\documentclass[letterpaper, 12pt]{article}
\usepackage[left=2cm,top=3cm,right=2cm]{geometry}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage[all]{xy}
\usepackage{fancyhdr}
\usepackage{enumerate}
\theoremstyle{definition}
\newtheorem{exercise}[]{Exercise}
\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 24, 2012}
\cfoot{}

\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}
  Show that a compact manifold cannot have a smooth function without
  critical points.
\end{exercise}

\begin{exercise}\mbox{}
  \begin{enumerate}
  \item Prove that the group $SO(3,\RR)$ of $3\times 3$ real special
    orthogonal matrices, i.e. $SO(3,\RR)=\{T\in SL(3,\RR)\ :\ TT ^\top
    = 1\}$ is a smooth submanifold of the vector space of $3\times 3$
    matrices.
  \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.
  \item Show that the intersection of the sphere $|z_1|^2 + |z_2|^2 +
    |z_3|^2 = 1$ in $\CC^3$ with the complex cone $z_1^2 + z_2^2 +
    z_3^2=0$ is a submanifold.
  \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.
  \end{enumerate}
\end{exercise}
\begin{bonus}
Show that $\RR P^3$ is null-cobordant.
\end{bonus}  
\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}
