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\title{De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds} %\maketitle

%
\begin{theorem}{Hironaka.}
Let $X$ be an affine Nash manifold. Then there exists a compact
affine nonsingular algebraic variety $Y\supset X$ s.t. $Y$ = $X
\udot D \udot U$ where $X$ and $U$ are open and $D = \bigcup
\limits _{i=1}^k D_i$ where $D_i\subset Y$ are closed Nash
submanifolds of codimension 1 and all the intersections are
normal, i.e. every $y\in Y$  has a neighborhood $V$ with a
diffeomorphism $\phi:V \rightarrow \R^n$ s.t. $\phi(D_i\cap V)$ is
either a coordinate hyperplane or empty.
\end{theorem}
%This is a consequence of Hironaka desingularization theorem
%\cite{Hir}. It implies the following interesting theorem.
%
\begin{theorem}{Local triviality of Nash manifolds.}
\label{loctriv}
Any Nash manifold can be covered by finite number of open
submanifolds Nash diffeomorphic to $\R^n$.
\end{theorem}
%
\begin{theorem}
Let $X$ be a Nash manifold. Then $H^i(X)$,$H_c^i(X)$,$H_i(X)$ are
finite dimensional.
\end{theorem}
%
\pagebreak




\begin{theorem} \label{quasiiso}
Let $X$ be an affine Nash manifold. Consider\\
the De-Rham complex of $X$ with compactly supported coefficients
$$DR_c(X): 0 \rightarrow C^{\infty}_c(X,{\Omega^0_X})
\rightarrow ... \rightarrow C^{\infty}_c(X,{\Omega^n_X})
\rightarrow 0,$$ the De-Rham complex of $X$ with Schwartz
coefficients
$$DR_S(X): 0 \rightarrow S(X,{\Omega^0_X})
\rightarrow ... \rightarrow S(X,{\Omega^n_X}) \rightarrow 0,$$ and
the natural map $i: DR_c(X) \rightarrow DR_S(X)$. \\
Then $i$ is a
quasiisomorphism, i.e. it induces an isomorphism on the
cohomologies.
\end{theorem}
%

\begin{theorem} \label{quasiisogen} Let $X$ be an affine Nash
manifold. Consider \\
the De-Rham complex of $X$ with coefficients in classical
generalized functions, i.e. functionals on compactly supported
densities $$DR_{G}(X): 0 \rightarrow C^{-\infty}(X,{\Omega^0_X})
\rightarrow ... \rightarrow C^{-\infty}(X,{\Omega^n_X})
\rightarrow 0,$$ the De-Rham complex of $X$ with coefficients in
generalized Schwartz functions  $$DR_{GS}(X): 0 \rightarrow
C^{-\infty}(X,{\Omega^0_X}) \rightarrow ... \rightarrow
C^{-\infty}(X,{\Omega^n_X}) \rightarrow 0,$$ and the natural map
$i:DR_{GS}(X) \rightarrow DR_G(X) $.  Then $i$ is a
quasiisomorphism.
\end{theorem}

\begin{theorem} \label{bottomline}
Let $X$ be an affine Nash manifold of dimension $n$. Then
\\ $H^i(DR_{GS}(X))\cong H^i(X)$
\\ $H^i(DR_{S}(X))\cong H_c^i(X)$
\\ $H^i(TDR_{S}(X))\cong H_i(X)$
\\and the standard pairing between
$S(X,T \Omega_X^{n-i})$ and $GS(X,{\Omega_X^i})$ gives an
isomorphism between $H^i(DR_{GS}(X))$ and $(H^i(TDR_{S}(X)))^*$.
\end{theorem}

\pagebreak


\begin{definition}
{\rm Let $X$ and $Y$ be Nash manifolds and let $E$ and $F$ be Nash
bundles over them. We define integration by fibers $IF:S(X \times
Y, E \boxtimes F)\times GS(Y,\widetilde{F}) \rightarrow
\Gamma(X,E)$ (where $\Gamma(X,E)$ is the space of all global
sections of $E$ over $X$) by $IF(\xi,\eta)(x)= \eta
(\xi|_{\{x\}\times Y}).$}
\end{definition}
%
\begin{proposition}
$Im(IF)\subset S(X,E)$, and $IF:S(X \times Y, E \boxtimes F)\times
GS(Y,\widetilde{F}) \rightarrow S(X,E)$ is continuous.
\end{proposition}
%
\begin{definition}
{\rm Let $X$ and $Y$ be Nash manifolds and let $E$ and $F$ be Nash
bundles over them. We define $IF':GS(X \times Y, E \boxtimes
F)\times S(Y,\widetilde{F}) \rightarrow GS(X,E)$ by
$IF'(\xi,\eta)(f)= \xi(\eta \boxtimes f).$}
\end{definition}
%
\begin{theorem} \label{reltriv}
Let $X$ be a Nash manifold and $Y$ be an affine Nash manifold of
dimensions $m$ and $n$ correspondingly. Denote $F=X\times Y$.
Identifying $ T \Omega^i_{F\rightarrow X}$ with $(X \times \R)
\boxtimes T \Omega^{i}_Y$ (where $X \times \R$ is interpreted as
the trivial bundle on $X$) and $\Omega^{n-i}_Y$ with $\widetilde{
T \Omega^{i}_Y}$ we get bilinear form $IF:S(F,T \Omega^i_{F
\rightarrow X})\times GS(Y,\Omega^{n-i}_Y) \rightarrow S(X)$. Then
$IF$ induces a non degenerate pairing between $H^i(TDR_{S}(F
\rightarrow X))$ and $H^{n-i}(Y)$ valued in $S(X)$ i.e. gives an
isomorphism between $H^i(TDR_{S}(F \rightarrow X ))$ and
$H^{n-i}(Y)^*\otimes S(X)$.
\end{theorem}
%
\begin{theorem} \label{genreltriv}
Let $X$ be a Nash manifold and $Y$ be an affine Nash manifold of
dimensions $m$ and $n$ correspondingly. Denote $F=X\times Y$. Then
the bilinear form $IF': GS(F, \Omega^i_{F \rightarrow X})\times
S(Y,T\Omega^{n-i}_Y) \rightarrow GS(X)$ gives an isomorphism
between $H^i(DR_{GS}(F \rightarrow X ))$ and $H_{n-i}(Y)^*\otimes
GS(X)$.
\end{theorem}

\pagebreak


\begin{theorem}
Let $p:F \rightarrow X$ be a Nash locally trivial fibration and
$E$ be a Nash bundle over $X$. Then $H^k(DR_S^E(F\rightarrow X))
\cong S(X,H^k_c(F\rightarrow X)\otimes E)$.
\end{theorem}
%
\begin{theorem} \label{relgenDeR}
Let $p:F \rightarrow X$ be a Nash locally trivial fibration and
$E$ be a Nash bundle over $X$. Then $H^k(DR_{GS}^E(F\rightarrow
X)) \cong GS(X,H^k(F\rightarrow X)\otimes E)$.
\end{theorem}
%
\begin{definition}
{\rm Let $f:X \rightarrow Y$ be a Nash map of Nash manifolds. It is
called a \emph{Nash locally trivial fibration} if there exist a Nash
manifold $M$ and surjective submersive Nash map $g:M \rightarrow Y$
s.t. the basechange $h:X \underset {Y}{\times} M \rightarrow M$ is
trivializable, i.e. there exists a Nash manifold $Z$ and an
isomorphism $k:X \underset {Y}{\times} M \rightarrow M \times Z$
s.t. $\pi \circ k = h$ where $\pi : M \times Z \rightarrow M$ is the
standard projection.}
\end{definition}
%
\begin{theorem} \label{sursec}
Let $M$ and $N$ be Nash manifolds and $s:M \rightarrow N$ be a
surjective submersive Nash map. Then locally it has a Nash
section, i.e. there exists a finite open cover $N= \bigcup \limits
_{i=1}^k U_i$ s.t. $s$ has a Nash section on each $U_i$.
\end{theorem}
%
\begin{proposition} \label{ext}
Let $M$ and $N$ be Nash manifolds and $f:M \rightarrow N$ be a
Nash submersion. Let $L \subset N$ be a Nash submanifold and $s:L
\rightarrow M$ be a section of $f$. Then there exist a finite open
Nash cover $L \subset \bigcup \limits _{i=1}^n U_i$ and sections
$s_i:U_i \rightarrow M$ of $f$ s.t. $s|_{L \cap U_i} = s_i|_{L
\cap U_i}$.
\end{proposition}
%
\begin{theorem} \label{semisec}
Any semi-algebraic surjection $f:X \rightarrow Y$ of
semi-algebraic sets has a semi-algebraic section.
\end{theorem}
%
\begin{theorem} \label{strat}
Let $f:M \rightarrow N$ be a semi-algebraic map of Nash manifolds.
Then there exists a finite stratification of $M$ by Nash manifolds
$M= \underset{i=1}{ \overset{k}{\udot}}  M_i$ s.t. $f|_{M_i}$ is
Nash.
\end{theorem}
\newpage
\begin{definition} %\label{}
{\rm Let $\mathfrak{g}$ be a Lie algebra of dimension $n$. Let
$\rho$ be its representation. Define $H^i(\mathfrak{g},\rho)$ to
be the cohomologies of the complex: $$C(\mathfrak{g},\rho): 0
\overset {d}{\rightarrow} \rho \overset {d}{\rightarrow}
\mathfrak{g}^* \otimes \rho \overset {d}{\rightarrow}
({\mathfrak{g}^*}) ^{\wedge 2} \otimes \rho \overset
{d}{\rightarrow}... \overset {d}{\rightarrow} ({\mathfrak{g}^*})
^{\wedge n} \otimes \rho \overset {d}{\rightarrow}
0$$ with the differential defined by \\
$$d\omega(x_1,...,x_{n+1}) = \sum _{i=1}^{n+1} (-1)^i \rho(x_i)
\omega(x_1,...,x_{i-1},x_{i+1},...,x_{n+1}) +$$
$$+\sum
_{i<j}(-1)^{i+j}\omega([x_i,x_j],x_1,...,x_{i-1},x_{i+1},...,x_{j-1},x_{j+1},...,x_{n+1})$$
where we interpret $({\mathfrak{g}}) ^{\wedge k} \otimes \rho$ as
anti-symmetric $\rho$-valued $k$-forms on ${\mathfrak{g}}$.}
\end{definition}
%
\begin{remark}
$H^i(\mathfrak{g},\rho)$ is the i-th derived functor of the functor
$\rho \mapsto {\rho}^{\mathfrak{g}}$.
\end{remark}
%
\begin{theorem}\label{recipe}
Let $G$ be a Nash group. Let $X$ be a Nash $G$-manifold and $E
\rightarrow X$ a Nash $G$-equivariant bundle. Let $Y$ be a
strictly simple Nash $G$-manifold. Suppose $Y$ and $G$ are
cohomologically trivial (i.e. all their cohomologies except $H^0$
vanish and $H^0=\R$) and affine. Denote $F=X\times Y$ . Note that
the bundle $E \boxtimes \Omega_Y^i$ has Nash $G$-equivariant
structure given by diagonal action. Hence the relative De-Rham
complex $DR_{GS}^E(F \rightarrow X)$ is a complex of
representations of $\mathfrak{g}$. Then $H^i(\mathfrak{g},
GS(X,E)) = H^i((DR_{GS}^E(F \rightarrow X))^{\mathfrak{g}})$.
\end{theorem}
%
\begin{proposition}
Let $G$ be a connected Nash group and $F$ be a Nash $G$ manifold
with strictly simple action. Denote $X:=G \setminus F$ and let $E
\rightarrow X$ be a Nash bundle. Then
$(GS(F,\pi^*(E)))^{\mathfrak{g}}\cong GS(X,E)$ where
$\pi:F\rightarrow X$ is the standard projection.
\end{proposition}
%
\begin{corollary}\label{H2G}
Let $G$ be a Nash group and $X$ be a transitive Nash $G$ manifold.
Let $x \in X$ and denote $H:=stab_G(x)$. Consider the diagonal
action of $G$ on $X\times G$. Let $E\rightarrow X\times G$ be a $G$
equivariant Nash bundle. Then $GS(X\times G,E)^\mathfrak{g}\cong
GS(\{x\} \times G,E|_{\{x\} \times G})^\mathfrak{h}$.
\end{corollary}

\begin{theorem}{Shapiro lemma.}
Let $G$ be a Nash group and $X$ be a transitive Nash $G$ manifold.
Let $x \in X$ and denote $H:=stab_G(x)$. Let $E\rightarrow X$ be a
$G$ equivariant Nash bundle. Let $V$ be the fiber of $E$ in $x$.
Suppose $G$ and $H$ are cohomologically trivial. Then
$H^i(\mathfrak{g},GS(X,E))\cong H^i(\mathfrak{h},V)$.
\end{theorem}

\end{document}
