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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf  Topologia das Variedades \\[15mm]
\small Cohomology, lecture 11: fundamental class }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

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

\newpage


\newpage

{\bf \blue Cohomology  with compact support (reminder)}

\definition
Let $\omega$ be a differential form on $M$ {\bf \blue Support}
$\Supp(\omega)$ of $\omega$ is the closure of the set 
of all points where $\omega\neq 0$.
We say that $\omega$ {\bf \blue has compact support}
if its support is compact.

\remark
Clearly, {\bf \purple a differential of a form with compact support 
has compact support.}

\definition
{\bf \blue Cohomology with compact support} of a manifold $M$
is cohomology of the complex $(\Lambda^*_c(M), d)$ 
of differential forms with compact support.


\theorem {\bf \blue (Poincar\'e duality theorem)}\\
Let $M$ be an $n$-dimensional oriented connected manifold.
Then the integration map $\alpha \arrow \int_M \alpha$
gives an isomorphism $H^n_c(M)=\R$. Moreover, {\bf \red the 
multiplication $H^i_c(M) \times H^{n-i}(M) \arrow H^n_c(M)=\R$
defines a non-degenerate pairing between $H^i_c(M)$
and $H^{n-i}(M)$.}


\newpage

{\bf \blue Fundamental class}

\definition
Let $M$ be an oriented $n$-manifold,
$X \subset M$ a $d$-dimensional compact oriented submanifold.
Consider a functional $H^d(M) \stackrel{\int_X} \arrow \R$
mapping a form $\alpha \in H^d(M)$ to $\int_X \alpha$.
Since $H^d(M)^*= H^{n-d}_c(M)$, the linear functional $\int_X$
is given by the integration with a cohomology class $[X]$:
\[
\int_X \alpha= \int_M [X] \wedge \alpha
\]
Then $[X]$ is called {\bf \blue the fundamental class of $M$}.

\definition
A de Rham cohomology class is called {\bf \blue integer}
if it is an image of an integer singular class under
the natural map $H^*(M, \Z) \arrow H^*(M, \R)$.

\claim 
{\bf \red Fundamental class $[X]$ is always integer.}

\proof
Partition $[X]$ into a union of simplices $\Delta_i$.
Then $\int_\alpha X = \sum_i \int_{\Delta_i}\alpha$, that
is, it is given by a pairing with an integer homology class.
\endproof

\newpage

{\bf \blue Intersection index}

\definition
Let $X, Y\subset M$ be two closed submanifolds
of an $n$-manifold with $\dim X + \dim Y= n+d$. We say that
$X$ and $Y$ {\bf \blue intersect transversally}
if for $d<0$ they do not intersect and for 
$d\geq 0$ and for each intersection point
$m \in X \cap Y$, there exists a neighbourhood
$U \ni m$ and a diffeomorphism to an open 
ball $U \arrow B$ which maps $X$ and $Y$
to linear subspaces, intersecting in 
a $d$-dimensional subspace.

\definition
Let $X, Y\subset M$ be two closed oriented submanifolds
of an oriented $n$-manifold with $\dim X=a$, $\dim Y= b$ and $a+b=n$.
Assume that $X$ and $Y$ intersect transversally.
Then $X \cap Y$ is a discrete set, and for any
$m \in X \cap Y$, one has $T_m M = T_m X \oplus T_m Y$.
Using orientation on $X$ and $Y$, we obtain the
volume forms $\omega_X \in \Lambda^a T_m^* X$
and $\omega_Y \in \Lambda^b T_m^* Y$, defined
up to a positive constant.
Then $\omega_X\wedge \omega_Y\in \Lambda^n T_m ^*M$
defines an orientation in $T_mM$. We say
that {\bf \blue $X$ and $Y$ intersect positively in $m$}
if this form defines the standard orientation on $M$,
and {\bf \blue intersect negatively} if it defines 
the opposite orientation.

\remark 
If $X$ and $Y$ are odd-dimensional,
$\omega_X\wedge \omega_Y=-\omega_Y\wedge \omega_X$,
hence {\bf \purple 
for odd dimensions the intersection index depends on order,}
otherwise it is independent.

\newpage

{\bf \blue Intersection form}

The main result of today's lecture.

\theorem
Let  $X, Y\subset M$ be two compact oriented submanifolds
of an oriented $n$-manifold with $\dim X + \dim Y= n$,
and $[X], [Y]$ their fundamental classes.
Assume that they intersect transversally in $x_1, ..., x_n$,
with intersection indexes $\sigma_1, ..., \sigma_n\in \{\pm 1\}$.
{\bf \red Then $\int_M [X]\wedge [Y]= \sum_i \sigma_i$.}

In other words, {\bf \purple Poincar\'e pairing is equal
to the intersection pairing}.

{\em \green It would be proven later today}.


\newpage

{\bf \blue A tubular neighbourhood}

\definition
Let $X \subset M$ be a smooth submanifold,
and $TM\restrict X$ the tangent
bundle to $M$ restricted to $X$. The {\bf \blue normal bundle}
$NX$ is defined as a quotient $TM\restrict X /TX$.
{\bf \blue A tubular neighbourhood} of $X$ in $M$
is a neighbourhood $U\supset X$ which is diffeomorphic
to a neighbourhood of $X$ in the total space of $NX$
which is convex in each fiber $N_x X$.

\claim 
Let $X\subset M$ be a compact submanifold.
{\bf \red Then a tubular neighbourhood always exists.}

\proof Introduce a Riemannian metric on $M$. Then
$NX$ can be identified with an orthogonal complement
to $TX$ in $TM\restrict X$. Consider the exponential
map $\exp:\; \Tot NX \arrow M$ mapping $v\in NX$
to $\gamma_v(1)$, where $\gamma:\; [0, 1]\arrow M$
is a geodesic tangent to $v$. It is well defined
when $v$ is sufficiently small, and its differential
in $v=0$ is bijective, because it is bijective
when $M$ is a vector space with a flat Riemannian metric 
and $X\subset M$ is an affine subspace.
{\bf \purple Then  $\exp:\; \Tot NX \arrow M$
is a diffeomorphism to its image in 
a neighbourhood of $X\subset \Tot NX$}
by the inverse function theorem.
\endproof

\newpage

{\bf \blue Realizing the fundamental class by a form}

\remark
Let $X\subset M$ be a submanifold and $U\supset X$
a tubular neighbourhood. Realizing $U$ as a 
fiberwise convex subset in $\Tot(NX)$ and using
the homothety map, we obtain that
{\bf \purple $U$ is homotopy equivalent
to $X\subset \Tot(NX)$.}

Poincar\'e duality immediately implies the following claim.

\claim 
Let $X\subset M$ be a submanifold and $U\supset X$
a tubular neighbourhood. {\bf \red Then the space $H^d(X)$ 
is dual to $H^{n-d}_c(U)$.}

\remark The $n$-th cohomology of a compact oriented
$n$-manifold $X$ is 1-dimensional, by Poincare duality. 
By definition, {\bf \purple the group 
$H^n(X)$ is generated by its own
funfamental class $[X]$.}

\proposition Let $X\subset M$ be a 
compact $d$-dimensional
submanifold, with $X$ and $M$ oriented and $X$ compact.
Consider a tubular neighbourhood  $U\supset X$.
{\bf \red Then the fundamental class of $X$ can be represented by
a compactly supported closed form $\tau_X\in \Lambda_c^{n-d}(U)$.}

\proof
Let $\alpha \in H^d(M)$ be any closed form.
Then $\int_X \alpha$ depends only on restriction
of $\alpha$ to $U$. Using Poincare duality in $H^*(U)$,
we find a class in $H^{n-d}_c(U)$ represented by
a form  $\tau_X\in \Lambda_c^{n-d}(U)$ such that
$\int_X \alpha = \int_U \tau_X\wedge \alpha$.
Then $\tau_X= [X]$.
\endproof

%\newpage
%
%{\bf \blue Spherical compactification  of a vector bundle}
%
%\definition
%Let $B$ be a vector bundle on a manifold $M$,
%and $L$ be a trivial bundle of rank 1.
%Consider the bundle $B_1:= B \oplus L$,
%put a smooth scalar product on its fibers,
%and let $\Sph(B)$ be the set of vectors of
%length 1 in $B_1$, considered as a spherical
%fibration over $M$. Then $\Sph(B)$ is called
%{\bf  \blue the spherical compactification of $B$}.
%
%\remark
%Let $\Sph(B)$ be the spherical compactification of $B$.
%For any $x\in M$, the fiber of $\Sph(B) \arrow M$
%in $x$ is obtained as a one-point compactification
%of the fiber $B_x$. Moreover, the complement
%to the set of vectors $(0, 1)\in \Sph(B)\subset B \oplus L$
%is naturally diffeomorphic to $\Tot(B)$.
%In other words, $\Sph(B)$ is obtained by
%gluing $M$ to $\Tot(B)$ at infinity.


\newpage

{\bf \blue Approximating the $\delta$-function}

\definition
Let $x\in M$ be a point.
{\bf \blue $\delta_x$-function} is a functional on
the space $C^0(M)$ of bounded functions such that
$\delta_x(f)=f(x)$.

\definition
Consider a sequence $\alpha_i$ of top forms
on $M$ with $\int_M \alpha_i=1$.
We say that $\alpha_i$ {\bf \blue approximates the
$\delta$-function} if $\lim_i \int \alpha_i f = f(x)$.

\proposition
Let $\pi:\; W \arrow M$ be a smooth fibration,
$\beta$ a form with compact support of dimension $\dim W - \dim M$, and
$\alpha_i$ approximates the delta-function $\delta_x$.
Then $\lim_i \int \pi^*\alpha_i \wedge \beta= \int_{\pi^{-1}(x)}\beta$.

\proof Follows from the Fubini theorem. \endproof


\newpage

{\bf \blue Intersection form and Poincar\'e duality}

\theorem
Let  $X, Y\subset M$ be two compact oriented submanifolds
of an oriented $n$-manifold with $\dim X + \dim Y= n$,
and $[X], [Y]$ their fundamental classes.
Assume that they intersect transversally in $x_1, ..., x_n$,
with intersection indexes $\sigma_1, ..., \sigma_n\in \{\pm 1\}$.
{\bf \red Then $\int_M [X]\wedge [Y]= \sum_i \sigma_i$.}

\pstep
Let $U$ be a tubular neighbourhood of $X$ in $M$,
and $\tau_X\in \Lambda_c^{n-d}(U)$ the form
representing the fundamental class.
We need only to show that $\int_Y \tau_X= \sum_i \sigma_i$.

{\bf \green Step 2:} 
Consider the projection $\pi:\; U \arrow X$ 
obtained from the identification between $U$ and an
open subset in $\Tot(NX)$. 
Applying an appropriate diffeomorphism, we may
assume that $Y\cap U$ is a union of fibers of $\pi$.
Then  $\int_M [X]\wedge [Y]= \int_M \tau_X\wedge [Y]= 
\int_Y \tau_X= \sum_i \int_{\pi^{-1}(x_i)}\tau_X$.
Therefore, the statement of the theorem
would follow if we prove that $\int_{\pi^{-1}(x)}\tau_X=1$
for any $x\in X$.

{\bf \green Step 3:} Assume that $X$ is $d$-dimensional.
For any $\alpha\in \Lambda^d(X)$, one has
$\int_U \pi^*\alpha \wedge\tau_X = \int_X \alpha$.
Consider a sequence of forms $\alpha_i\in \Lambda^d(X)$
approximating the $\delta$-function; then 
$\lim_i \int_X \pi^*(\alpha_i)\wedge\beta=\int_{\pi^{-1}(x)}\beta$
for all $\beta\in \Lambda^{n-d}_c(U)$.
This gives  $\int_{\pi^{-1}(x)}\tau_X= \lim_i\alpha_i\wedge\tau_x= \int_X \alpha_i=1$.
\endproof



\end{document}
