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\begin{document}
\pagestyle{fancy} \lhead{Differential Topology, Assignment 3}
\rhead{Due date: Friday Nov 8, 2013} \cfoot{} 
% \begin{quote}
% \emph{Treat this
% as a take-home exam.  Don't discuss problem sets until after they are
% handed in.  If you need clarification or help, ask me after class.  }
% \end{quote}

\section*{Vector fields}

\begin{exercise}
  Consider the smooth vector field $v = x^k\tfrac{\partial}{\partial
    x}$, $k\geq 0$ on the real line $X = \RR$. The flow $\Phi_v(x,t)$
  of the vector field $v$ for time $t$ starting at $x\in X$ is defined for
  $(x,t)$ in an open subset $U\subset X\times \RR$.  Determine this
  open set precisely for each $k$.
\end{exercise}

\begin{exercise}
   Let $v$ be a vector field on the manifold $M$, and suppose it vanishes 
at the point $p\in M$.  In coordinates $(x^1,\ldots, x^n)$ centered at $p$, we may write
\[
v =  \sum_{i=1}^nv^i \frac{\partial}{\partial x^i},
\]
and the following expression defines an endomorphism of $T_p M$:
\[
d_pv = \sum_{i=1}^n (dv^i)|_p\otimes \left.\frac{\partial}{\partial x^i}\right|_p\in T^*_p M\otimes T_pM.
\]
Prove that $d_p v$ does not depend on the choice of coordinates centered at $p$.
\end{exercise}

\begin{exercise}
  Let $v$ be a vector field on $M=\RR^2$ with an isolated zero at the
  origin.  For a sufficiently small circle $\gamma(t) =
  \varepsilon e^{it}$, the normalized vector field 
  \begin{equation}\label{winding}
  \sigma(t) = \frac{v(\gamma(t))}{|v(\gamma(t))|}
  \end{equation}
  defines a map $S^1\to S^1$.  The winding number of this map is
  called the \emph{index} of the vector field at the origin.  
\begin{enumerate}
\item Provide an explicit family of vector fields $v_k$ on the plane
  with index $k$ at the origin for $k\in\ZZ$.
\item Given a continuous family $v_t$ of vector fields on $\RR^2$
  parametrized by $t\in \RR$, such that $v_t$ always has a single zero in the
  unit disc at the origin, prove that the index remains constant in
  the family. [This requires a basic understanding of what the
  fundamental group is, in particular the fact $\pi_1(S^1)=\ZZ$.]
\item Suppose that the vector field $v$ on $\RR^2$ is nonvanishing on the unit circle
  $\gamma(t)=e^{it}$, and suppose that the winding number of the map~\eqref{winding} is nonzero. Prove that $v$ must have a zero somewhere in the unit disc.
\item Use the above to prove that $S^2$ cannot have a
  nowhere-vanishing vector field. Use the description of $S^2$ and its
  tangent bundle in terms of a pair of stereographic charts.
\end{enumerate}
\end{exercise}

\pagebreak


\section*{Transversality}

Vector subspaces $U,V$ of $W$ are \emph{transverse} when $U+V=W$. Two
submanifolds $K,L$ of the manifold $M$ intersect \emph{transversally} if at 
each point $p\in K\cap L$, the tangent spaces $T_pK$ and $T_pL$ are transverse in $T_p M$.

\begin{exercise}
  Prove that if the submanifolds $K, L$ of $M$ intersect
  transversally, then $K\cap L$ is also a submanifold. Also, determine
  the dimension of the intersection.

  For each $k=0,1,\ldots$ give an example of two transversally
  intersecting submanifolds $L,K$ of $S^1\times S^1$ which intersect
  in exactly $k$ points.
\end{exercise}

% \begin{exercise}\mbox{}
%   Let $K,L$ be submanifolds of a manifold $X$, and suppose that their
%   intersection $K\cap L$ is a submanifold.  Then $K,L$ are said to
%   have \emph{clean} intersection when, for each $p\in K\cap L$, we
%   have $T_p(K\cap L) = T_pK\cap T_pL$.  
% \begin{enumerate}
% \item Can the intersection of embedded submanifolds be transverse but
%   not clean? Can it be clean but not transverse? Give examples or
%   proofs as necessary.
% \item If $K,K'$ are cleanly intersecting submanifolds of $X$, of
%   codimension $k,k'$ respectively, show that there is a neighbourhood
%   $U$ of any point $p\in K\cap K'$, and coordinates $x^1,\ldots, x^n$
%   on $U$ such that $K\cap U=\{x^1=\cdots=x^k=0\}$ and $L\cap
%   U=\{x^j=\cdots=x^{j+k'-1}=0\}$, for some $j\leq k+1$.
% \item Let $M,N$ intersect cleanly in $X$, and let $g:G\rightarrow M$ and
%   $h:H\rightarrow N$ be submersions.  Is $G\times_X H :=G {_{\iota_M\circ
%       g}}\times_{\iota_N\circ h} H$ an embedded submanifold of
%   $G\times H$? (here $\iota_M,\iota_N$ are the inclusions). Justify
%   your claim.

% \end{enumerate}
  
  

% Show that there are
%   coordinates near $p\in K\cap L$ such that $K$, $L$, and $K\cap L$
%   are given by linear subspaces of $\RR^n$ (of the form
%   $V(x^{i_1},\ldots, x^{i_k})$ for some subset of the coordinates).
% \end{exercise}

\begin{exercise}
  Sard's theorem states that for any smooth map, the set of critical
  \emph{values} has measure zero in the codomain.  In other words, the
  regular values are dense.  Recall that for a point $y$ in the
  codomain of $f$ to be regular, each point in the preimage
  $f^{-1}(y)$ must be regular, i.e. have surjective derivative.
  (Important point: if $f^{-1}(y)$ is empty, then $y$ is regular!).
  \begin{enumerate}
  \item If $f:M\to M$ is a smooth map from a compact manifold to
    itself, prove that there must be a point $y\in M$ with $f^{-1}(y)$ finite.
  \item If $f:M\to S^n$ is a smooth map and $\dim M<n$, prove that $f$
    is smoothly homotopic to a constant map.  `Smoothly homotopic' in this case would 
    mean that you have a smooth map
    \[
    F:M\times[0,1]\to S^n
    \]
    with $F(-,0) = f(-)$ and $F(-,1)$ being a constant map.
  \end{enumerate}
  \end{exercise}

\begin{exercise}
  We say that a smooth map $f:K\to M$ is transverse to the submanifold
  $L\subset M$ if $Df(T_p K) + T_{f(p)}L = T_{f(p)}M$ for all $p\in
  f^{-1}(L)$.  If $f$ were an embedding of the submanifold $K$, we
  would recover the usual notion of transversality.

  Let $S$ be another manifold (think of it as a parameter space) and
  suppose that $F:K\times S\to M$ is a smooth map which is transverse
  to $L$. We would like to know if the individual maps $F(-,s):K\to
  M$, where $s$ is fixed, are transverse to $L$.
  \begin{enumerate}
    \item Prove that $Q=F^{-1}(L)$ is a smooth submanifold of $K\times S$.
    \item Let $\pi:Q\to S$ be the projection map.  Prove that if $s$ is a regular value
      for $\pi$, then $F(-,s):K\to M$ is transverse to $L$.
     Conclude that $F(-,s):K\to M$ is transverse to $L$ for almost all $s$. 
  \end{enumerate}


\end{exercise}



\begin{exercise}
  Let $f$ be a smooth real-valued function on the compact manifold $M$
  such that $df$ is transverse to the zero section, meaning that the
  image of the section $df\in \Gamma(M, T^*M)$ in $T^*M$ defines a
  submanifold which intersects the image of the zero section transversally.
Prove that $f$ has finitely many critical points, at each of which
  its Hessian is nondegenerate.
\end{exercise}











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