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\lhead{\tiny Topologia das Variedades, IMPA, 2018} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  IMPA,\ \ \sc\version}
\rhead{{\tiny  Misha Verbitsky}}

\begin{document}

\listok{2}{Topologia das Variedades: class exercise 2}

\exercise
Let $M$ be a smooth manifold,
and $\Tot TM$ the total space of its tangent bundle.
Prove that the manifold 
$\Tot TM$ is orientable, or find a counterexample.
\ez


\exercise
Let $M$ be an orientable compact $n$-manifold.
Prove that $H^n(M)=\R$.
\ez

\exercise
Let $G$ be a finite group acting
freely on a compact manifold $M$,
and let $\Lambda^*(M)^G$ denote the
$G$-invariant differential forms.
Prove that the cohomology of the
complex
\[
\Lambda^0(M)^G \arrow \Lambda^1(M)^G \arrow \Lambda^2(M)^G \arrow ...
\]
are equal to the de Rham cohomology of the quotient $M/G$.
\ez

\exercise
Let $T^n$ be $n$-dimensional compact torus.
Consider the action if $T^n$ on itself,
$x(y)=x+y$. Denote by $\Lambda^k(T^n)^{T^n}$ 
the space of $T^n$-invariant $k$-forms
on $T^n$. Prove that $\Lambda^k(T^n)^{T^n}= \Lambda^k(\R^n)= H^k(T^n)$.
\ez

\exercise
Let $k$ be an odd number.
Recall that {\bf an odd derivation}
of the de Rham algebra $\Lambda^*(M)$
is a map $\delta:\;  \Lambda^*(M)\arrow \Lambda^{*+k}(M)$
which satisfies $\delta(\alpha \wedge \beta)=
\delta(\alpha) \wedge \beta + (-1)^{\tilde \alpha} \alpha \wedge \delta(\beta)$.
Prove that an anticommutator $\delta_1 \delta_2 + \delta_2\delta_1$
of two odd derivations $\delta_1, \delta_2$ of
$\Lambda^*(M)$ is an even derivation, that is,
satisfies the usual Leibitz identity.
\ez


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