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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{4}{Topologia das Variedades: class exercise 4}

\definition
Unitary group $U(n)$ is the group of complex isometries
of a complex Hermitian space $\C^n$. {\bf Special unitary}
group $SU(n)$ is all $g\in U(n)$ with determinant 1.
\ed

\exercise
Find a cell decomposition
for a Grassmann manifold $\frac{U(4)}{U(2)\times U(2)}$
of all 2-dimensional complex subspaces in $\C^4$
such that all cells are even-dimensional.
\ez

\exercise
Prove that a standard embedding $\C P^{n-k} \hookrightarrow \C P^n$
induces a surjective map on cohomology.
\ez

\exercise
Prove that $SU(2)$ acts on a unit sphere $S^3\subset \C^2$ freely and transitively.
\ez

\remark This implies that $SU(2)=S^3$.
\er

\exercise
Prove that $SU(n)$ is fibered over $S^{2n-1}$ with fiber $SU(n-1)$.
\ez


\exercise
Let $U({\Bbb H}, 1)$ be a group of unitary quaternions.
Prove that $U({\Bbb H}, 1)$ is isomorphic to $SU(2)$.
\ez

\exercise
Construct a double cover $SU(2) \arrow SO(3)$.
\ez


\exercise
Construct a double cover $SU(2)\times SU(2) \arrow SO(4)$.
\ez




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