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%version 1.0,\ \   01.07.2018
%version 2.0,\ \   02.07.2018 some problems by M. B. added
%version 2.1,\ \   03.07.2018 the map in 2.2 should be inverted; 2.5 should be even

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\input{defs-listki-en.tex}
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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 (2)}}

\begin{document}

\listok{0}{Topologia das Variedades: July 03 exam}

{\small Each student gets a list of 6 problems chosen randomly.
Regular ones are worth 1 point, with ``2 points'' problems where 
indicated. Please write the solutions, and be prepared to explain them 
orally using your notes.

The final mark is given as follows: A+ for score of 5,
A for 4, B for 3, C for 2, and modified (increased or 
decreased) depending on results of the oral examination
(with additional questions if required).
}

%\subsection{Homological algebra}
%
%\definition
%Let $(F^i, d_F) \stackrel {\psi_i}\arrow (G^i, d_G)$
%be a morphism of complexes. {\bf \blue The cone} $C(\psi)$
%is a complex $F^{i+1}\oplus G^{i}$, with differential given
%by $d_F+d_G + (-1)^{i} \psi_{i+1}$.
%\ed
%
%\exercise
%Let $(F^i, d_F) \stackrel {\psi_i}\arrow (G^i, d_G)$ be a morphism
%of complexes. Prove that the complex $C(\psi)$ has trivial cohomology if and only if $\psi$ induces 
%an isomorphism on cohomology.
%\ez
%
%\exercise
%Let $(F^i, d_F) \stackrel {\psi_i}\arrow (G^i, d_G)$ be an injective morphism
%of complexes. Prove that $C(\psi)$ has the same cohomology as 
%\ez


\section{Homology and cohomology}


\exercise
Let $K$ be a Klein bottle, that is, a manifold
obtained by gluing two ends of a cylinder with opposite orientation. 
Find $H^*(K, \Z)$.
\ez

\exercise[2 points]
Let $M$ be a compact, smooth $n$-manifold, and
$f:\; M \arrow S^n$ a smooth map of degree 0. Prove that
$f$ is homotopic to a projection to a point.
\ez

\exercise
Prove that any homology class in $H_2(S^2 \times S^2, \Z)$ can be represented 
by a smooth submanifold.
\ez

\exercise
Prove that any homology class in $H_2(\C P^2, \Z)$ can be represented 
by a smooth submanifold.
\ez

\exercise
Find an automorphism $A$ of an n-torus $T^n$
such that any smooth curve fixed by $A$ is
homologous to 0.
\ez

\exercise
Let $X$ be a connected CW-space, and $SX$ its suspension. 
Prove that $H_i(X) = H_{i+1}(SX)$ for all $i>0$.
\ez

\exercise
Compute cohomology groups of the space $S^1 \vee S^1 \vee S^1$
and of its universal cover.
\ez


\exercise
Compute cohomology of $S^1\times (S^1 \vee S^2)$.
\ez

\exercise
Show that the map $S^1 \times S^1 \arrow (S^1 \times S^1 /S^1 \vee S^1) \cong S^2$
is not homotopic to a constant map.
\ez



\section{De Rham cohomology}

\exercise
Prove that any continuous map from $S^{2n}$ to $\C P^n$  has degree 0.
\ez

\exercise
Prove that any continuous map from $S^2 \times S^2$ to $\C P^2$  has degree 0.
\ez

\exercise
Prove that any continuous map from $\C P^n$  to a compact
Lie group of dimension $2n$ has degree 0.
\ez

\exercise
Prove that any continuous map from  a compact, simply connected 
Lie group of dimension $2n$ to $\C P^n$ has degree 0.
\ez


\exercise
Prove that any continuous map from a sphere $S^{2n}$ to a compact, simply connected 
Lie group of dimension $2n$ has degree 0.
\ez

\exercise[2 points]
Let $X, Y \subset \C P^2$ be complex algebraic curves,
that is, submanifolds given by a homogeneous polynomial equations.
Assume that $X$ and $Y$ are non-empty; prove that $X\cap Y\neq \emptyset$.
\ez

\exercise[2 points]
Let $\omega\in \Lambda^2(V)$, $V=\R^{2n}$ be a non-degenerate 2-form.
Prove that multiplication by $\omega$ induces an isomorphism
$\Lambda^{n-1}(V) \stackrel{\wedge \omega} \arrow \Lambda^{n+1}(V)$.
\ez

\exercise
Let $\omega\in \Lambda^2(V)$, $V=\R^{2n}$ be a non-degenerate 2-form.
Prove that multiplication by $\omega^{n-1}$ induces an isomorphism
$\Lambda^{1}(V) \stackrel{\wedge \omega^{n-1}} \arrow \Lambda^{2n-1}(V)$.
\ez

\exercise
Let $A$ be an automorphism of $\C P^3$ preserving the orientation.
Prove that $A$ acts as identity on its de Rham cohomology.
\ez


\section{Homotopy groups}

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

\exercise[2 points]
Calculate $\pi_3(S^3 \vee \R P^4)$.
\ez

%\exercise[2 points]
%Prove that $\pi_1(\R P^2 \vee \R P^2)$ is infinite.
%\ez


\exercise[2 points]
Prove that $\pi_1(\R P^2 \vee \R P^2)$ is non-commutative.
\ez

\exercise[2 points]
Let $L_1, L_2$ be non-intersecting lines in $\R^3$,
and $M:= \R^3\backslash (L_1 \cap L_2)$ the complement.
Prove that $\pi_2(M)=0$.
\ez


\exercise
Let $M\subset \R^3$ be the set of all points with at least two 
of three coordinates irrational. Prove that it is connected.
\ez

\exercise
Let $M= \R^2 \backslash \{0, 1\}$. Probe that $\pi_2(M)=0$.
\ez

\exercise
Find a compact 3-dimensional manifold $M$
such that $\pi_3(M)=0$.
\ez


\exercise[2 points]
Construct a compact manifold $M$ such that $\pi_1(M)$
is a free group ${\Bbb F}_2$.
\ez

\exercise
Find all $n$ such that there exists a
surjective map $S^n \arrow S^n$ of degree 0.
\ez


\end{document}
