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\title{Homework 5 --- Surfaces and fundamental groups}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

This homework is due Wednesday November 22nd at the start of class. Remember
that the notation
$$\langle e_1,e_2,\dots,e_n | w_1,w_2,\dots,w_m \rangle$$
denotes the group whose generators are equivalence classes of words
in the generators $e_i$ and $e_i^{-1}$, where two words are equivalent
if one can be obtained from the other by inserting or deleting
$e_i e_i^{-1}$ or $e_i^{-1} e_i$ or by inserting or deleting a word
$w_j$ or its inverse.

\begin{prob}
Let $S_n$ be a surface obtained from a regular $4n$--gon by identifying
opposite sides by a translation. What is the Euler characteristic of
$S_n$? What surface is it? Find a presentation for its fundamental
group.
\end{prob}

\begin{prob}
Let $S$ be the surface obtained from a hexagon by identifying opposite
faces by translation. Then the fundamental group of $S$ should be
given by the generators $a,b,c$ corresponding to the three pairs of
edges, with the relation $abca^{-1}b^{-1}c^{-1}$ coming from the
boundary of the polygonal region.

What is wrong with this argument? Write down an actual presentation for
$\pi_1(S,p)$. What surface is $S$?
\end{prob}

\begin{prob}
The {\em four--color theorem} of Appel and Haken
says that any map in the plane can be
colored with at most four distinct colors so that two regions which
share a common boundary segment have distinct colors. Find a cell
decomposition of the torus into $7$ polygons such that each two
polygons share at least one side in common.
\end{prob}

\begin{rmk}
It turns out that only $7$ colors are required for any map on the
torus. In general, for a genus $g$ surface,
the number of necessary colors is the smallest integer greater than or
equal to $\frac{7 + \sqrt{1 + 48 g}} 2$.
\end{rmk}

\begin{prob}
Define the {\em connect sum} of two surfaces $S_1,S_2$ to be a new
surface obtained by removing a small disk from each of $S_1,S_2$ and
sewing the two boundary components together. The surface so obtained
is denoted $S_1 \# S_2$. 
Show for orientable $S_1,S_2$
that this operation is well--defined. That is,
the resulting surface does not depend on which disks are removed from
$S_1,S_2$ or how the boundaries are glued. What is the formula
for $\chi(S_1 \# S_2)$ in terms of $\chi(S_1)$ and $\chi(S_2)$?
\end{prob}

\begin{prob}
If $F,G$ are two orientable surfaces, show that $F$ is a covering
space of $G$ of degree $n$ if and only if 
$$\chi(F) = n \chi(G)$$
In particular, there exist degree $n$ covers by the torus of itself
of arbitrary degree.
\end{prob}

\begin{prob}
Show that every nonorientable surface $S$ has a degree $2$ orientable cover.
Hint: show that there is a homomorphism $\pi_1(S,p) \to \Z/2\Z$ whose
kernel consists of the loops which are orientation--preserving.
\end{prob}

\begin{prob}
The {\em Klein bottle} $K$ is the surface obtained from a square by
gluing top and bottom sides by translation, and gluing left
and right sides with a ``twist''. Show that a presentation for
the fundamental group is given by
$$\pi_1(K,p) \cong \langle a,b | aba^{-1}b \rangle$$
Find another cell decomposition of $K$ with one polygon and one vertex
that gives another presentation for the fundamental group as
$$\pi_1(K,p) \cong \langle a,b | a^2b^2 \rangle$$
\end{prob}

\begin{prob}
What orientable surfaces without boundary
can be built from heptagons, where $3$ of these are
required to meet around every vertex? What surfaces can
be built from triangles, where $7$ of these are
required to meet around every vertex?

Note: a {\em heptagon} is a $7$--sided polygon.
\end{prob}

\begin{prob}
A {\em pair of pants} is another word for a disk with two holes (can
you see why?). Show that a surface of genus $g$ can be decomposed into
$2g-2$ pairs of pants. Show that the number of such decompositions,
up to combinatorial equivalence, is equal to the number of graphs
with $2g-2$ vertices, and $3$ edges at every vertex. Enumerate the
number of such graphs for $g\le 5$.
\end{prob}

\begin{prob}[hard]
Recall the notation $[a,b] = aba^{-1}b^{-1}$ for elements
$a,b$ in a group.

The {\em abelianization} of a group $G$ is an abelian
group $A$ and a homomorphism $\varphi:G \to A$ such that if 
$B$ is any abelian group, and $\phi:G \to B$ is any homomorphism,
there is a unique homomorphism $\psi:A \to B$ (which might depend on $\phi$)
such that $\psi\varphi = \phi$.

\begin{itemize}
\item{Show that if $A$ exists, then it is unique in the sense that
for any other $\varphi':G \to A'$ with the properties above,
there is an isomorphism $\rho:A' \to A$ such that
$\rho\varphi' = \varphi$.}
\item{If $G=\langle e_1,e_2,\dots,e_n|w_1,w_2,\dots,w_m \rangle$ is
a finitely presented group, show
$$A=\langle e_1,e_2,\dots,e_n|w_1,\dots,w_m,[e_1,e_2],\dots,[e_i,e_j],\dots,
[e_{n-1},e_n]\rangle$$
is a presentation of the abelianization of $G$, where the homomorphism
$\varphi:G \to A$ sends the equivalence class of $w$ in $G$ to the
equivalence class of $w$ in $A$ for each word $w \in G$.}
\item{Calculate presentations for
the abelianization of the fundamental groups of the
Klein bottle and of the genus $2$ surface. Identify these abelian
groups explicitly.}
\end{itemize}
\end{prob}
\end{document}


