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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
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\begin{document}
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\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}