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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf  Topologia das Variedades \\[15mm]
\small Cohomology, lecture 6: de Rham theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small\bf Misha Verbitsky  
\\[2mm]
May 14, 2018\\[4mm]

{\tiny\bf IMPA, Tuesdays and Thursdays, 10:30, Sala 224}
}
\end{center}


\newpage

{\bf \blue Long exact sequence (reminder)}

\definition
{\bf\blue A complex} is a sequence of vector spaces
and homomorphisms
$... \stackrel d\arrow C^{i-1} \stackrel d\arrow
C^i\stackrel d\arrow C^{i+1} \stackrel d\arrow...$
satisfying $d^2=0$.  {\bf \blue
Homomorphism} $(C^*, d)\arrow (C_1^*, d)$ of complexes 
is a sequence of homomorphism $C^i \arrow C_1^i$
commuting with the differentials.

\definition
An element $c\in C^i$ is called {\bf\blue closed}
if $c\in \ker d$ and {\bf \blue exact} if $c\in \im d$.
{\bf \blue Cohomology} of a complex
is a quotient $\frac{\ker d}{\im d}$.
One denotes the $i$-th group of cohomology 
of a complex by $H^i(C^*)$

\remark A homomorphism of complexes induces a natural
homomorphism of cohomology groups.

\definition
{\bf \blue Short exact sequence of complexes}\\
$0 \arrow A^* \arrow B^* \arrow C^* \arrow 0$
is a sequence of morphisms of complexes
$A^* \stackrel x \arrow B^* \stackrel y\arrow C^*$
such that $x:\; A^i \arrow B^i$ is injective,
$y:\; B^i \arrow C^i$ is surjective (for all $i$),
and $\ker y= \im x$.

\theorem
Let  $0 \arrow A^* \arrow B^* \arrow C^* \arrow 0$
be an exact sequence of complexes. Then 
there exists a {\bf \blue long exact sequence of cohomology}
\[... \arrow H^{i-1}(C^*) \arrow H^i(A^*) \arrow  H^i(B^*) 
\arrow H^{i}(C^*) \arrow H^{i+1}(A^*) \arrow ...
\]

\newpage

{\bf \blue Differential forms on closed subsets (reminder)}

\definition
Let $M$ be a manifold, and $Z\subset M$
a closed subset. Let $\alpha, \beta$
be two differential forms defined in 
open sets $U_\alpha$ and $U_\beta$ containing $Z$.
We say that $\alpha$ and $\beta$ are equivalent
if $\alpha=\beta$ on $U_\alpha\cap U_\beta$.
The space of equivalence classes is denoted
$\Lambda^*(Z)$ and called {\bf \blue the space of
differential forms on $Z$} or {\bf \blue
the space of germs of differentials forms in $Z$}.

\remark
{\bf \purple If $Z$ is a manifold with boundary, 
$\Lambda^*(Z)$ is the space of differential
forms on $Z$,} by definition of differential
forms on manifolds with boundary.

\exercise
Using partition of unity, {\bf \red prove that
the natural restriction map $\Lambda^*(M)\arrow \Lambda^*(Z)$
is surjective for any closed $Z\subset M$.}

\remark
The de Rham differential is well defined on $\Lambda^*(Z)$,
allowing us to {\bf \purple define the de Rham cohomology of $Z$ as usual.}
Poincar\'e lemma holds in this situation, too.

\claim
Let $X\subset \R^n$ be a closed starlike subset.
{\bf \red Then $H^i(X)=0$ for all $i>0$.}

\proof Same as for open $X$. \endproof

\newpage

{\bf \blue Mayer-Vietoris long exact sequence (reminder)}

\claim
Let $X, Y\subset M$ be closed subsets. Then 
{\bf \red the restriction maps define an exact sequence of complexes:}
\[ 
0 \arrow \Lambda^*(X\cup Y)\stackrel \phi \arrow \Lambda^*(X)
\oplus  \Lambda^*(Y) \stackrel \psi \arrow \Lambda^*(X\cap Y)\arrow 0. \ \ (*)
\]
Here $\phi$ is restriction to both components, and
$\psi$ is restriction $|_{X\cap Y}$
on the first component and $-|_{X\cap Y}$ on the second component.

\proof
The map $\phi$ is clearly injective;  $\psi$
is surjective because all forms in $\Lambda^*(X\cap Y)$
can be smoothly extended to $M$. Now, $\ker \psi$ 
is pairs $\alpha \in \Lambda^*(X),  \beta \in \Lambda^*(Y)$
such that $\alpha|_{X\cap Y}=- \beta|_{X\cap Y}$,
and this is precisely pairs which agree on an open neighbourhood of
$X\cap Y$, that is, obtained by restriction from some 
$\gamma\in \Lambda^*(X\cup Y)$.
\endproof

\corollary {\bf \blue (Mayer-Vietoris long exact sequence)}
Let $X, Y\subset M$ be closed subsets.
{\bf \red Then there is a long exact sequence associated with (*):}
\begin{multline*}... \arrow H^{i-1}(X)
\oplus H^{i-1}(Y) \arrow H^{i-1}(X\cap Y) \arrow H^i(X\cup Y) \arrow \\ \arrow  H^i(X)
\oplus H^i(Y)  
\arrow H^{i}(X\cap Y) \arrow H^{i+1}(X\cup Y) \arrow ...
\end{multline*}
\endproof

\newpage

{\bf \blue Singular cohomology}

\definition
Let $M$ be a topological space and $C_i$ {\bf \blue the group of 
singular $i$-chains,}
that is, the group freely generated by continuous maps $f:\; \Delta^k \arrow M$,
where $\Delta^k$ is $i$-simplex. {\bf \blue Boundary operator}
maps a $k$-simplex $f$ to a sum $\sum_{i=0}^k (-1)^i f_i$,
where $f_i\in C_{i-1}$ is $i$-th face of $f$.
Cohomology of this complex are called {\bf \blue singular
homology} of $M$.

\definition
{\bf \blue The group of $i$-cochains} is $C^i:=\Hom(C_i, \Z)$.
The boundary operator $\6:\; C_i\arrow C_{i-1}$
defines {\bf \blue the coboundary operator}
(denoted by the same letter) $\6:\; C^i\arrow C^{i-1}$.
The cohomology of the complex
\[ ... \stackrel \6 \arrow C^i \stackrel \6 \arrow C^{i+1}  \stackrel \6 \arrow...
\]
are called {\bf \blue singular cohomology of $M$}.

\definition
Let $R$ be a ring (typically, $R$ will be a field $\R$).
{\bf \blue The group of $i$-cochains with values in $R$} 
is $C^i_R:=\Hom_\Z(C_i, R)$, where $\Hom_\Z$ denotes
the homomorphism of abelian groups. Cohomology
of the corresponding cochain complex are called
{\bf \blue cohomology with coefficients in $R$}.

\newpage

{\bf \blue Properties of singular cohomology}

\remark
Singular homology are functorial.
Indeed, for any continuous map $f:\; X \arrow Y$,
the map $f$ maps any chain in $X$ to a chain in $Y$.
Then any cochain on $Y$ gives a cochain on $X$.
This operation is called {\bf \blue the pullback} of a cochain.

\claim {\bf \red Singular cohomology are homotopy invariant,}
that is, for any map $F_t:\; X \times [0,1] \arrow Y$,
the map $F_0:\; X \arrow Y$, $F_0(x) = F_t(x\times \{0\})$
acts on the cohomology in the same way as 
$F_1:\; X \arrow Y$, $F_1(x) = F_t(x\times \{1\})$.

\proof Left as an exercise for now (we shall return to it later).
\endproof

\newpage

{\bf \blue Homology are dual to cohomology}

\definition
Define {\bf \blue singular chain with coefficients in a field $R$}
as a formal linear combination of maps $f:\; \Delta^k \arrow M$,
with coefficients in $R$. The boundary map $\6$ is defined 
the space $C_k(R)$ of such chains in a natural way. 
{\bf \blue homology with coefficients in $R$}
is $\frac{\ker\6\restrict C_k(R)}{\6(C_{k+1}(R))}$.

\remark
{\bf \purple There is a natural pairing between homology and 
cohomology.} Indeed, a coboundary vanishes on cycles,
hence the natural pairing between a cycle and a cocycle
descends to cohomologies.

\definition
{\bf \blue A non-degenerate}, or {\bf \blue perfect} pairing between two vector spaces
$V$ and $W$ over a field $R$ is a bilinear map 
$B:\; V\times W \arrow R$ such that for any non-zero
vector $v\in V$ there exists $w\in W$
such that $B(v,w)\neq 0$, and for all non-zero $w\in W$
there exists $v\in V$ with such a property.

\theorem
Let $R$ be a field. {\bf \red Then the natural pairing
$H_i(M, R) \times H^i(M, R) \arrow R$ is non-degenerate,} 
and, moreover, {\bf \red  $H^i(M, R)=H_i(M, R)^*$.}

{\em \green Proof: see the next slide}

\newpage

{\bf \blue Homology are dual to cohomology (2)}

\theorem
Let $R$ be a field. {\bf \red Then the natural pairing
$H_i(M, R) \times H^i(M, R) \arrow R$ is non-degenerate,} 
and, moreover, {\bf \red  $H^i(M, R)=H_i(M, R)^*$.}

\pstep
Let $A:\; V \arrow W$ be a linear map,
and $A^*:\; W^*\arrow V^*$ the dual map.
Then the natural pairing between $\ker A$ and $V^*/\im A^*$
and $\im A$ and $W^*/\ker A^*$ is non-degenerate,
and, moreover, $(\im A)^*= W^*/\ker A^*$.
Indeed, $x\in \ker A \Leftrightarrow \langle Ax, y\rangle=0
\Leftrightarrow  \langle x, A^* y\rangle$, hence
$\im A^*$ is precisely the space of functionals
vanishing  on $\ker A$. Similarly, 
$\langle A(z), y\rangle=0
\Leftrightarrow  \langle z, A^* (y)\rangle$,
hence one $y\in W^*$ vanishes on $\im A$ if and only if
$A^*(y)=0$. 

{\bf \green  Step 2:} 
Let $d:\; C \arrow C$ be a map which satisfies $d^2=0$. Step 1 gives
an isomorphism $(\ker d)^*=C^*/\im d^*$ and 
$(\im d)^*=C^*/\ker d^*$. Then 
$(\ker d/\im d)^* = \frac{\ker d^*}{\im d^*}$.
\endproof

\remark This statement makes no sense 
 when the coefficients are not a field. In this case
{\bf \blue the universal coefficients theorem}
gives a precise formula relating homology and cohomology.

\newpage

{\bf \blue Cohomology of smooth chains}

\proposition
Let $C_k$ be the space of $k$-chains on a smooth manifold $M$
with coefficients in $\R$, and $C_k^{sm}$ the space of smooth chains.
Assume that homology of $M$ are finite-dimensional.
{\bf \red Then cohomology of the complex $C_k$ and cohomology of the complex
$C_k^{sm}$ are equal, and the cohomology of the dual complexes
are also equal.}

\proof
Fix a complete Riemannian metric on $M$.
Consider the ``uniform norm'' $\nu$ on the space of chains, 
defined as follows: it is the maximal norm such that for any $f, g:\; \Delta^k \arrow M$,
one has $\nu(\lambda f) \leq |\lambda|$ and $\nu(f-g) \leq \sup_{x\in \Delta^k}d(f(x), g(x))$.
Clearly, the space $C_k^{sm}$ is dense in $C_k$, hence the
cohomology space of $C_k^{sm}$ is dense in cohomology of $C_k$,
which is equal to homology of $M$. However, cohomology of $C_k$ is
finitely-dimensional, hence $H_i(C_k^{sm})= H_i(M)$.
\endproof

\remark Integration over a smooth chain defines
a linear map $\Lambda^k(M)\arrow  (C_k^{sm})^*$.
Stokes' theorem gives $\int_Cd\alpha=\int_{\6 C}\alpha$,
hence {\bf \purple this map commutes with the differential.}


\newpage

{\bf \blue De Rham theorem}


The main result of this lecture: de Rham cohomology are equal to the singular cohomology.

\theorem {\bf \blue (de Rham theorem)} \\
Let $M$ be a smooth manifold (compact or with a finite polyhedral structure),
$H^*_{DR}(M)$ its de Rham cohomology and $H^*_\sing(M)$ its
singular cohomology. {\bf \red Then the map $H^*_{DR}(M)\arrow H^*_\sing(M)$
constructed above is an isomorphism.}
 
{\it \green It will be proven later today.}

\newpage

{\bf \blue Mayer-Vietoris theorem for singular homology}

\claim
Let $M= U\cup V$ be a metrizable space, where $U$ and $V$ are open.
Denote by $C^k_{U, V}(M)$ the space of chains generated by
simplices $f:\; \Delta^k\arrow M$ which are contained in $V$ or $U$.
{\bf \red Then the following sequence of chain complexes is exact:}
$0 \arrow C_*(U\cap V) \arrow C_*(U)\oplus C_*(V) \arrow  C_*^{U, V}(M)\arrow 0$.
\endproof

\claim {\bf \red The natural embedding $C_*^{U, V}(M)\arrow C_*(M)$
of complexes induces an isomorphism on cohomology of these complexes.}

\proof Fix a metric on $M$.
Any symplex $\Delta$ in $M$ can be partitioned onto smaller simplices
which lie in $U$ or $V$. Indeed, let $S$ be a simplex in $M$, and
$\epsilon$ the distance between $S\backslash U$ and $S\backslash V$.
These are two non-intersecting compact sets, hence $\epsilon >0$.
Clearly, any symplex in $M$ of diameter  $< \epsilon$ belongs to
$U$ or $V$ or both. Now, if we partition $\Delta$ onto smaller
simplices of diameter $<\epsilon$, we obtain a chain ${\cal D}\in C_*^{U, V}(M)$.
However,  $\Delta-{\cal D}$ is a boundary. {\bf \purple This construction implies
that the map $C_*^{U, V}(M)\arrow C^*(M)$ is surjective on cohomology} of complexes.
It is injective on cohomology, because 
for any boundary $x\in C_*^{U, V}(M)$, $x=\6(y)$, with $y\in C_*(M)$,
$y$ can be partitioned in a similar way, giving $y'\in  C_*^{U, V}(M)$
with $d(y')=x'$, where $x'$ is obtained from $x$ by partitionining it onto
smaller simplices.
\endproof

\newpage

{\bf \blue Mayer-Vietoris theorem for singular homology and cohomology}

\corollary {\bf \blue (Mayer-Vietoris exact sequence for homology)}\\
Let $M= U\cup V$ be a metrizable space, where $U$ and $V$ are open.
{\bf \red Then there exists a long exact sequence of homology}
\begin{multline*}... \arrow H_{i+1}(U)
\oplus H_{i+1}(V) \arrow H_{i+1}(U\cup V) \arrow 
H_i(U\cap V) \arrow \\ \arrow  H_i(U) \oplus H_i(V)  
\arrow H_{i}(U\cup V) \arrow H_{i-1}(U\cap V) \arrow ...
\end{multline*}
\proof We obtain this long exact sequence from the exact 
sequence of complexes
$0 \arrow C_*(U\cap V) \arrow C_*(U)\oplus C_*(V) \arrow  C_*^{U, V}(M)\arrow 0$.
\endproof

Dualizing this sequence, we obtain

\corollary {\bf \blue (Mayer-Vietoris exact sequence for cohomology)}\\
Let $M= U\cup V$ be a metrizable space, where $U$ and $V$ are open.
Then {\bf \red there exists a long exact sequence of singular
cohomology with real coefficients.}
\begin{multline*}... \arrow H^{i-1}(U,\R)
\oplus H^{i-1}(V,\R) \arrow H^{i-1}(U\cap V,\R) 
\arrow H^i(U\cup V,\R) \arrow \\ \arrow  H^i(U,\R)
\oplus H^i(V,\R)  
\arrow H^{i}(U\cap V,\R) \arrow H^{i+1}(U\cup V,\R) \arrow ...
\end{multline*}
\endproof

\remark {\bf \red Mayer-Vietoris exact sequence exists for cohomology
with any coefficients.} The proof is very similar to the one for homology.


\newpage

{\bf \blue Voronoi partitions}

\definition
Let $M$ be a metric space, and $S\subset M$
a finite subset. {\bf \blue Voronoi cell} associated
with $x_i \in S$ is $\{z\in M \ \ | \ d(z, x_i) \leq d(z, x_i)
\forall j\neq i\}$. {\bf \blue Voronoi partition}
is partition of $M$ onto its Voronoi cells.

\begin{center}
\epsfig{file=voronoi.jpg,width=0.35\linewidth}\\
{\em\green Voronoi partition}
\end{center}

\theorem 
Let $M$ be a complete Riemannian manifold.
{\bf \red Then, for an appropriate choice of the points $x_i$,
Voronoi cells are polyhedral. }

\newpage

{\bf \blue Convex sets}

\definition
Let $M$ be a metric space, and $S\subset M$
a discrete subset. {\bf \blue The Voronoi cell} associated
with $x_i \in S$ is $\{z\in M \ \ | \ d(z, x_i) \leq d(z, x_i)
\forall j\neq i\}$. {\bf \blue The Voronoi partition}
is partition of $M$ onto its Voronoi cells.

\definition
A subset $S\subset M$ of a Riemannian manifold is called
{\bf \blue convex} if for any two points $x, y\subset S$ the
minimal geodesic connecting $x$ to $y$ is unique, and
it belonds to $S$. 

\remark 
{\bf \purple Any convex set $S$ is starlike.} Indeed. 
fix an ``origin'' $x\in S$. For any $y\in S$,
denote by $\gamma_y:\; [0, d(x, y)]\arrow S$ the minimal geodesic.
Then $F_t(y) =\gamma_y(t)$, $t\in[0,1]$ induces a smooth homotopy between
identity map and the projection mapping $S$ to the point $\{x\}$.

\definition
Let $M$ be a complete Riemannian manifold.
{\bf\blue Injectivity radius} is the supremum of all numbers
$\epsilon >0$ such that any closed ball of radius $\epsilon$ is convex.

\theorem Let $M$ be a compact Riemannian manifold.
{\bf \red Then the injectivity radius of $M$ is always positive.}

\proof Uses basic differential geometry (left as an exercise).
\endproof

\newpage

{\bf \blue Convex covers}

\remark
Let $M$ be a 
compact manifold, and $\epsilon$ less than its injectivity radius.
{\bf \blue An $\epsilon$-net} is a finite subset
$S\subset M$ such that $M$ is contained in 
the union of $\epsilon$-balls
with centers in $S$. {\bf \red Then these $\epsilon$-balls are
convex, as well as all their intersections}
{\bf \purple (an intersection of several convex sets is 
clearly convex).}

\definition
A {\bf \blue convex cover} of a Riemannian manifold is a cover 
$\{U_i\}$ such that all $U_i$ and all the closures ${\overline U_i}$
are convex.

\remark Clearly, {\bf \purple any cover of $M$ by $\epsilon$-balls 
is convex,} if $\epsilon$ is less that the injectivity radius of $M$.

\remark If $M$ is a polyhedral manifold, we could take
for $U_i$ small neighbourhoods of the polyhedra.
Even without the metric, it is clear that
the intersections of $U_i$ are starlike for an appropriate 
choice of neighbourhoods.
Indeed, these intersections are neighbourhoods
of the corresponding faces of the polyhedra.
{\bf \purple This gives a cover with the same properties
as a convex cover.}


\newpage

{\bf \blue The proof of de Rham theorem}



\theorem  {\bf \blue (de Rham theorem)} 
Let $M$ be a smooth manifold (compact or with a finite polyhedral structure),
$H^*_{DR}(M)$ its de Rham cohomology and $H^*_\sing(M)$ its
singular cohomology. {\bf \red Then the map $H^*_{DR}(M)\arrow H^*_\sing(M)$
constructed above is an isomorphism.}

\proof
Let $M$ be a manifold and $\{U_i\}$ a finite open cover such that
all $U_i$, all the closures ${\overline U_i}$ 
and all their intersections are starlike (such a cover
can be obtained from a polyhedral structure or from
a convex cover as above).
Poincar\'e lemma implies that $H^*_{DR}(\bar K)\arrow H^*_\sing(K)$
is an isomorphism for any $K$ which is starlike, where $\bar K$ is the closure of $K$.
Using induction in $n$, we may assume that $H^*_{DR}(\bar K)\arrow H^*_\sing(K)$
is an isomorphism for any $K$ which is obtained as a union 
of $n-1$ or less of $U_i$. Let us prove it for $K=\bigcup_{i=1}^n U_i$.
Using the Mayer-Vietoris exact sequences for de Rham and singular
cohomology, applied to $X=U_0$ and $Y=\bigcup_{i=2}^n U_i$, 
we obtain the following diagram
\[\small
\begin{CD}
... @>>> H^{i-1}_{DR}(\bar X\cap \bar Y) @>>>  H^{i}_{DR}(\bar X\cup \bar Y) 
@>>>  H^{i}_{DR}(\bar X)\oplus  H^{i}_{DR}(\bar X)@>>> H^{i+1}_{DR}(\bar X\cap \bar Y)@>>>...\\
&& @VVV @VVV @VVV @VVV \\
... @>>> H^{i-1}_{\sing}(X\cap Y) @>>>  H^{i}_{\sing}(X\cup Y) 
@>>>  H^{i}_{\sing}(X)\oplus  H^{i}_{\sing}(X)@>>> H^{i+1}_{\sing}(X\cap Y)@>>>...
\end{CD}
\]
By induction assumption,
the vertical arrows in this diagram are isomorphisms for all terms
except for $H^*_{DR}(\bar X \cup \bar Y)\arrow H^*_\sing(X \cup Y)$.
{\bf \purple The 5-lemma implies that they are isomorphisms in all terms.}
\endproof.

\end{document}
