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\newtheorem{bonus}{Bonus}[exercise]




% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}

\begin{document}
\pagestyle{fancy}
\lhead{1300F Geometry and Topology, Assignment 2}
\rhead{Due date: October 21, 2016}
\cfoot{}


%\begin{exercise}
%  Let $ M = \CC^n\setminus\{0\}$ and let the generator of $\Gamma=\ZZ$
%  act via $x\mapsto 2x$, for $x\in M$.  Show that the quotient
%  $M/\Gamma$ is a smooth manifold diffeomorphic to $S^{2n-1}\times S^1$.
%\end{exercise}  

\begin{exercise}
  Construct, using the stereographic charts for $S^2$ given in class,
  a smooth vector field on $S^2$ which vanishes exactly at 2 points,
  and another vector field which vanishes at exactly 1 point.
\end{exercise}


\begin{exercise}
Consider the Hopf map $p:S^3\to S^2$ given by composing the
embedding $S^3\subset \CC^2\backslash\{0\}$ with the projection
$\pi:\CC^2\backslash\{0\}\to \CC P^1\cong S^2$.  In the first assignment, you computed this map explicitly in the given coordinate charts for $S^3$ and $S^2$.  Now compute the derivative of this map in two distinct sets of charts and verify that these derivatives are indeed glued together by the gluing data for the tangent bundles of $S^3$ and $S^2$. 
\end{exercise}


\begin{exercise}\mbox{}
  \begin{enumerate}
\item Prove that the orthogonal group $O(n)=\{X\in GL(n,\RR) :  XX^\top = 1\}$ is a smooth submanifold of $M(n,\RR)$, the $n\times n$ matrices.  To show this, consider the map $f:M(n,\RR)\to S(n,\RR)$ to the symmetric matrices $S(n,\RR)$ given by $f(X) = XX^\top$.

  \item Prove that the special orthogonal group $SO(n,\RR)$  is a smooth submanifold of $M(n,\RR)$.  
  \item Prove that the unitary and special unitary groups $U(n)$, $SU(n)$ are smooth submanifolds of $M(n,\CC)$.
  \item Prove that $SU(2)$ is diffeomorphic to $S^3$ and that $U(2)$ is diffeomorphic to $S^3\times S^1$.
\end{enumerate}
\end{exercise}
\begin{exercise}\mbox{}
\begin{enumerate}
      \item Consider the subset of $T\RR^3$ consisting of the vectors
    tangent to the 2-sphere $S^2\subset \RR^3$ and of unit length (we
    use the usual Euclidean length on $\RR^3$, and the fact that
    $T\RR^3\cong \RR^3 \times \RR^3$).  Prove this subset is a
    submanifold, and prove it is diffeomorphic to $SO(3)$.
%  \item Show that the intersection of the sphere $|z_1|^2 + |z_2|^2 +
%    |z_3|^2 = 2$ in $\CC^3$ with the complex cone $z_1^2 + z_2^2 +
%    z_3^2=0$ is a submanifold.
%   \item Prove that all the above are diffeomorphic to $\RR P^3$.
%  \item Are any of the above three manifolds diffeomorphic to each
%    other? Prove such assertions.
%  \item Are any of the above manifolds diffeomorphic to $\RR
%    P^3$? Prove your claim.
\item Deduce from the above that $SO(3)$ is null-cobordant.
  \end{enumerate}
\end{exercise}
\noindent In fact $SO(3)$ is isomorphic to $SU(2)/\{\pm 1\}$ and so is diffeomorphic to $S^3/\ZZ_2\cong \RR P^3$, so the above shows that $\RR P^3$ is null-cobordant, explaining the exclusion of $x_3$ in Thom's theorem.

%\begin{bonus}
%Show that $\RR P^3$ is null-cobordant.
%\end{bonus}  


%\begin{exercise}
%  Show that a compact manifold cannot have a smooth real-valued
%  function without critical points.  Can it have a complex-valued function without critical points? Justify your response.
%\end{exercise}

\begin{exercise}\mbox{}
  Prove that if $K$ is a submanifold\footnote{We always mean
    \emph{embedded} or \emph{regular} submanifold when we say
    submanifold.} of $L$ and $L$ is a submanifold of $M$, then $K$ is
  a submanifold of $M$.
\end{exercise}

\begin{exercise}
  Let $K, K'$ be transverse submanifolds of codimension $k, k'$ in the
  $n$-manifold $M$.  Prove that each point $p\in K\cap K'$ has a
  neighbourhood $U\subset M$ and a diffeomorphism from $U$ to a
  neighbourhood of the origin in $\RR^n$ which takes $K$ and $K'$ to
  the coordinate planes $V(x_1,\ldots, x_k)$ and $V(x_{n-k'+1},\ldots,
  x_n)$, respectively (Here $V$ denotes the common zero set of its arguments).
\end{exercise}

\begin{exercise}
Let $f:M\to M$ be a smooth map and suppose $p$ is a fixed point
of $f$, i.e. $f(p)=p$.  The point $p$ is called a \emph{Lefschetz
fixed point} when the derivative map $f_*:T_pM\to T_pM$ does not
have $+1$ as an eigenvalue.

Show that if $M$ is compact and all fixed points for $f$ are
Lefschetz, then there are only finitely many fixed points for $f$.
\end{exercise}


%\begin{exercise}\label{jay}\mbox{}
%  For any vector space $V$, we have a natural diffeomorphism $TV\cong
%  V\times V$, where the projection $\pi_V: TV\to V$ corresponds to
%  the first projection $\pi_1:V\times V\to V$ given by $(a,b)\mapsto
%  a$.
%
%  Let $M$ be a smooth manifold, and choose a chart $(U,\varphi)$ on
%  $M$.  By applying the tangent functor, and using the canonical
%  isomorphism $T\RR^n\cong \RR^n\times \RR^n$, we obtain a chart
%  $(TU,T\varphi)$ on the manifold $TM$.  Repeating this procedure, we
%  obtain a chart $(T(TU),T(T\varphi))$ on the manifold $T(TM)$.
%
%  Now define a diffeomorphism $J_U:T(TU)\to T(TU)$ by the composition
%  $(T(T\varphi))^{-1}\circ j_U\circ T(T\varphi)$, where $j_U$ is the
%  automorphism of $(\varphi(U)\times \RR^n)\times (\RR^n\times\RR^n)$
%  given by
%  \begin{equation}
%    j_U:((x,v),(u,w))\mapsto((x,u),(v,w)).
%  \end{equation}
%  \begin{enumerate}
%  \item Show that for any atlas $\{(U_i, \varphi_i)\}$, we have $J_{U_i}
%    = J_{U_j}$ on the overlap $T(T(U_i\cap U_j))$.  Deduce that this
%    defines a global diffeomorphism $J:T(TM)\to T(TM)$ and show that it
%    is independent of the atlas used to construct it.
%  \item Consider the tangent bundle projection $\pi_M: TM\to M$.
%    Applying the tangent functor, we obtain the smooth map $T\pi_M:
%    T(TM)\to TM$.  What is the relationship between this map and the
%    tangent bundle projection $\pi_{TM}: T(TM)\to TM$? 
%  \item Let $X:M\to TM$ be a vector field on $M$, and let its tangent
%    mapping be $TX:TM\to T(TM)$.  Is $TX$ a vector field on the manifold $TM$?
%  \item Is $J\circ TX$ a vector field on the manifold $TM$?
%  \end{enumerate}
%\end{exercise}
%\begin{bonus}
%  Show that $J$ defines a natural transformation from the
%  functor $T\circ T$ to itself, and that this natural transformation
%  is an equivalence.
%\end{bonus}




\end{document}
