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\lhead{\tiny Topologia das Variedades, IMPA, 2018} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  IMPA,\ \ \sc\version}
\rhead{{\tiny  Test problems (1)}}

\begin{document}

\listok{1}{Topologia das Variedades: June 29 exam}

{\small The final mark is given as follows: A+ for score of 6,
A for 4, B for 3, C for 2, and modified (increased or 
decreased) depending on results of the oral examination.

The first student (of two) is offered all odd problems
and all problems with ``2 points'' mark, the second student
even problems and  with ``2 points'' mark.}


\exercise[2 points]
Let $M$ be a compact simply connected manifold,
and $Z \subset M$ a closed smooth hypersurface. Prove that the
complement $M \backslash Z$ is disconnected, or find
a counterexample
\ez

\exercise[2 points]
Prove that $\R^5$ is not homeomorphic to $\R^4$.
\ez

\exercise
Prove that a smooth map $\phi:\; S^4 \arrow S^5$ cannot
be surjective.
\ez

\exercise
Let $D\subset \R^2$ be an open disk.
Prove that a smooth map $\phi:\; S^4 \arrow D$
cannot be surjective.
\ez

\exercise
Let $G$ be a topological group. Prove that its
fundamental group is commutative.
\ez

\exercise
Find a compact manifold with fundamental group $\Z/n\Z$,
for any given number $n \in \Z$.
\ez

\exercise
Let $G$ be a finite group which freely acts
on a manifold $M$, and $M_1= M/G$. Prove that
the pullback map of de Rham cohomology 
$H^*(M_1) \arrow H^*(M)$ is injective.
\ez

\exercise
Let $A\in GL(n, \R)$ be a linear automorphism
with all eigenvalues $|a_i|>1$, and $H:= \R^n\backslash 0/\langle A\rangle$
be a quotient of $\R^n\backslash 0$ by the group generated by $A$.
Prove that $H$ is diffeomorphic to $S^1 \times S^{n-1}$.
\ez

\exercise
Let $V=\R^{2n}$, and $\alpha\in \Lambda^n V$ be a non-zero element
of the corresponding Grassmann algebra.
Prove that there exists $\beta \in \Lambda^n V$ such that 
$\alpha \wedge \beta\neq 0$.
\ez


\exercise
Let $\omega\in \Lambda^2V$ be a non-degenerate 
anti-symmetric 2-form on a vector space. Prove that
the multiplication by $\omega$ 
defines an injective map $\Lambda^{1}(V)\arrow \Lambda^{3}(V)$.
\ez

\exercise
Calculate $\pi_3(S^3 \times RP^4)$.
\ez

\exercise
Show that a closed surface whose fundamental group $G$ is infinite
is $K(G,1)$.
\ez


\end{document}
