliated cohomology of
$\mathcal F$.
\end{abstract}
\maketitle
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\newcommand\diff{\operatorname{diff}}
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\section{Introduction}
Let $(M,\mathcal F)$ be a $C^\infty$ foliated
manifold. Associated to $\mathcal F$, there is a
filtration of the de Rham complex of $M$, $A^\ast(M)$: a
smooth form of degree
$i$ is said to be of filtration $\ge p$ if it vanishes whenever
$i-p+1$ of the vectors are
tangent to the foliation. The associated spectral
sequence $E_{2,dR}({\mathcal F})$ is called the \emph{de Rham
spectral sequence}, and converges to
the de Rham cohomology of
$M$ (see (\ref{sedR}) below). This spectral sequence is an important
$C^\infty$ invariant of the
foliation. It relates the basic cohomology and the foliated
cohomology of $\mathcal F\/$ with
the cohomology of $M$. The goal of this paper is to prove that, for Riemannian
foliations, the de Rham spectral sequence is a topological invariant
from $E_2$ onwards. For basic
cohomology this is a result by El~Kacimi and Nicolau \cite{AM}. J.A.
\'Alvarez L\'opez and
the author \cite{sm}
have given a different proof by using approximations of continuous
foliated maps by smooth maps.
Here we construct two new spectral sequences associated to $\mathcal
F$. One starts
with two real Alexander-Spanier cochain complexes of $M$, the
continuous one, $_{\cont}AS^\ast (M)$,
and the differentiable one, $_{\diff}AS^\ast (M)$, with a filtration similar to that considered
above (see (\ref{fAS})). In this way, one gets two spectral
sequences, $E_{2,\cont}({\mathcal
F})\/$ and $E_{2,\diff}({\mathcal F})$,
respectively. As every smooth Alexander-Spanier
cochain is also continuous, the inclusion induces an homomorphism of
spectral sequences
\[
J_r\,\colon E_{r,\diff}({\mathcal
F})\,\longrightarrow\, E_{r,\cont}({\mathcal
F})\, .
\]
There is also a homomorphism
\[
\Lambda_r\,\colon E_{r,\diff}({\mathcal
F})\,\longrightarrow\, E_{r,dR}({\mathcal F})\,
\]
between the differentiable Alexander-Spanier spectral sequence and
the de Rham spectral
sequence (see (\ref{morEE})). Such a homomorphism will be called
\emph{quasi-isomorphism} if the homomorphism induced on the
level of spectral sequences is an isomorphism of the terms $E_r\/$
for $r$ bigger or equal than $2$.
It seems reasonable to conjecture that $\Lambda$ is a
quasi-isomorphism for an arbitrary $C^\infty$ foliation. The
Main Theorem of this paper states that $J$ and $\Lambda$ are
quasi-isomorphisms if
$\mathcal F\/$ is a Riemannian foliation on a closed manifold
(Theorem~\ref{dRth}).
The proof has two steps. In the first and fundamental step, the
theorem is proved for
the particular case of Lie foliations. If $\mathcal F\/$ is the
foliation by points on a Lie group, the assertion that $J$ is
a quasi-isomorphism reduces to the classical theorem that the
continuous and differentiable cohomologies
of a Lie group coincide (cf.~\cite{Mw}). The proof for a Lie
foliation is a generalization of the classical proof. To
prove that
$\Lambda$ is a quasi-isomorphism, the second terms of the spectral
sequences are computed (Proposition~\ref{gcoh} and
Proposition~\ref{Gcoh}) and they are presented as a Lie group
cohomology or a Lie algebra cohomology,
\[
E_{r,\diff}^{p,q}({\mathcal F})\, =\,
H^p_{\Box}(G,\overline{H}^q_{\mathcal
F})\, ,\quad
E_{r,dR}^{p,q}({\mathcal F})\, =\, H^p(\mathfrak{g},\overline{H}^q_{\mathcal
F})\, ,
\]
where $\overline{H}^q_{\mathcal
F}\/$ stands for the \emph{reduced foliated cohomology}, the quotient
of the space $H^q_{\mathcal
F}$, equipped with the $C^\infty$-topology, by the closure of $0$,
and $H^p_{\Box}(G,\; )\/$ refers to the group
cohomology defined by Hu \cite{Hu}. Finally, a theorem by
S.~\'Swierczkowski \cite{sw} is used to conclude that they
are isomorphic.
The second
step takes into account Molino's structure theorem to reduce the
general case of a Riemannian foliation to the
particular case above. It reduces essentially to technicalities about
spectral sequences.
Throughout this paper we use the terminology of sheaves. In fact,
spectral sequences are
constructed from resolutions of the constant sheaf $\mathbb R_M$,
which are made up of sheaves
of \emph{basic forms} or \emph{basic Alexander-Spanier cochains}. The
homomorphisms between spectral
sequences are also defined by homomorphisms between these resolutions.
The paper is
organized as follows.
In Section 2, we introduce the Alexander-Spanier spectral sequence in
the very general framework of
a
space with two topologies, one finer than the other. We relate the
spectral sequence with the
\emph{continuous cohomology\/} defined by Bott and Haefliger
\cite{b} in this setting.
Section 3 deals with foliated manifolds and their associated spectral
sequences. Some properties
of foliated homotopy are proved and the homomorphism between the
Alexander-Spanier and
de Rham cohomology is defined.
Section 4 is devoted to the proof of the Main Theorem for a Lie foliation.
In Section 5 we use Molino's structure theorem to reduce the general
case of Riemannian
foliations to the particular case of Lie foliations.
Codimension one foliations without holonomy provide examples of
foliations whose continuous
spectral sequence is not isomorphic to the de~Rham spectral sequence.
The results of this work were announced in \cite{EuASL}.
%%%%%%%%%%%%%%%
\section{Alexander-Spanier spectral sequence}
Let $X\/$ be a topological space, and let $X^\prime\/$ be a space with
the same set as
$X$, but with a finer topology. Let $U\/$ be an open set in $X$. A
map
\[
\varphi\colon\, U^{p+1}\,\longrightarrow\,\mathbb R
\]
is said to be a \emph{basic Alexander-Spanier
$p$-cochain\/} in $U\/$ if it is locally constant when one
considers in $U^{p+1}\/$ the topology induced by $X^\prime$.
The reason for the term ``basic'' is that if one considers
the decomposition of $X\/$ by the connected components of $X^\prime\/$ and
the equivalence relation that this partition defines on $X$, then, under
suitable hypotheses, these cochains correspond to the cochains on the
quotient space.
For each $U$, the vector space of basic
Alexander-Spanier cochains in $U$, with the obvious restriction
maps, defines a presheaf, which generates the
\emph{sheaf of basic Alexander-Spanier cochains\/}
$AS^\ast_{(X^\prime\mid X)}$. With the usual differential
\begin{equation}\label{differential}
\delta\,\varphi (x_0,\ldots ,x_p)\, =\, \sum_{i=0}^p\,
(-1)^i\,\varphi (x_0,\ldots ,\hat x_i,\ldots ,x_p)
\end{equation}
we have a resolution
\begin{equation}\label{resol}
AS_{(X^\prime\mid X)}^0\,\stackrel{\delta}{\longrightarrow}\,
AS_{(X^\prime\mid X)}^1\,\stackrel{\delta}{\longrightarrow}\,
AS_{(X^\prime\mid X)}^2\,\stackrel{\delta}{\longrightarrow}\,\cdots
\end{equation}
of the constant sheaf $\mathbb
R_X$. In fact, the sequence (\ref{resol}) is
pointwise homotopically trivial: let $\epsilon\colon\mathbb
R_X\rightarrow AS_{(X^\prime\mid X)}^0\/$ be the
obvious map, let
$\eta_x\colon {AS_{(X^\prime\mid X)}^0}_x\rightarrow
\mathbb R_X\/$ be the map that assigns to
a cochain $\varphi\/$ the constant function with
value $\varphi (x)$, and let $D_x\colon {AS_{(X^\prime\mid X)}^p}_x\rightarrow
{AS_{(X^\prime\mid X)}^{p-1}}_x\/$ be given by
\[
D_x(\varphi )(x_0,\ldots ,x_{p-1})\, =\, \varphi (x,x_0,\ldots
,x_{p-1})\, .
\]
We have
\begin{equation}\label{pht}
\left\{%
\begin{array}{ll}
\delta D_x\, +\, D_x\delta\, =\, 1\, ,&\text{in positive
degrees,}\\
D_x\delta\, =\, 1-\epsilon\eta_x\, ,&\text{in degree zero,}\\
\eta_x\epsilon\, =\, 1\, ,&\text{on }\mathbb
R_X\, .
\end{array}
\right.%
\end{equation}
As a consequence, there is a spectral sequence
\begin{equation}
E_2^{p,q}(X^\prime\mid X)\, =\, H^pH^q(X,AS_{(X^\prime\mid
X)}^{\star})\,\Rightarrow\, H^{p+q}(X,\mathbb R)\, .
\end{equation}
$E_2^{p,0}(X^\prime\mid X)\/$ is the cohomology of the sections
of the sheaves $AS^\ast_{(X^\prime\mid X)}\/$ and will be called the
\emph{basic cohomology\/} of $(X^\prime\mid X)$.
In the definition
of a cochain, we can consider continuous (or smooth, in the
appropriate case) rather than arbitrary functions. In this case we
use the terms
\emph{continuous\/} or \emph{differentiable\/} Alexander-Spanier cohomology.
The above constructions for the resolution and
spectral sequence hold also in the continuous and differentiable cases. In
this work, we are mainly concerned with the continuous and the
differentiable cohomologies. If it is necessary to avoid confusion, we shall
write
\[
_dAS_{(X^\prime\mid X)}\, ,\qquad
_{\cont}AS_{(X^\prime\mid X)}\, ,\qquad
_{\diff} AS_{(X^\prime\mid X)}\, ,
\]
for the discrete (arbitrary functions), continuous or differentiable
Alexander-Spanier sheaves.
%%%%%%%%%%%%
\begin{remark}%
Bott and Haefliger \cite{b} define \emph{continuous cohomology\/} of
spaces with two topologies. Let $\Delta^q\/$ be the Euclidean
$q$-simplex. One considers on
$\operatorname{Map}\, (\,\bigtriangleup^q,\, X^\prime\, )\/$ the
pull back of the compact open topology on
$\operatorname{Map}\, (\,\bigtriangleup^q,\, X\, )\/$ by the map
induced by the identity
$X^\prime\rightarrow X$. A \emph{continuous cochain\/} is a continuous map
from
$\operatorname{Map}\, (\,\bigtriangleup^q,\, X^\prime\, )\/$ to
$\mathbb R$. Mostow \cite{Mo}
proves that the continuous cohomology is the cohomology of $X\/$ with
values in the sheaf of
continuous functions on $X$, locally constant in $X^\prime$; i.e.,
$H^q(X,\,{}_{\cont}AS_{(X^\prime\mid X)}^0)\/$ is the continuous cohomology of
Bott and Haefliger.
\end{remark}
\begin{example}
Let $G\/$ be a topological group and let $BG\/$ be its Milnor
classifying space. Denote by
$G_{\delta}\/$ the group $G\/$ with the discrete topology. Then
$BG_\delta\/$ is the same set
as $BG$, but with a finer topology. As $BG\/$ is the semi-simplicial
space associated to the
nerve $NG$ of $G$, to compute the spectral sequence
\begin{equation}\label{bg}
E_r(BG_\delta\mid BG)\,\Rightarrow\, H(BG)
\end{equation}
one can use a theorem by Segal (\cite[Proposition 5.1]{sg}), which
asserts that
\[
E_1(BG_\delta\mid BG)\, =\, H^q(BG,AS^p_{(BG_\delta\mid
BG)})\,\cong\,H^q_\delta (
AS^p(N^\ast G))\, ,
\]
to conclude that this
spectral sequence is associated to the double complex
\[
AS^p(N^qG)
\]
with $p$ as filtrant degree. The differential is
$D=\delta_{1,0}+\delta_{0,1}$, where
\[
\delta_{1,0}\colon AS^p(N^qG)\rightarrow AS^{p+1}(N^qG)\/
\]
is the differential
of Alexander-Spanier cochains and
\[
\delta_{0,1}\colon AS^p(N^qG)\rightarrow
AS^p(N^{q+1}G)\/
\]
is induced by the simplicial structure of $NG$. For
a Lie group, this spectral
sequence is very close to that considered by Bott and Hochschild
(cf.~\cite{bot}), constructed from the
\v{C}ech-de~Rham complex of
$G$,
\[
A^p(N^qG)\, .
\]
Bott and Hochschild proved that the $E_1\/$ term of this spectral sequence is isomorphic to
\[
H_c^{q-p}(G,S^p{\mathfrak g}^\ast )\, ,
\]
where $\mathfrak g\/$ is the Lie algebra of $G\/$ considered as a
$G$-module under the
adjoint action, $S^q{\mathfrak g}^\ast\/$ denotes the $q$-th
symmetric power, and the
subscript $c$ denotes the smooth (or equivalent continuous)
cohomology of $G\/$ with values
in $S^q{\mathfrak g}^\ast$, as defined by van Est~\cite{vE}. (For
another construction of
this spectral sequence see \cite{k-t1}.) The Bott spectral sequence
is a direct summand of
(\ref{bg}), and its terms are isomorphic from the term
$E_2$ onwards. In the particular case
$q=0$, as $A^0(NG)=AS^0(NG)$, we have
\[
H^\ast(BG,{}_{\cont}AS^0_{(BG_\delta\mid BG)})\,\cong\, H_c(G)\, .
\]
(For this isomorphism, see also
\cite[Corollary 7.6]{Mo}.)
\end{example}
%%%%%%%%%%%%%
\begin{example}[cf. \cite{arf2}]
If $f\colon X\rightarrow Y\/$ is a continuous and closed map such that each
$f^{-1}(y)\/$ is compact and relatively Hausdorff in $X$, and
\[
X^\prime = \coprod_{y\in Y}f^{-1}(y)\, ,
\]
then $E_r^{p,q}(X^\prime\mid X)\/$ is the Leray spectral
sequence of $f$.
In fact, since $X^\prime\/$ is defined by a continuous map $f$, there is
an isomorphism
\[
AS^\ast_{(X^\prime\mid X)}\,\cong\, f^\ast AS_Y\, .
\]
One can compute the Leray spectral sequence from the fine resolution
\[
{\mathcal H}^q(f,\mathbb R_X)\otimes AS^0_Y\,\longrightarrow\,
{\mathcal H}^q(f,\mathbb
R_X)\otimes AS^1_Y\,\longrightarrow\,\cdots
\]
of ${\mathcal H}^q(f,\mathbb R_X)$, the Leray sheaf of $f$. So for
the second term of the Leray
spectral sequence we have
\[
H^p(Y,{\mathcal H}^q(f,\mathbb R_X))\,\cong\, H^p(\Gamma (Y,{\mathcal
H}^q(f,\mathbb
R_X)\otimes AS^\ast_Y))\, .
\]
Now under the above hypothesis on $f$ (see~\cite[Proposition
4.6]{bre}), we have
\[
{\mathcal H}^q(f,\mathbb R_X)\otimes AS^\ast_Y\,\cong\,{\mathcal
H}^q(f,{\mathcal
H}^q(f,AS^\ast_{(X^\prime\mid X)}))\, ,
\]
and, finally,
\[
\begin{array}{ll}
H^p(\Gamma (Y,{\mathcal H}^q(f,\mathbb R_X)\otimes AS^\ast_Y))&\cong\,
H^p(\Gamma (Y,{\mathcal H}^q(f,AS^\ast_{(X^\prime\mid X)})\\
\\
&\cong\, H^pH^q(M,AS^\ast_{(X^\prime\mid X)})\, .
\end{array}
\]
\end{example}
%%%%%%%%%%%
It is possible to give an alternative description of the spectral
sequence. To do that, we
define a filtration $\{ F^pAS^i(X)\}\/$ of the Alexander-Spanier cochains of
$X$, $AS^\ast (X)$. We say that
$\varphi\in F^pAS^i(X)\/$ if there exists an open cover $\mathcal U\/$ of
$X\/$ such that
\begin{equation}\label{fAS}%
\varphi (x_0,\ldots ,x_{p-1},x_p,\ldots ,x_i)\, =\, \varphi (y_0,\ldots
,y_{p-1},x_p,\ldots ,x_i)
\end{equation}
if $x_j,y_j,\; 0\le j\le p-1$, belong to the same connected component
of $U\/$ in $X^\prime$,
$U\in\mathcal U$. Obviously
\[
F^0AS^i(X)=AS^i(X)\, ,\quad F^pAS^i(X)\supset
F^{p+1}AS^i(X)
\]
and
\[
\delta\left( F^pAS^i(X)\right)\subset F^pAS^{i+1}(X)\, .
\]
A $p$-cochain $\varphi\/$ is basic if $\varphi\in F^pAS^p(X)\/$ and
$\delta\varphi\in F^{p+1}AS^{p+1}(X)$. Let $F^pAS^i_{X}\/$ be the
sheaves of germs of cochains of filtrant degree $p$. The link between
the spectral sequence
defined by the filtration and that initially defined is given by the
following resolution of
$AS^p_{X}$:
\[
F^pAS^p_{X}\,\stackrel{\overline\delta}{\longrightarrow}\,\displaystyle\frac{F^pAS^{p+1}_{X}}{F^{p+1}AS^{p+1}_{X}}\,
\stackrel{\overline\delta}{\longrightarrow}\,\displaystyle\frac{F^pAS^{p+2}_{X}}{F^{p+1}AS^{p+2}_{X}}\,
\longrightarrow\,\cdots,
\]
where $\overline\delta\/$ is induced by $\delta$. Let us check the
exactness of this
sequence for $q>0$. We define a map
\[
E_x\colon\left(F^pAS^{p+q}_{X}\right)_x\,\longrightarrow\,\left(F^pAS^{p+q-1}_{X}\right)_x\,
,
\]
$E_x(\varphi )=\varphi_x$, by $\varphi_x(x_0,\ldots
,x_{p+q-1})=\varphi (x_0,\ldots ,x_{p+q-1},x)$.
One gets
\[
\varphi\, +\, (-1)^{p+q+1}\delta (\varphi _x)\, =\,
(-1)^{p+q+1}(\delta\varphi )_x\, .
\]
Let us briefly discuss the naturality of the Alexander-Spanier
spectral sequence.
Let $(Y^\prime\mid Y)\/$ be another space with two topologies. A
map
\[
f\colon (X^\prime\mid X)\,\longrightarrow\, (Y^\prime\mid Y)
\]
is a continuous map $f\colon X\rightarrow Y\/$ that is also
continuous as a map from
$X^\prime\/$ to $Y^\prime$. Such a map $f\/$ defines a differential morphism
\[
f^\ast\colon\, AS^i_{(Y^\prime\mid Y)}\,\longrightarrow\,
AS^i_{(X^\prime\mid X)}
\]
by $f^\ast (\varphi )=\varphi\circ f^{i+1}$, and
a homomorphism of spectral sequences
\begin{equation}\label{he}
f_r^\ast\colon\, E_r^{p,q}(Y^\prime\mid Y)\,\longrightarrow\,
E_r^{p,q}(X^\prime\mid X)\, .
\end{equation}
\section{Foliated manifolds}
A foliation $\mathcal F\/$ on a topological manifold $M\/$ is a
decomposition of
the manifold into connected topological submanifolds, with the leaves
$L_x$, $x\in M$, all of the same dimension, the dimension of $\mathcal
F$, and the
additional condition that, locally, the decomposition is modeled on
the decomposition of
$\mathbb R^n\/$ into the cosets $x+\mathbb R^l\/$ of the standardly
embedded subspace
$\mathbb R^l$. An open set with this condition is said to be a
\emph{distinguished open
set}. The number $k=n-l\/$ is the codimension of
$\mathcal F$. We can consider smooth foliations,
$C^{r,s}$-foliations, singular foliations, where leaves of several dimensions
are permitted, or just a
\emph{lamination}, a metric space decomposed into leaves (see \cite{Caco} for
precise definitions).
Let $(M,{\mathcal F})\/$ be a foliated manifold, $M=\bigcup_{x\in M}
L_x$. Denote by
$M^{\mathcal F}\/$ the set
$M\/$ with the leaf topology, for which a basis
is formed by the connected components of
intersections of open sets of $M$ with leaves. The Alexander-Spanier
sheaf and the spectral
sequence of the foliated manifold will be that associated to
$(M^{\mathcal F}\mid M)$. We use the notations
\[
_{\cont}AS^\ast_{\mathcal F}\, ,\qquad _{\diff} AS^\ast_{\mathcal F}
\]
for the continuous and differentiable Alexander-Spanier sheaves,
respectively, and
\[
E_{r,\cont}^{p,q}({\mathcal F})\qquad \text{and}\qquad
E_{r,\diff}^{p,q}({\mathcal
F})
\]
for the corresponding spectral sequences.
\subsection*{The de~Rham spectral sequence}
For smooth foliations (or $C^r$-fo\-lia\-tions, with $r\ge 1$) one can
construct the \emph{de~Rham
spectral sequence\/} of $\mathcal F$. Let
$(A^\ast (M),d)\/$ be the de~Rham complex of $M\/$ and denote by
$A^\ast_M\/$ the de Rham sheaf of $M$.
A smooth form $\eta $
is said to be {\em basic} if it satisfies
\begin{equation}\label{basic}
i_Y\,\eta =0\;\text{ and }\; i_Y\,d\eta =0
\end{equation}
for all $Y\in\Gamma\mathcal F$, the algebra of vector fields tangent
to the foliation,
where
$i_Y$ is the interior product by $Y$. The algebra $A^\ast_{\mathcal
F}(M)\/$ of basic forms is a differential subcomplex of the de~Rham
complex $A^\ast (M)$.
The sheaves $A^\ast_{\mathcal F}\/$ of germs of basic forms define a
resolution of
${\mathbb R}_M$, the constant sheaf on $M$,
\begin{equation}\label{dResol}
A^0_{\mathcal F}\,\stackrel{d}{\longrightarrow}\,A^1_{\mathcal
F}\,\stackrel{d}{\longrightarrow}\,\cdots\,\stackrel{d}{\longrightarrow}
A^l_{\mathcal F}\,\longrightarrow\, 0\, ,
\end{equation}
where $d\/$ is the exterior derivative and $l\/$ is the dimension of
$\mathcal F$.
Associated to this resolution, we have the de~Rham spectral sequence
of $\mathcal F$,
\begin{equation}\label{sedR}%
E_{2,dR}^{p,q}({\mathcal F})\, =\, H^p(H^q(M,A_{\mathcal
F}^{\star}))\,\Rightarrow\,
H^{p+q}(M,\mathbb R)\, .
\end{equation}
$E_{2,dR}^{p,0}({\mathcal F})\/$ is the basic de~Rham cohomology of
$\mathcal F$. We
denote by
\begin{equation}\label{folcoh}%
H^q_{\mathcal F}\, =\, H^q(M,A^0_{\mathcal F})
\end{equation}
the differentiable foliated cohomology of the foliation. Note that
\[
_{\diff} AS^0_{\mathcal F}\, =\,
A^0_{\mathcal F}\, ,
\]
the sheaf of smooth functions on $M\/$ that are locally
constant along the leaves.
To describe the de~Rham spectral sequence of $\mathcal F\/$ it is
also possible to start with
a filtration of the de~Rham complex $A^\ast (M)$. A smooth form of
degree $i$ is said to be
of filtration
$\ge p$ if it vanishes whenever $i-p+1$ of the vectors are tangent
to the foliation. We
shall denote the ideal of all forms of filtration degree $\ge p$ by
$F^pA(M)$. We will use further
the sheaves
\begin{equation}\label{A}%
A^{p,q}\,\colon\!\! =\, F^pA^{p+q}/F^{p+1}A^{p+q}
\end{equation}
with the operator
\[
d_{\mathcal F}\,\colon\, A^{p,q}\,\longrightarrow\, A^{p,q+1}
\]
induced by the exterior derivative.
Checking that it is a resolution of $A_{\mathcal F}\/$
reduces to a form of the Poincar\'e Lemma (cf.~\cite{vai}). As
$A^0_M$-modules, these sheaves are fine.
\subsection*{Homotopy}
Let $(M^\prime,{\mathcal F}^\prime )\/$ be another foliated manifold.
By a foliated map
\[
f\colon\, (M, {\mathcal F})\,\longrightarrow\, (M^\prime ,{\mathcal
F}^\prime )
\]
we understand a smooth map which takes leaves into leaves, or, more
precisely, is such that the induced map $df\colon
TM\rightarrow TM^\prime\/$ satisfies $df(T\mathcal F)\subset
T\mathcal F^\prime$. Now the map $f^\ast\colon A^i(M^\prime
)\rightarrow A^i(M)\/$ is a filtration preserving
homomorphism, and
we have induced
homomorphisms
\[
f_r^{p,q}\colon\, E_r^{p,q}({\mathcal F}^\prime )\,\longrightarrow\,
E_r^{p,q}({\mathcal F})
\]
between the de~Rham (or Alexander-Spanier) spectral sequences of the
foliations.
We need two different types of homotopy in this framework. One can consider
in $M\times I\/$ two foliations: the first one is ${\mathcal F}\times I$, with
leaves
$L\times I$, where $L$ is a leaf of $\mathcal F$, the second one is
${\mathcal F}\times I_\delta$,
where $I_\delta\/$ is the closed unit interval with the
discrete topology, with leaves $L\times\{ t\}$. An $s$-homotopy,
for $s=1\/$ or $2$, will be a
smooth function
$H\colon M\times I\rightarrow M^\prime\/$ such that the image of each
leaf is contained in a
leaf, given the first or the second foliation in $M\times I$, respectively.
\begin{lemma}
Let $H\colon M\times I\rightarrow M^\prime\/$ be an $s$-homotopy between
foliated maps $h_0,h_1\/$
from
$(M, {\mathcal F})\/$ to $(M^\prime , {\mathcal F}^\prime )$. Then
the induced homomorphisms in
the spectral sequences are identical, i.e.,
\[
h_0^\ast =h_1^\ast\colon\, E_r({\mathcal F}^\prime )\,\longrightarrow\,
E_r({\mathcal F})\, ,
\]
for $r\ge s$.
\end{lemma}
\begin{proof} It is enough to consider the maps
\[
i_t\colon M\,\longrightarrow\, M\times I\, ,
\qquad i=0,1\, ,
\]
given by $i_t(x)=(x,t)$. The identity map of $M\times I\/$ gives an
$s$-homotopy between $i_1\/$
and $i_2$, for $s=1$ or $2$,
as we consider the foliation ${\mathcal F}\times I\/$ or ${\mathcal
F}\times I_\delta$,
respectively.
For the de~Rham spectral sequence, the result is well
known (cf.~\cite{Sark-t}): one can construct a cochain homotopy as usual,
starting with a smooth
homotopy, and check the filtration requirements.
For $s=1$, in any case, the lemma follows from the general homotopy
invariance of sheaf
cohomology.
In fact, for the de~Rham sheaves, it is
\[
A_{{\mathcal F}\times I}^\ast\,\cong\,\pi^\ast A_{\mathcal F}^\ast\, ,
\]
where $\pi\colon M\times I\rightarrow M\/$ is the projection.
Analogously, for the Alexander-Spanier sheaves we have
\[
AS_{({\mathcal F}^\prime\times I\mid {\mathcal F}\times I)}\,\cong
\pi^\ast AS_{({\mathcal
F}^\prime\mid {\mathcal F})}\, .
\]
A general proof of the homotopy axiom for the Alexander-Spanier
coho\-mo\-lo\-gy was given by
Spanier \cite{Sp}. For $s=2\/$ we must check the compatibility of the
homotopy $S\colon
AS^{i+1}(M\times I)\rightarrow AS^i(M)\/$ with the filtration, that is,
\[
S(F^pAS(M\times I))\,\subset\, F^{p-1}(AS(M))\, ,
\]
where the filtration of $AS(M\times I)\/$ is defined from the foliation
${\mathcal F}\times I_\delta$.
This is a consequence of the fact that, locally, the homotopy is
induced by maps like
\[
(x_0,\ldots ,x_q)\,\mapsto\,\sum_{i=0}^q(-1)^i\left((x_0,t_0),(x_1,t_0),\ldots
,(x_i,t_0),(x_i,t_1),\ldots ,(x_q,t_1)\right),
\]
with $t_0,t_1\in I$, $t_0\/$ close to $t_1$.
\end{proof}
\subsection*{A spectral sequences morphism}
There exists an onto morphism of differential sheaves
\[
\Lambda\,\colon\,_{\diff}
AS^\ast_{\mathcal F}\,\longrightarrow\, A^\ast_{\mathcal F}\, .
\]
If we take for an
open set $U\/$ of $M\/$ a $p$-cochain $\varphi\/$ given by the product of
$p+1\/$ smooth functions $f_i\colon U\rightarrow\mathbb R$, $0\le i\le p$,
\[
\varphi (x_0,x_1,\ldots ,x_p)=f_0(x_0)f_1(x_1)\ldots f_p(x_p)\, ,
\]
then
\[
\Lambda (\varphi )\, =\, f_0\, df_1\wedge\ldots \wedge df_p\, .
\]
In the general case, for $x\in U\/$ and $Z_1,\ldots , Z_p\in T_xM$,
\begin{align}
\label{Landa}
&\Lambda (\varphi )_x(Z_1,\ldots ,Z_p)\,
\\
&\
=\frac{1}{p!}\sum_{\tau\in{\mathcal
S}_p}\operatorname{sgn}(\tau )\frac{\partial}{\partial\varepsilon_1}\cdots
\frac{\partial}{\partial\varepsilon_p}\varphi (x,\exp_x\varepsilon_1 Z_{\tau
(1)},
\ldots ,\exp_x\varepsilon_p Z_{\tau
(p)})\bigg|_{\varepsilon_i =0},
\notag
\end{align}
where $\varepsilon_i\in\mathbb R,\, 1\le i\le p$.
As a morphism of resolutions of $\mathbb R_M$, $\Lambda\/$ defines
a spectral sequence
homomorphism
\begin{equation}\label{morEE}
\Lambda_r^{p,q}\,\colon\, E_{r, \diff}^{p,q}({\mathcal F})\,\longrightarrow\,
E_{r,dR}^{p,q}({\mathcal F})
\end{equation}
that converges to an isomorphism.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cohomology of Lie foliations}
Let $\mathcal F\/$ be a Lie foliation on $M\/$ with dense leaves. A suitable
description of this structure is the following one: there exists a
homomorphism
\[
\Pi_1\colon\,\pi_1(M)\,\longrightarrow\, G\, ,
\]
where $G$ is a
simply connected Lie group, a covering map
$\pi\colon \tilde M\rightarrow M\/$ associated to the homomorphism,
with group of
deck transformations $\Gamma$, and a locally trivial fibration
$\Pi\colon\tilde
M\rightarrow G$, equivariant with respect to the action of $\Gamma\/$ over
$\tilde M\/$ and over $G\/$ by the left product, if we identify $\Gamma\/$
with the image of $\Pi_1\/$ in $G$. The fibers of
$\Pi\/$ are the leaves of the lifting foliation $\tilde{\mathcal
F}\/$, and $\Gamma\/$ is dense in $G$ \cite{Fed}.
A vector field $X\/$ on $M\/$ is said to be \emph{foliated\/} if, for
every vector field
$Y\in\Gamma\mathcal F$, the Lie bracket $[X,Y]\/$ is also tangent to $\mathcal
F$. Denote by ${\mathfrak{X}}(M,{\mathcal F})\/$ the algebra of foliated
vector fields of $\mathcal
F$. A Lie foliation is
\emph{transitive}, that is, at each point of $M$ the complete
foliated vector fields generate the whole tangent space (cf.~\cite{Mol}).
In fact,
\begin{equation}\label{grealiz}
{\mathfrak{X}}(M,{\mathcal F})/\Gamma{\mathcal F}\,\cong\,\mathfrak{g},
\end{equation}
where $\mathfrak{g}\/$ is the Lie
algebra
of
$G$, realized as global foliated vector fields on $M$, and
one takes vector fields
tangent to
$\mathcal F\/$ to generate $T_x\mathcal F\/$ at each point. We will
call such a foliation a $G$-Lie foliation or $\mathfrak{g}$-Lie
foliation.
The inclusion $J\colon _{\diff}
AS_{\mathcal F}\rightarrow {}_{\cont}AS_{\mathcal F}\/$ induces a
spectral sequence homomorphism
\begin{equation}\label{incl}%
J_r^{p,q}\colon E_{r, diff}^{p,q}({\mathcal F})\,\longrightarrow\,
E_{r,\cont}^{p,q}({\mathcal F})\,.
\end{equation}
We now prove that $J$ is a \emph{quasi-isomorphism}, i.e., that $J\/$
induces isomorphisms of the terms $E_r\/$ for $r\ge 2$.
\begin{proposition}
Let $\mathcal F\/$ be a Lie foliation on a compact manifold $M$. The
inclusion $J\colon _{\diff}
AS_{\mathcal F}^\ast\rightarrow _{\cont}AS_{\mathcal F}^\ast\/$
induces an isomorphism
\[
J_2\,\colon\, E_{2,\diff}({\mathcal
F})\,\cong\, E_{2,\cont}({\mathcal F})\,
.
\]
\end{proposition}
\begin{proof}
We construct morphisms
\begin{equation}\label{oper}%
s\colon_{\cont}AS^i(M)\,\rightarrow\,_{\diff} AS^i(M)\, ,\quad
h\colon_{\cont}AS(M)^i\,\rightarrow\,_{\cont}AS^{i-1}(M)
\end{equation}
such that
\begin{equation}
\left\{%
\begin{array}{l}
s(F^p\,_{\cont}AS(M))\,\subset\, F^p\,_{\diff} AS(M)\, ,\\
\\
h(F^p\,_{\cont}AS(M))\,\subset\, F^{p-1}\,_{\cont}AS(M)\, ,\\
\\
h(_{\diff} AS(M))\,\subset\,
_{\diff} AS(M)\, ,
\end{array}\right.%
\end{equation}
and satisfying the relation
\begin{equation}
\delta h\, +\, h\delta\, =\, \pm (1-s)\, .
\end{equation}
So $s\/$ will be a cochain morphism compatible with the filtrations,
which defines a
morphism of spectral sequences, and
$h\/$ is a homotopy between the identity and $s$, which decreases the
filtration degree at most
by one. Then
\[
s_r\colon\,E_{r,\cont}({\mathcal F})\,\longrightarrow\,
E_{r,\diff}({\mathcal F})\
\]
is an isomorphism for $r\ge 2$ (cf.~\cite{ch}).
In order to define $s$ and $h$ let us fix a finite dimensional vector space
$V\/$ of foliated vector fields such
that $V(x)=T_xM$ for all $x\in M$. A vector space of foliated vector fields
satisfying such a property will be called {\em transitive}. Since $M$
is compact and ${\mathcal F}$ is transversally complete, we can
always find a transitive
finite dimensional vector space. We choose a Riemannian metric on $V$ with
volume element $dX$, and a smooth function $\rho $ on $V$ supported in a
compact neighborhood of $0$. We define $s\/$ by
\begin{align}
\label{sdef}%
&(s\varphi)(x_0,\ldots ,x_i)\, \\
&\quad
=\int _V\cdots\int_V\varphi
(\phi^\ast_{X_0}(x_0),\ldots ,\phi^\ast_{X_i}(x_i))\cdot\rho
(X_0)\ldots \cdot\rho
(X_i)\cdot dX_0\ldots dX_i\notag
\end{align}
where $\phi_{tX}:M\longrightarrow M$ denotes the flow of the vector
field $X\in V$ and $\phi_X$ is the diffeomorphism corresponding to $t=1$.
The function $\rho $ can be normalized by $\int _V\rho (X)\, dX=1$,
and $h$ is
defined by
\begin{align*}
&(h\varphi)(x_1,\ldots ,x_i)\, \\
&\qquad=\sum_{j=1}^i(-1)^j\int
_V\cdots\int_V\varphi (x_1,\ldots ,x_j,\phi^\ast_{X_j}(x_j),\ldots
,\phi^\ast_{X_i}(x_i))\cdot\rho (X_j)\cdot\\
&\hspace{2in}
\ldots \cdot\rho (X_i)\cdot dX_j\ldots dX_i
\qedhere
\end{align*}
\end{proof}
See \cite{Sark} and \cite{EuMin} for similar constructions. One could
also conclude that
$E_2(AS_{\mathcal F})\/$ is finite dimensional, but this will be a
consequence of
Theorem~\ref{dRth} below.
\begin{example}
In general there is no isomorphism between the $E_1\/$ terms. The
torus $T^2\/$ foliated by
lines of constant irrational slope provides a counter-example. As it
is equal to the de~Rham
foliated cohomology,
$H^1 (T^2,{}_{\diff} AS^0_{\mathcal
F})\/$ has either infinite dimension or dimension one,
depending upon whether the irrational slope is Liouville or
diophantine, while $H^1 (T^2,\,{}
_{\cont}AS^0_{\mathcal F})\/$ has always infinite dimension (cf. \cite{Mo}).
\end{example}
To compare the differentiable Alexander-Spanier spectral sequence
with the de~Rham spectral sequence, we
start by computing the term $E_2$ for a
$G$-Lie foliation.
We consider first the de Rham spectral sequence of a Lie foliation.
The sheaves of basic forms
for a Lie foliation with Lie algebra
$\mathfrak{g}\/$ are
\[
A^p_{\mathcal F}\,\cong\,\underset{\sim\;}{\textstyle
\bigwedge\nolimits^p}{\mathfrak{g}}^\ast\/\otimes A^0_{\mathcal F}\, ,
\]
where $\underset{\sim\;}{\textstyle
\bigwedge\nolimits^p}{\mathfrak{g}}^\ast\/$ denotes the constant sheaf over $M\/$ with stack
the vector space
$\bigwedge^p{\mathfrak{g}}^\ast$. Then, by the universal coefficient
theorem, we have
\[
E_{1,dR}^{p,q}({\mathcal F})\,\cong\, \bigwedge\nolimits^p
{\mathfrak{g}}^\ast\otimes H^q_{\mathcal F}\, ,
\]
where $H^q_{\mathcal F}\/$ denotes the foliated cohomology (\ref{folcoh}).
The Lie algebra ${\mathfrak{X}}(M,\mathcal F)\/$ acts over $A^\ast(M)\/$
by the Lie derivative,
\[
(X,\alpha )\,\longrightarrow\, {\mathcal L}_X\alpha\, .
\]
This action is
compatible with the exterior derivative and the filtration, so it
defines an action over
$H^i_{\mathcal F}$. Since
$\Gamma\mathcal F\/$ acts on $H^i_{\mathcal F}\/$ as the identity, we
get, by (\ref{grealiz}),
an action of $\mathfrak{g}\/$ on
$H^i_{\mathcal F}$. If $X_1,\ldots ,X_k\/$ is a basis for a
realization of $\mathfrak{g}\/$
on $M\/$ and $\omega^1,\ldots ,\omega^k\/$ are dual $1$-forms, the
differential $d_1\/$ is
\begin{equation}
d_1(\eta\otimes [\alpha ])\, =\, d_{\mathfrak{g}}\eta\otimes [\alpha]\, +\,
(-1)^p\sum_{j=1}^k\eta\wedge\omega^j\otimes\theta_j[\alpha ]\, ,
\end{equation}
where $d_{\mathfrak{g}}\/$ is the differential in
$\bigwedge{\mathfrak{g}}^\ast\/$ and $\theta_j[\alpha
]=[{\mathcal L}_{X_j}\alpha]$. So we have proved (cf.~\cite{EuLie})
that for a $\mathfrak{g}$-Lie
foliation $\mathcal F\/$ there is an isomorphism
\[
E_{2,dR}^{p,q}({\mathcal F})\,\cong\, H^p({\mathfrak{g}},\, H^q_{\mathcal F})\, .
\]
We need a finer result. With the notation introduced in (\ref{A}),
$H^q_{\mathcal
F}\/$ is the cohomology of the complex $(A^{0,q}(M),d_{\mathcal F})$.
With the $C^\infty\/$
topology, $\operatorname{Im}\, d_{\mathcal F}\/$ is not always a
closed subspace of
$\operatorname{Ker}\,d_{\mathcal F}$, and the spaces
$H^q_{\mathcal F}\/$ are not necessarily Hausdorff. But to compute the
$E_2\/$ term of the
spectral sequence of a Lie foliation one can use
$\overline{H}^q_{\mathcal F}$, the \emph{reduced foliated
cohomology}, the quotient space of $H^q_{\mathcal F}\/$ over the
closure of its trivial subspace; i.e., to compute the $E_2$-term of
the spectral sequence, one can use
$\overline{E}_{1,dR}^{p,q}({\mathcal F})$, the quotient space of the
$E_1$-term over the
closure of $0$. This fact was proved in \cite{EuMin}. This result is
also true for the
differentiable Alexander-Spanier spectral sequence, with the same
proof, by using the
compact operator
\[
s\colon\,_{\diff}
AS^i(M)\,\longrightarrow\,_{\diff} AS^i(M)
\]
defined in (\ref{oper}). So
we get the following expression for the second term of the de Rham
spectral sequence of a Lie foliation:
\begin{proposition}\label{gcoh} %
For a $\mathfrak{g}$-Lie foliation $\mathcal F\/$ there is an isomorphism
\[
E_{2,dR}^{p,q}({\mathcal F})\,\cong\, H^p({\mathfrak{g}},\,
\overline{H}^q_{\mathcal F})\, .
\]
\end{proposition}
We now derive a similar expression for the differentiable
Alexander-Spanier spectral sequence of a Lie
foliation.
To begin with, we recall a definition by Hu \cite{Hu}. Let $E\/$ be a left
$G$-module. Let $\left(C^p(G,E),\delta\right)\/$ be the complex of
homogeneous cochains of
$G\/$ over $E$. A $p$-dimensional cochain $\varphi\in C^p(G,E)\/$ is a map
\[
\varphi\colon\, G\times\,\stackrel{\stackrel{p+1}{\smile}}{\ldots}\,\times
G\,\longrightarrow\, E
\]
satisfying the following homogeneity condition:
\[
\varphi (gx_0,\ldots ,gx_p)\, =\, g\varphi (x_0,\ldots ,x_p)\, .
\]
We assume that $\varphi\/$ is smooth. (The same cohomology is
obtained if one assumes that
$\varphi\/$ is continuous.)
The differential $\delta\/$ is defined by
(\ref{differential}). A cochain
$\varphi\in C^p(G,E)\/$ is called
\emph{locally trivial\/} if there is a neighborhood $U\/$ of $e\/$ in
$G\/$ such that
$\varphi (x_0,\ldots ,x_p)=0\/$ whenever all $x_0,\ldots ,x_p\/$ are
in $U$. The locally
trivial cochains form a subcomplex of $C^\ast (G,E)$. Let $\left(G_{\Box}^\ast
(G,E),\delta\right)\/$ be the quotient complex. Its cohomology
$H_{\Box}(G,E)\/$ is, by
definition,
the \emph{reduced cohomology\/} of $G$.
Corresponding to the $\mathfrak{g}$-action, there is an action of $G$ over
$H^i_{\mathcal F}$. The
action can be defined geometrically, by lifting paths of $G$ into
$\tilde M$, or algebraically,
as we do in the following proposition.
\begin{proposition}\label{Gcoh}%
Let $\mathcal F\/$ be a $\mathfrak{g}$-Lie foliation with dense leaves.
There exists an isomorphism
\[
E_{2,\diff}^{p,q}({\mathcal F})\,\cong\,
H^p_{\Box}(G,\overline{H}^q_{\mathcal F})\, .
\]
\end{proposition}
\begin{proof} To compute $H^q(M,{}_{\diff} AS^p_{\mathcal F})\/$ one can use of the following
resolution of
$_{\diff} AS^p_{\mathcal F}$:
\[
\begin{CD}
_{\diff} AS^p_{\mathcal F}\otimes
A^0_M@>{id\otimes d_{\mathcal
F}}>>{}_{\diff} AS^p_{\mathcal F}\otimes
A^{0,1}_M@>{id\otimes d_{\mathcal
F}}>>{}_{\diff} AS^p_{\mathcal F}\otimes
A^{0,2}_M@>>>\cdots
\end{CD}
\]
We have
\[
\Gamma({}_{\diff} AS^p_{\mathcal
F}\otimes A^{0,q})\,\cong\,{}_{\diff} AS_{\mathcal F}^p(M)\hat\otimes A^{0,q}(M)\, ,
\]
the complete tensor product of $_{\diff}
AS_{\mathcal F}^p(M)\/$ and $A^{0,q}(M)$.
As $A^{0,q}(M)\/$ are nuclear spaces, from the exact sequence
\[
0\,\rightarrow\,\overline{\operatorname{Im}\, d_{\mathcal
F}}\,\longrightarrow\,
\operatorname{Ker}\, d_{\mathcal F}\,\longrightarrow\, \overline{H}_{\mathcal
F}\,\rightarrow\, 0
\]
we deduce the isomorphism
\[
\overline{E}_{1,\diff}^{p,q}({\mathcal
F})\,\cong\,{}_{\diff}
AS^p_{\mathcal F}(M)\hat\otimes \overline{H}^q_{\mathcal F}\, .
\]
The $d_1^{0,q}\/$ differential defines an action of $G$ over
$\overline{H}^q_{\mathcal F}$,
given by
\[
g\cdot [\alpha ]\, =\, d_1^{0,q}[\alpha ](e,g)\, .
\]
Finally, as the leaves of $\mathcal F\/$ are dense, via the lift of
cochains to $\tilde{M}$,
we get
\begin{equation*}
_{\diff} AS^p_{\mathcal F}(M)\hat\otimes
\overline{H}^q_{\mathcal
F}\,\cong\, C^p_{\Box}(G,\overline{H}^q_{\mathcal F})\, .
\qedhere
\end{equation*}
\end{proof}
The spectral sequence homomorphism (\ref{morEE}) induced by
$\Lambda\/$ defines,
in particular, a differential complex map
\[
\Lambda_1\colon\, (C^p_{\Box}(G,\overline{H}^q_{\mathcal F})\, ,\,
d_1)\,\longrightarrow\,
(\bigwedge\nolimits^p{\mathfrak{g}}^\ast\otimes \overline{H}^q_{\mathcal
F}\, ,\, d_1)\, .
\]
Then the $G$-module structure of $\overline{H}^q_{\mathcal F}\/$ corresponds,
by $\Lambda$, to the $\mathfrak{g}$-action
on $\overline{H}^q_{\mathcal F}$.
We need the following result:
\begin{theorem}\label{swi}%
Let $E\/$ be a $G$-module. Assume that $E$ is a Hausdorff locally convex complete
topological vector space. There is an isomorphism
\[
H_{\Box}^p(G,E)\,\cong\, H^p({\mathfrak{g}},E)\, .
\]
\end{theorem}
If $E\/$ is finite dimensional (a very unusual property for
$\overline{H}^q_{\mathcal F}$),
this is a theorem by \'Swierczkowski~\cite{sw}. The same proof works
in this case; all one needs
is a Poincar\'e lemma for $E$-valued forms. We indicate here a slightly
different proof.
\begin{proof}
To begin with, we define a double complex
\begin{equation}\label{act}
C^{p,q}\, =\, C_\Box^p(G,\, \underset{\sim{}^{\scriptstyle e}}{A^q}(G,E))\,
,
\end{equation}
where $\underset{\sim{}^{\scriptstyle e}}{A^q}(G,E)\/$ is the
space of germs at $e\/$ of de~Rham forms on $G\/$ with values on $E$,
and $C_\Box^p(G,\, \underset{\sim{}^{\scriptstyle e}}{A^q}(G,E))\/$
is the space of
homogeneous cochains of $G\/$ with values on
$\underset{\sim{}^{\scriptstyle e}}{A^q}(G,E)$, modulo the locally
trivial
ones. If the vector space (\ref{act}) is written as
$C_\Box^p(G,\, C_\Box^\infty (G,\,\Lambda^q\mathfrak{g}^\ast\otimes E)$,
an element $\varphi\/$ will be a map
\[
\varphi\,\colon\, G\times\,\stackrel{\stackrel{p+1}{\smile}}{\ldots}\,\times
G\,\longrightarrow\, C_\Box^\infty (G,\,\Lambda^q\mathfrak{g}^\ast\otimes E)
\]
satisfying
\begin{equation}
\varphi (gx_0,\ldots ,gx_p)(gy)\, =\, g\cdot \varphi (x_0,\ldots ,x_p)(y)\, ,
\end{equation}
where on the right-hand side we have the $G$-action on $E$.
The two differentiation operators $d_1\/$ and $d_2\/$ of degree
$(1,0)\/$ and $(0,1)$,
respectively, are defined as follows. Let $d_1=\delta$, the usual
differentiation as defined in
(\ref{differential}). The operator $d_2\/$ can be given by
\[
d_2(\eta )\, =\, d_G\eta\, +\,\sum_{j=1}^k\omega^j\wedge\theta_j\circ\eta\, ,
\]
where $d_G\/$ is the exterior derivative on $G$, $\omega^1,\ldots
,\omega^k\/$ is a basis
of the left invariant $1$-forms on $G$, $\xi_1,\ldots ,\xi_k\/$ is a
basis of $\mathfrak{g}\/$ dual
to $\{ \omega^i\}\/$, and
$\theta_j\/$ stands for the action of
$\xi_j\in\mathfrak{g}\/$ on $E$. It is straightforward to check the identities
$d_1^2=0=d_2^2\/$ and
$d_1d_2+d_2d_1=0$.
The two spectral sequences associated to this double complex, $I_r^{p,q}\/$
and
$I\! I_r^{p,q}$, converge to the same graded space and satisfy
\[
I_1^{p,q}\, =\, 0\, =\, I\! I_1^{q,p}\qquad \text{if}\quad q\ne 0\, .
\]
It will follow that $I_2^{p,0}\cong I\! I_2^{0,p}$, which is the
statement of the
proposition.
In fact, the space $I_1^{p,q}\/$ may be identified with
$H^q(C^{p,\star},d_2)$. Then the
Poincar\'e lemma implies $I_1^{p,q}=0$, $q>0$, and $I_1^{p,0}\cong
C^p_{\Box}(G,\, E)$. So
\begin{equation}\label{i}
I_2^{p,0}\,\cong\, H_{\Box}^p(G,E)\, .
\end{equation}
On the other hand, $I\! I_1^{p,q}\cong H^p(C^{\star ,q},d_1)$. The
complex $(C^{\star
,q},d_1)\/$ admits the homotopy $h\colon C^{p,q}\rightarrow
C^{p-1,q}\/$ defined by
\begin{equation}
h(\varphi )(x_0,\ldots ,x_{p-1})(y)\, =\,\varphi (y,x_0,\ldots ,x_{p-1})(y)\,
.
\end{equation}
For $p>0$, $h\delta\varphi +\delta h\varphi =\varphi$, and $I\!
I_1^{p,q}=0$. For $p=0$,
$h\delta\varphi =\varphi -c(\varphi )$, with the constant chain
$c(\varphi )$ defined by
\[
c(\varphi )(x)(y)\, =\, y\cdot [\varphi (e)(e)]\, .
\]
If we introduce the transposed double complex $^tC$, as usual, to
compute $I\!
I_r^{p,q}$, we get $I\! I_1^{p,0}\cong
\bigwedge^p{\mathfrak{g}}\otimes E\/$ and
\begin{equation}\label{ii}
I\! I_2^{p,0}\,\cong\, H^p({\mathfrak{g}},E)\, .
\qedhere
\end{equation}
\end{proof}
Finally, Theorem
\ref{swi} and Propositions
\ref{gcoh} and
\ref{Gcoh} prove the following result:
\begin{proposition}\label{isoLie}%
Let $\mathcal F\/$ be a Lie foliation. There exists an isomorphism
\[
E_{2,\diff}({\mathcal F})\,\cong\,
E_{2,dR}({\mathcal F})\, .
\]
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We are now interested in getting an explicit isomorphism. To do that, we
recall first the
definition of \emph{Riemannian foliation}.
A foliation can be defined by local submersions on some open subsets
of ${\mathbb R}^k$. We
can assume that these subsets
are open balls with center the origin. For a
Riemannian foliation there is a Riemannian metric on
$M$, a
\emph{bundle-like\/} metric (cf.~\cite{re}), such that these
submersions are Riemannian
submersions.
For Riemannian foliations one can construct a map
\[
\Phi\colon\, A^p_{\mathcal
F}\,\longrightarrow\,{}_{\diff}
AS_{\mathcal F}^p
\]
between differential sheaves that defines a morphism of spectral
sequences and which will be an
inverse of the homomorphism
$\Lambda$, defined in (\ref{morEE}), from $E_2\/$ onwards.
Let us fix a bundle-like metric on $M$. Let $f\colon U\rightarrow
\mathcal O\/$ be a
Riemannian submersion defining $\mathcal F\/$ in $U$. If $\eta\/$ is
a basic $p$-form, then
$\eta\mid_U=f^\ast\gamma$, with $\gamma\in A^p(\mathcal O)$. Let
$x\in U\/$ and satisfying
$\chi_{_U}\colon U\rightarrow\mathbb R\/$ be a smooth function with
support in $U\/$ and satisfying
$\chi_{_U}\equiv 1\/$ on a neighborhood of $x$.
Given $\eta\in A_{\mathcal F}^p(U)$, we define $\Phi(\eta )\/$ by
\[
\Phi (\eta )(x_0,\ldots ,x_p)\,
=\,\chi_{_U}(x_0)\cdots\chi_{_U}(x_p)\,\int_{\Delta
[\overline x_0,\ldots ,\overline x_p]}\gamma\, ,
\]
where $\overline x_0,\dots ,\overline x_p\/$ are the images of
$x_0,\ldots ,x_p$ in
$\mathbb R^k\/$ and $\Delta [\overline x_0,\ldots ,\overline x_p]\/$
is the simplex defined
by these points. The germ of $\Phi (\eta )\/$ at $x\/$ does not
depend on the function
$\chi_{_U}\/$ either on the open neighborhood $U$, and so we have a
well defined map between
the sheaves. To check that it is a homomorphism of complexes one uses
the Stokes' theorem for
chains and the fact that the boundary of $\Delta [\overline
x_0,\ldots ,\overline x_p]\/$ can be
expressed as follows:
\[
\partial\Delta [\overline x_0,\ldots ,\overline x_p]\, =\,
\sum_{j=0}^p(-1)^j\Delta
[\overline x_0,\ldots ,\overline x_{j-1},\overline x_{j+1},\ldots
,\overline x_p]\, .
\]
Finally, we have
\[
\Lambda\circ\Phi\, =\, \operatorname{id}_{A_{\mathcal F}}\, .
\]
The computation can be done in $\mathcal O\subset\mathbb R^k\/$ and
it is the same
as in
\cite{cm}.
If $S_{\mathcal F}^\ast\/$ denotes the kernel of $\Lambda$, we have
a split exact
sequence
\[
0\,\longrightarrow\, S^p_{\mathcal F}\,\longrightarrow\,
_{\diff} AS^p_{\mathcal
F}\,\stackrel{\Lambda}{\longrightarrow}\, A^p_{\mathcal
F}\,\longrightarrow\, 0\, .
\]
Thus
\[
_{\diff} AS^p_{\mathcal F}\,\cong\,
A^p_{\mathcal F}\oplus S^p_{\mathcal F}
\]
and
\[
E_{2,\diff}^{p,q}({\mathcal F})\,\cong\,
E_{2,dR}^{p,q}({\mathcal F})\oplus
E_2^{p,q}(S_{\mathcal F})\, ,
\]
where $E_2^{p,q}(S_{\mathcal F})\/$ is the spectral sequence
associated to the exact
sequence of sheaves
$(S^\ast_{\mathcal F},\delta )$, which converge to $0$.
As a consequence, we deduce that the isomorphism in
Proposition~\ref{isoLie} between
$E_{2,\diff}^{p,q}({\mathcal F})\/$ and
$E_{2,dR}^{p,q}({\mathcal F})\/$ is induced by
$\Lambda$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Riemannian foliations}
To prove the next theorem, we will use the description of the
structure of a Riemannian foliation
given by Molino \cite{Mol} to reduce the question to the case,
already proved, of Lie foliations
with dense leaves.
To begin with, we consider a particular type of Riemannian foliation.
\begin{proposition}\label{tp}%
Let $\mathcal F_0\/$
be a Lie foliation on a closed manifold $N$, let $M\/$ be a bundle of fiber
$N\/$ with structure group $\operatorname{Aut}(N,\mathcal F_0)\/$, and
let $\mathcal
F\/$ be the foliation induced on $M$. For this foliation $\mathcal
F\/$ the homomorphisms $J$
and $\Lambda$ are quasi-isomorphisms.
\end{proposition}
\begin{proof}
To prove the proposition we will use the following well known theorem
(see \cite[Theorem IV.2.2.]{bre}):
\begin{auxiliar} Let $h\colon{\mathcal L^{\star}}\rightarrow{\mathcal
M^{\star}}\/$ be a homomorphism of differential
sheaves on a topological space $B$ and assume that both $\mathcal
L^{\star}\/$ and $\mathcal M^{\star}\/$ are bounded below.
Also assume that the induced map $h^\ast\colon\mathcal H^q(\mathcal
L^\ast)\rightarrow\mathcal H^q(\mathcal M^\ast)\/$ of
derived sheaves is an isomorphism for all $q\/$ and that
\[
H^\ast(H^q(B;\mathcal L^\ast))\, =\, 0\, =\, H^\ast(H^q(B;\mathcal M^\ast))
\]
for $q>0$.
Then the induced map $H^n(\mathcal L^\ast (B))\,\rightarrow\,
H^n(\mathcal M^\ast (B))\/$ is an isomorphism for
all $n$.
\end{auxiliar}
Let $p\colon M\rightarrow B\/$ be the bundle of fiber $N$. To prove that
$\Lambda$ is a quasi-isomorphism, for $j\ge 0\/$
fixed, we consider the differential sheaves defined as follows:
for each open subset $U\/$ of $B$, set
\[
{\mathcal L}_j^{\ast }(U)\, =\,
H^j(p^{-1}U,\,_{{\diff}}AS_{\mathcal
F}^\ast\mid p^{-1}U)
\]
and
\[
{\mathcal M}_j^{\ast }(U)\, =\, H^j(p^{-1}U,\, A_{\mathcal F}^\ast\mid p^{-1}U)
\]
The sheaf ${\mathcal L}^0_0\/$ is soft, i.e., every section defined
on a closed set can be
extended to $B$. Then all sheaves ${\mathcal L}_j^{\ast}\/$ are soft,
as they are modules
over ${\mathcal L}^0_0$. The same holds for ${\mathcal M}_j^{\ast}$.
Since these sheaves are soft, they are acyclic, and thus
\[
H^q(B,\mathcal L_j^\ast )\, =\, 0\, =\, H^q(B,\mathcal M_j^\ast )
\]
for $q>0$.
We prove now that the homomorphism $\Lambda^\ast\colon\mathcal
H(\mathcal L^\ast)\rightarrow
\mathcal H(\mathcal M^\ast)\/$ induced by $\Lambda$ is an
isomorphism. In fact, as a consequence of the theorem for Lie
foliations, this homomorphism will be an isomorphism over each stack
of the sheaves.
We compute the stack of $H^\ast ({{\mathcal L}_j^{\ast}} )\/$ at a
point $x\in B$. Let $U\/$ be a contractible open
subset of $B\/$ such that
$p^{-1}(U)\cong U\times N$.
$N\/$ and $p^{-1}(U)\/$ are homotopically equivalent, by foliated
$2$-homotopies. Then
\begin{equation}\label{xtubo}
E_{2,{\diff}}^{p,q}({\mathcal L}_{p^{-1}(U)})\,\cong\,
E_{2,{\diff}}^{p,q}({\mathcal F_0})\, ,
\end{equation}
and, finally,
\begin{equation}\label{xtallo}
H^p({\mathcal L}_q^{\ast})_x\, =\, E_{2,{\diff}}^{p,q}({\mathcal F_0})\, .
\end{equation}
Analogously, $H^p({\mathcal M}_q^{\ast})_x\, =\,
E_{2,dR}^{p,q}({\mathcal F_0})$, and the proof that
$\Lambda$ is a quasi-iso\-mor\-phism is completed.
The proof for $J\/$ is similar.
\end{proof}
Let $\mathcal F\/$ be a Riemannian
foliation on a compact manifold $M$. Let
$Q$ be the principal bundle of transverse frames of $\mathcal F$.
Molino defined a lifting
foliation
$\tilde{\mathcal F}\/$ on
$Q$, with the same dimension as $\mathcal F$. For
such a foliation, the closures of the leaves are the fibers of a
locally trivial fibration, the
\emph{basic fibration}. The foliation induced in each fiber is a Lie
foliation, so we are under the hypotheses of
Proposition~\ref{tp}.
\begin{theorem}\label{dRth}%
For a Riemannian foliation $\mathcal F\/$ on a compact manifold $M$
the homomorphism $J$ and
$\Lambda\/$ are quasi-isomorphisms.
\end{theorem}
\begin{proof}
Let $\mathcal F\/$ be a Riemannian foliation on a compact manifold.
We assume, for
simplicity, that it is transversally oriented. Let $Q\/$ be the
$SO(k)$-principal bundle of
transverse frames of $\mathcal F$ equipped with the lifting foliation
$\tilde{\mathcal F}$. We use the notation
$E_2(\mathcal F)\/$ for the de~Rham and the continuous or
differentiable Alexander-Spanier spectral sequences of $\mathcal
F$. Associated to the action of
$SO(k)\/$ on $Q\/$ there are, for each $q$, spectral sequences
\[
E_2^{r,s}(q)\, =\, E_2^{s,q}(\mathcal F)\otimes H^r(SO(k),\mathbb
R)\,\Rightarrow\,
E_2^{r+s ,q}(\tilde{\mathcal F})\, .
\]
Now, $J\/$ and $\Lambda\/$ induce homomorphisms between the spectral
sequences $E_2(q)$,
corresponding to the differentiable Alexander-Spanier and to the
continuous Alexander-Spanier and
de~Rham cohomologies, respectively. These homomorphisms are isomorphisms
over
$E_2^{s ,q}(\tilde{\mathcal F})\/$ and over
$H^r(SO(k))$. Now the result follows by the Zeeman's comparison theorem.
\end{proof}
%%%%%%%
\begin{corollary}\label{inv}
The de~Rham spectral sequence $E_{r,dR}({\mathcal F})\/$ of a
Riemannian foliation is a topological
invariant for $r\ge 2$.
\end{corollary}
For the basic cohomology, $E_{2,dR}^{p,0}({\mathcal F})$, this result
was proved by El Kacimi and
Nicolau \cite{AM}. In the general case, it was also proven in
\cite{sm}, by a different
method.
For an arbitrary foliation Corollary~\ref{inv} is not true.
\begin{example}
We consider foliations of codimension $1$,
without compact leaves. There are well known examples of such
foliations in the torus that are
topologically equivalent, but have different de~Rham spectral sequences
(cf.~\cite{re2}, \cite{Ar}).
All these foliations, if they are transversally orientable, can be
defined by a nonsingular
closed $1$-form, but to do that it is sometimes necessary to change
the smooth structure of $M$.
(This is a well known theorem by Sacksteder; cf.~\cite{Caco}.) This
change does not modify the
continuous cohomology, but it certainly changes the de~Rham
spectral sequence:
$E_{2,dR}^{1,0}({\mathcal F})\/$ is isomorphic to $\mathbb R$, in the
new smooth structure, and
vanishes in the old one. But for a codimension one foliation
we always have
\[
E_{r,\diff}^{0,q}({\mathcal F})\,\cong\,
E_{r,dR}^{0,q}({\mathcal
F})
\]
for $r\ge 1$,
and
\[
E_{r,\diff}^{1,0}({\mathcal F})\,\cong\,
E_{r,dR}^{1,0}({\mathcal
F})\, ,
\]
for $r\ge 2$. In fact, for codimension one, the sequence
\[
0\,\longrightarrow\, S^\ast_{\mathcal F}
\,\longrightarrow\, AS^\ast_{\mathcal F}\,\longrightarrow\,
A^\ast_{\mathcal F}\,\longrightarrow\, 0
\]
always splits. So these foliations provide examples, where continuous
and differentiable
Alexander-Spanier spectral sequences are different.
\end{example}
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\end{document}