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\def\classname{Math 6210 --- Fall 2012}

\hw{6}
\due{Choose $5$ problems to submit by Fri., Nov.\ 30.  Remember to cite your soources.}

\section{Fundamental groups of graphs and an application to group theory}

\begin{exercise}
\begin{enumerate}[label=(\alph{*})]
\item Let $X = S^1 \vee \cdots \vee S^1$ be a finite wedge of~$n$ circles.  Demonstrate that $\pi_1(X, \ast)$ is a free group generated by the maps $S^1 \rightarrow X$ corresponding to the inclusions of the~$n$ circles.
\item Suppose that $X$ is a wedge of $n$ circles and $Y$ is a wedge of $m$ circles.  Show that $X \simeq Y$ if and only if $n = m$.
\end{enumerate}
\end{exercise}

\begin{exercise}
Suppose that $X_1 \subset X_2 \subset \cdots$ is an increasing union of subspaces of a topological space $X$.  Assume that all $X_i$ contain the basepoint $x$ of $X$.
\begin{enumerate}[label=(\alph{*})]
\item Construct a homomorphism $\varinjlim \pi_n(X_i, x) \rightarrow \pi_n(X,x)$ for each $n$.
\item Assume that the $X_i$ are all open in $X$ and their union is $X$.\marg{originally forgot to include the assumption that the union was $X$}  Prove that the map constructed in the last section is an isomorphism.
\item Let $X = \bigvee_{\alpha \in A} S^1$ be a wedge of circles indexed by a set $A$.  Prove that $\pi_1(X, \ast)$ is a free group generated by the maps $S^1 \rightarrow X$ corresponding to the elements of $A$.
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $G$ be a graph.  We allow $G$ to have edges connecting vertices to themselves.  Let $V$ be the set of vertices of $G$ and $E$ the set of edges.  Define a topological space $X(G)$:  Begin with the set $V$ as a discrete topological space.  For each edge $e$ in $E$, adjoin to $V$ a copy of $[0,1]$ connecting the two vertices to which $e$ is incident.
\begin{enumerate}[label=(\alph{*})]
\item Prove that $X(G)$ is homotopy equivalent to a wedge of circles.
\item Conclude that $\pi_1(X(G),v)$ is a free group for any graph $G$ and any vertex $v$ of $G$.
\item Show that any covering space of $X(G)$ is homeomorphic to $X(G')$ for some graph $G'$.
\item Conclude that any subgroup of a free group is a free group.
\end{enumerate}
\end{exercise}

\section{Eilenberg--Mac Lane spaces}

A based topological space $(X, x)$ is said to be a $K(G,1)$ if $X$ is path connected, $\pi_1(X,x) \cong G$, and $\pi_n(X,x) = 0$ for all $n \geq 2$.

\begin{exercise}
Let $p : E \rightarrow B$ be a covering space and $b$ a point of $B$.  Prove that the map 
\begin{equation*}
p_\ast : \pi_n(E, e) \rightarrow \pi_n(B,b)
\end{equation*}
is an isomorphism for all $n \geq 2$.
\end{exercise}

\begin{exercise}
\begin{enumerate}[label=(\alph{*})]
\item Prove that $S^1$ is a $K(\bZ, 1)$.
\item Prove that if $X_i$ is a $K(G_i, 1)$ for all $i$ in some set $I$ then $\prod_{i \in I} X_i$ is a $K(\prod_{i \in I} G_i, 1)$.
\item Conclude that the $n$-torus is a $K(\bZ^n, 1)$.
\end{enumerate}
\end{exercise}

\begin{exercise}
Let $S^\infty$ be the set of sequences $x_1, x_2, \ldots$ of real numbers such that
\begin{enumerate}[label=(\roman{*})]
\item all but finitely many $x_i$ are zero, and
\item $\sum x_i^2 = 1$.
\end{enumerate}
Let $\RP^\infty = S^\infty / \set{\pm 1}$.
\begin{enumerate}[label=(\alph{*})]
\item Prove that $S^\infty$ is contractible.
\item Prove that $\RP^\infty$ is a $K(\bZ/2\bZ,1)$.
\end{enumerate}
\end{exercise}

\section{Siefert--van Kampen}

\begin{exercise}
Let $(X,\ast)$ and $(Y,\ast)$ be locally contractible topological spaces.  Prove that $\pi_1(X \vee Y, \ast) \cong \pi_1(X) \ast \pi_1(Y)$.
\end{exercise}

\begin{exercise} \cite[\S1.2, \# 10]{Hatcher}
Let $X$ be the cube $[-2,2] \times [-2,2] \times [-2,2]$.  Let\marg{correction in the equations below; thanks Megan}
\begin{gather*}
A = X \cap \set{(x,y,0) \: \Big| \: \frac{1}{9} (x + 2)^2 + \frac{1}{4} y^2 = 1 } \\
B = X \cap \set{(x,0,z) \: \Big| \: \frac{1}{9} (x - 2)^2 + \frac{1}{4} z^2 = 1 } \\
C = \set{(x,y,z) \: \Big| y^2 + z^2 = 4 } \subset X
\end{gather*}
Let $Y = X \smallsetminus (A \cup B)$.
\begin{enumerate}[label=(\alph{*})]
\item Compute $\pi_1(Y)$.
\item Show that $C$ represents a non-zero element of $\pi_1(Y)$.
\end{enumerate} 
\end{exercise}

\section{Covering spaces}

\begin{exercise}
Let $p : E \rightarrow B$ and $p' : E' \rightarrow B$ be two covering spaces.  Assume that $B$ is path connected and locally contractible and let $b$ be a basepoint of $B$.  Let $S$ and $S'$ be the two $\pi_1(B,b)$-sets corresponding to $E$ and $E'$.  We can form a new $\pi_1(B,b)$-set by taking the product $S \times S'$.  Prove that the corresponding covering space of $B$ is $E \mathop{\times}_B E'$.
\end{exercise}

\begin{exercise} \cite[\S1.3, \#14]{Hatcher}
Describe all connected covering spaces of $\RP^2 \vee \RP^2$.
\end{exercise}

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