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\newtheorem{exercise}[]{Exercise}
\newtheorem{bonus}{Bonus}[exercise]

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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 3}
\rhead{Due date: November 4, 2016}
\cfoot{}

\begin{exercise}\mbox{}
Prove that, using the definition of submanifold given in class, any submanifold $L\subset M$ inherits a smooth structure from $M$. Also show that using this smooth structure, the inclusion map $i:L\to M$ is an embedding. 
\end{exercise}

\begin{exercise}\mbox{}
Prove Proposition 3.8 in the notes: \\

\noindent {\bf Proposition 3.8}
Let $K$ be a manifold with boundary where $L,M$ are without
boundary.  Assume that $f:K\lra M$ and $g:L\lra M$ are smooth maps
such that both $f$ and $\del f$ are transverse to $g$.  Then the
fiber product $K\times_M L$ is a manifold with boundary equal to
$\del K\times_M L$.

\end{exercise}

\begin{figure}[h!]
\centering
\includegraphics{planar-arm.jpg}
\end{figure}

\begin{exercise}\mbox{}
Consider an planar arm, consisting of two rigid unit lengths joined with a hinge at endpoints.  Assume that the shoulder is fixed at the origin, as shown above.
\begin{enumerate}
\item What is the space of configurations $X$ of this planar arm?
\item Suppose we fix the extremity $p\in\RR^{2}$ of the arm. What is the configuration space then? How does this depend on the particular placement of $p$?
\item Consider the map $\pi:X\to \RR^{2}$ given by $p$, the position of the extremity.  Prove that it is a smooth map and determine the critical points and critical values of this map. Describe the manifold $\pi^{-1}(p)$ for $p$ lying in each connected component of the set of regular values.

\item Now let $X$ be the configuration space for a planar arm made of three segments rather than the above two; repeat the analysis in question 3.
\item Finally, let $X$ be the configuration space for a planar arm made of four segments; repeat the analysis in question 3.  
\end{enumerate} 

\end{exercise}



\begin{bonus}\mbox{}
  Let $K,L$ be submanifolds of a manifold $M$, and suppose that their
  intersection $K\cap L$ is also a submanifold.  Then $K,L$ are said
  to have \emph{clean} intersection when, for each $p\in K\cap L$, we
  have $T_p(K\cap L) = T_pK\cap T_pL$.  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.  It
  is useful to use the algebraic geometry notation $V(x^1,\ldots,
  x^k)$ to mean the ``vanishing'' subspace $x^1 = \cdots = x^k = 0$.

%  Also, can the intersection of submanifolds be transverse but not
%  clean? Can it be clean but not transverse? Give examples or proofs
%  as necessary.
\end{bonus}

\end{document}
