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\begin{document}
\pagestyle{fancy} \lhead{Differential Topology, Assignment 5}
\rhead{Due date: Monday Dec 9, 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*{Degree}
% Let $M$ be a compact $n$-manifold and $f:M\to N$ a smooth map to an
% $n$-manifold $N$. The degree mod 2, $\deg_2(f)\in\ZZ/2\ZZ$ is defined as the
% cardinality of $f^{-1}(y)$ modulo $2$, for $y$ any regular value of
% $f$.  We first show that $\deg_2(f)$ is independent of $y$ and that it
% depends only on the homotopy class of $f$.



% \begin{exercise}
%   Let $X$ be a manifold with boundary and let $F:X\to N$ be a smooth
%   map.  Now choose $q\in N$ to be a regular value for both $F$ and
%   $\partial F = F|_{\partial X}$. Prove that $F^{-1}(q)$ is a
%   submanifold with boundary, and that this boundary is given by
%   $F^{-1}(q)\cap \partial X$.  Note that a submanifold with boundary
%   is defined to be locally modeled on the inclusion of half-spaces
%   $H^k\subset H^n$ given by
%   \[
%   (x_1,\ldots, x_k)\mapsto (0,\ldots,0,x_1,\ldots, x_k),
%   \]
%   where $x_k\geq 0$. To illustrate this result, provide one explicit
%   example of the situation in question.
% \end{exercise}

% \begin{exercise}
%   Let $g$ be a map smoothly homotopic to $f$, meaning that there
%   exists a smooth map
%   \begin{equation}
%     \label{eq:1}
%     F:[0,1]\times M\to N
%   \end{equation}
%   with $H(0,-)=f$ and $H(1,-)=g$. Choose a value $y\in N$ which is
%   regular for both $f$, $g$.  By studying the preimage $F^{-1}(y)$,
%   conclude that $\#f^{-1}(y)=\#g^{-1}(y)\pmod 2$. [First prove under the
%   assumption that $y$ is regular for $F$]
% \end{exercise}

% \begin{exercise}
%   Let $\gamma:[0,1]\to N$ be a smooth path with $\gamma(0)=y_0$ and
%   $\gamma(1)=y_1$.  Prove there exists a vector field $V$ on $N$ whose
%   flow $\Phi_t:N\to N$ is a diffeomorphism which moves $y_0$ to $y_t =
%   \gamma(t)$, for all $t\in [0,1]$.
% \end{exercise}

% \begin{exercise}
%   Any two regular values $y_0,y_1\in N$ of $f$ may be joined by a smooth path
%   $\gamma$. Use the results above to show that $\#f^{-1}(y_0)=\#f^{-1}(y_1)\pmod 2$.
% \end{exercise}

% % \vspace{2em}
% % \noindent An orientation on an n-dimensional real vector space $V$ is
% % an equivalence class of nonzero elements in $\wedge^n V$, where $v,v'$
% % are equivalent if $v'=\lambda v$ with $\lambda\in\RR_{> 0}$.  An
% % $n$-manifold $M$ is \emph{orientable} if $\wedge^n TM$ is isomorphic to the trivial
% % line bundle

% \pagebreak


\section*{Kirby Calculus}

\begin{exercise}
  Show that the blackboard framing of a knot depends on the way the
  knot is drawn on the blackboard -- i.e. show that it is not an
  isotopy invariant.
\end{exercise}

\begin{exercise}
  Prove that 1-surgery applied to $S^3$ along the unknot with the canonical framing yields $S^1\times S^2$.  
\end{exercise}
\begin{exercise}
  Prove that 1-surgery applied to $S^3$ along the unknot with either
  framing $+1$ or $-1$ gives back $S^3$.  Think about how you could
  define a diffeomorphism between two manifolds, each of which is
  described as a gluing of pieces.
\end{exercise}

\begin{exercise}
  Show how to use Kirby 1 and Kirby 2 to prove that the following link
  surgeries give equivalent 3-manifolds.  The vertical curve
  represents a part of a link component which continues outside our
  field of view.
\begin{center}
 \includegraphics[scale=0.35]{kirby1}
\end{center}
\end{exercise}
\begin{exercise}
  Show how to use Kirby 1 and Kirby 2 to prove that the following link
  surgeries give equivalent 3-manifolds.  In this case, we have two vertical 
curves -- we do not specify the framing of these since it does not change.
\begin{center}
 \includegraphics[scale=0.35]{kirby2}
\end{center}
\end{exercise}






\end{document}
