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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{1}{Topologia das Variedades: class exercise 1}


\exercise
Let $T S^2 \oplus \R$ be a direct sum of
a tangent bundle $TS^2$ and a trivial 1-dimensional bundle.
Is the bundle $T S^2 \oplus \R$ trivial?
\ez

\exercise
Let $M$ be a simply connected manifold.
Prove that any real rank 1 bundle on $M$ is trivial.
\ez

\exercise
Let $M_1\stackrel \phi \arrow M$ be a surjective, smooth
map without critical points, $M$, $M_1$ compact manifolds, 
and  $B$ a non-trivial bundle on $M$.  Can the pullback 
bundle $\phi^* B$ be trivial?
\ez

\exercise
Construct a non-trivial rank 2 vector bundle which does not have
any non-trivial sub-bundles.
\ez

\exercise
Let $V=\R^{4}$, and $\alpha\in \Lambda^2 V^*$.
Assume that $\alpha \wedge \alpha \in \Lambda^4 V^*$
is non-zero. Prove that $\alpha$ 
is a symplectic  2-form.
\ez


\end{document}
