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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due April 26 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

\item Define the triangle $T_{ijk}\subset\R^n$ by 
  \[
  T_{ijk}=\{(x_1,\dots,x_n)\mid x_i+x_j+x_k=1,\;x_i,x_j,x_k\geq
  0,\;x_\ell=0\mbox{ for }\ell\neq i,j,k\}\,.
  \]
  In this problem, we prove that in $\R^n$, the intersections of the
  $T_{ijk}$ are precisely given by the set intersections of their indices. 
  \begin{enumerate}
  \item Prove that $T_{123}\cap T_{456}=\emptyset$.
  \item Prove that $T_{123}\cap T_{345}$ is a common vertex of the two
    triangles.
  \item Prove that $T_{123}\cap T_{234}$ is a common edge of the two trianges. 
  \end{enumerate}


\item Let $S_1$ and $S_2$ be surfaces. Use the cut-and-paste
  description of the connected sum to prove that
  $\chi(S_1\#S_2)=\chi(S_1)+\chi(S_3)-2$.


\item Recall that the fundamental moves on polygon complexes are
  adding/removing midpoints, adding/removing dividing edges in
  polygons, and adding/removing hanging edges.
  \begin{enumerate}
  \item Prove that an $n$-gon may be subdivided into triangles using a
    sequence of fundamental moves.
  \item Prove that a triangle may be barycentrically subdivided using
    a sequence of fundamental moves.
  \end{enumerate}


\item An \emph{abstract regular polyhedron} is a way of constructing
  the $2$-sphere, $S^2$, as a polygonal complex such that each
  polygonal has $n$ sides each vertex is incident to $m$ polygons. Use
  the combinatorial Gauss-Bonnet theorem to classify the regular
  polyhedra, with proof.


\item A \emph{trivalent polyhedron} is a polygon complex homeomorphic
  to $S^2$ such that exactly three polygons meet at any vertex. Prove
  that there are only finitely many trivalent polyhedra that can be
  made from regular triangles, squares, and pentagons.


\item A triangle is \emph{commensurable} if its three angles are of the
  form $\pi/n_i$, where $n_i\in\N$. 
  \begin{enumerate}
  \item Find (with proof) the area of the smallest possible
    commensurable hyperbolic triangle.
  \item Let $S$ be a surface of genus $g\geq 2$. Prove that $S$ cannot
    be smoothly triangulated by either spherical or Euclidean
    triangles.
  \item Prove that a smooth triangulation of a surface $S$ of genus $g\geq
    2$ by commensurable triangles contains at most $168(g-1)$
    triangles.
  \end{enumerate}


\end{enumerate}

\end{document}
