\chapter{Covering Spaces: Layering It On}

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$. If $x,y \in X$, then $|p^{-1}(x)|=|p^{-1}(y)|$.  
\end{thm}

\oneperpage

\begin{exercise}{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\begin{enumerate}
\item Describe two non-homeomorphic 2-fold covers of the Klein bottle.
\item Describe all non-homeomorphic 2-fold covers of the figure eight.
\item Describe all non-homeomorphic 3-fold covers of the figure eight.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{thm}
Let $(\R^1,\omega)$ be the standard wrapping map covering of
$\Sph^1$.  Then any path $f:[0,1]\to\Sph^1$ has a lift $\wt f:[0,1] \to \R^1$.
\end{thm}

\oneperpage

\begin{thm}
If $(\wt X,p)$ is a cover of $X$, $Y$ is connected, and $\wt f$, $\wt g:Y\to
\wt X$ are continuous functions such that $p\circ \wt f= p\circ \wt g$, then
$\{ y\mid \wt f(y) = \wt g(y)\}$ is empty or all of $Y$.
\end{thm}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a cover of $X$ and let $f$ be a path in $X$. Then
for each $x_0\in \wt X$ such that $p(x_0) = f(0)$, there exists a
unique lift $\wt f$ of $f$ satisfying $\wt f(0) = x_0$.
\end{thm}

\oneperpage

\begin{exercise}
Let $p$ be a $k$-fold covering of $\Sph^1$ by itself and $\alpha$ a
loop in $\Sph^1$ which when lifted to $\R^1$ by the standard lift
satisfies $\wt{\alpha}(0)=0$ and $\wt{\alpha}(1)=n$. What are the
conditions on $n$ under which $\alpha$ will lift to a loop?
\end{exercise}

\oneperpage

\begin{thm}[Homotopy Lifting Lemma]
\index{Homotopy Lifting Lemma} Let $(\wt X,p)$ be a cover of $X$ and
$\alpha$, $\beta$ be two paths in $X$. If $\wt \alpha$, $\wt \beta$
are lifts of $\alpha$, $\beta$ satisfying $\wt\alpha (0) =\wt\beta
(0)$, then $\wt \alpha\sim\wt\beta$ if and only if
$\alpha\sim\beta$.
\end{thm}

\oneperpage

\begin{thm}
If $(\wt X,p)$ is a cover of $X$, then $p_*$ is a monomorphism (i.e.
an injective homomorphism) from $\pi_1(\wt X)$ into $\pi_1(X)$.
\end{thm}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a cover of $X$, $\alpha$ a loop in $X$, and $\wt
x_0 \in\wt X$ such that $p(\wt x_0) =\alpha (0)$. Then $\alpha$
lifts to a loop based at $\wt x_0$ if and only if $[\alpha]\in p_*
(\pi_1(\wt X,\wt {x}_0))$.
\end{thm}

\oneperpage

\begin{exercise}
Recast a proof of the fact that $\pi_1(\Sph^1)\cong \Z$ using
the language of covering spaces.
\end{exercise}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$ and let $x_0\in X$. Fix
$\wt x_0\in p^{-1}(x_0)$. Then a subgroup $H$ of $\pi_1(X,x_0)$ is
in $\{ p_* (\pi_1(\wt X,\wt x))\}_{p(\wt x)=x_0}$ if and only if $H$
is a conjugate of $p_* (\pi_1(\wt X,\wt x_0))$.
\end{thm}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$. Choose $x\in X$, then
$|p^{-1}(x)| = [ \pi_1(X): p_* (\pi_1(\wt{X}))]$, where the equation
has the obvious interpretation if either side is infinite.
\end{thm}

\oneperpage

\begin{exercise} Give a covering space of $\Sph^1$ that corresponds
to a subgroup of index 3.  If $p$ is the covering map, describe
$p_*$.
\end{exercise}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a covering space of $X$ and $\wt x_0\in\wt X$,
$x_0\in X$ with $p(\wt x_0)=x_0$. Also let $f: Y\to X$ be continuous
where $Y$ is connected and locally path connected and $y_0 \in Y$
such that $f(y_0) =x_0$. Then there is a lift $\wt f:Y\to \wt X$
such that $p\circ \wt f=f$ and $f(y_0) = \wt x_0$ if and only if
$f_* (\pi_1(Y,y_0))\subseteq p_* (\pi_1(\wt X,\wt x_0))$.
Furthermore, $\wt f$ is unique.

\end{thm}

\oneperpage

\begin{exercise}
Let $X= \Sph^1$, $\wt X = \R$, $(\wt X,\omega)$ be the covering
space of $X$ given by the standard wrapping map, and $Y$ as in
Figure \ref{counterexample}. When does a map $f:Y\to X$ not have a
lift? Why is this example here?
\end{exercise}

\oneperpage

\begin{exercise}
Show that $\pi_2 (\T^2) = 0$, i.e., every map of a sphere $\Sph^{2}$ into $\T^{2}$ is null homotopic.
\end{exercise}

\oneperpage

\begin{thm}
Let $(\wt X_1,p_1)$ and $(\wt X_2,p_2)$ be covering spaces of $X$.
Let $\wt x_1\in \wt X_1$ and $\wt x_2\in \wt X_2$ such that $
p_1(\wt x_1) = p_2(\wt x_2)$. Then there is a cover isomorphism
$f:\wt X_1\to \wt X_2$ with $f(\wt x_1)= \wt x_2$ if and only if
$p_* (\pi_1 (\wt X_1,\wt x_1)) = p_* (\pi_1(\wt X_2,\wt x_2))$.
\end{thm}

\oneperpage

\begin{exercise}\label{figeightex}
What is ${\mathcal{C}}(\wt X,p)$ for the covering space of the
figure eight shown in Figure \ref{4foldcw2c}?

\begin{figure}
\begin{center}
%\includegraphics[width = 2 in]{Figures/4foldcw2c}
\caption{\label{4foldcw2c} A covering of the figure eight.}
\end{center}

\end{figure}

\end{exercise}

\oneperpage

\begin{thm}
If $(\wt X,p)$ is a covering space of $X$ and $f\in
{\mathcal{C}}(\wt X,p)$, then $f= Id_{\wt X     }$ if and only if
$f$ has a fixed point.
\end{thm}

\oneperpage

\begin{exercise}

Consider the second three-fold covering space of the figure eight
discussed in Exercise \ref{figeightex}. Find an element of $p_*(\pi
_1(\wt X))$ which, when conjugated, is not in $p_*(\pi _1(\wt X))$.
Conclude that the covering space is not regular.
\end{exercise}

\oneperpage

\begin{thm}
If $(\wt X,p)$ is a regular covering space of $X$ and $x_1,x_2\in
\wt X$ such that $p(x_1) = p(x_2)$, then there exists a unique $h\in
{\mathcal{C}}(\wt X,p)$ such that $h(x_1)= x_2$.
\end{thm}

\oneperpage

\begin{exercise}
Do such covering transformations necessarily exist in irregular covering spaces?
\end{exercise}

\oneperpage

\begin{thm}
A covering space is regular if and only if for every loop in the base space either all
its lifts are loops or all its lifts are paths that are not loops.
\end{thm}

\oneperpage

\begin{exercise} Find a covering space $p:{\wt X}\to
X$ and generators $e_1,\ldots,e_n$ of $\pi_1(X)$ such that each
$e_i$ satisfies the criteria of the previous theorem but the cover
is not regular.\end{exercise}

\oneperpage

\begin{exercise}
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm} \be
\item Describe all regular 3-fold covering spaces of the figure eight.
\item Describe all irregular 3-fold covering spaces of the figure eight.
%\item Describe all regular 4-fold covering spaces of the figure eight.
%\item Describe all irregular 4-fold covering spaces of the figure eight.
\item Describe all regular 3-fold covering spaces of the bouquet of 3 circles.
%\item Describe all regular 4-fold covering spaces of the bouquet of 3 circles.
\ee
\end{exercise}

\oneperpage

\begin{thm}
Let $(\wt X,p)$ be a regular covering space of $X$. Then
${\mathcal{C}}(\wt X,p) \cong \pi_1 (X)/p_* (\pi_1(\wt X))$. In
particular, ${\mathcal{C}}(\wt X,p) \cong \pi_1 (X)$ if $\wt{X}$ is
simply connected.
\end{thm}

\oneperpage

\begin{exercise}
Observe that the standard wrapping map is a regular covering map of
$\Sph^1$ by $\R^1$.  Describe the covering transformations for this
covering space. Describe the covering map that maps $\R^2$ to the
torus $\T^2$ and describe the covering transformations for this
covering space.
\end{exercise}

\oneperpage

\begin{thm}[Existence of covering spaces]
Let $X$ be connected, locally path connected, and semi-locally
simply connected. Then for every $G< \pi_1 (X,x_0)$ there is a
covering space $(\wt X,p)$ of $X$ and $\wt x_0\in\wt X$ such that
$p_* (\pi_1(\wt X,\wt x_0)) =G$. Furthermore, $(\wt X,p)$ is unique
up to isomorphism.
\end{thm}

\oneperpage

\begin{cor} Let $X$ be connected, locally path connected, and semi-locally
simply connected. Then there is a one-to-one correspondence between
the subgroups of $\pi_1(X)$ and the collection of isomorphism
classes of covering spaces of $X$ where the covering space
$p:\tilde{X}\to X$ corresponds to $p_*(\pi_1({\wt X}))$.  
%Hence, the
%degree of a covering space is equal to the index of the
%corresponding subgroup and normal subgroups correspond to regular
%covers.
\end{cor}

\oneperpage

\begin{cor}
Every connected, locally path connected, semi-locally simply
connected space admits a unique universal covering space.
\end{cor}

\oneperpage

\begin{exercise}
Find a universal cover ${\wt X}$ for each of the Klein bottle, the
torus, and the projective plane.  In each case,  show explicitly
that ${\mathcal{C}}(\wt X,p) \cong \pi_1 (X)$.
\end{exercise}

\oneperpage

\begin{thm} A finite tree is contractible.\end{thm}

\oneperpage

\begin{thm} Let $G$ be a finite graph, and $T$ be a maximal tree in $G$.
Then if $\{e_1,\ldots e_n\}$ is the set of edges that are not in
$T$, $\pi_1(G)=F_n$, the free group on $n$ generators; and there is
a system of generators that are in one-to-one correspondence with
the edges $\{e_1,\ldots e_n \}$.
\end{thm}

\oneperpage

\begin{lem} Let $X$ be the bouquet of $n$ circles.  Every
finite cover of $X$ is homeomorphic to a finite graph.\end{lem}

\oneperpage

\begin{thm} Let $F_n$ be the free group on $n$ letters.  Then every
subgroup of $F_n$ of finite index is isomorphic to a free group on
finitely many letters.\end{thm}

\oneperpage

\begin{lem} Suppose that $G$ is a graph and that $K\subset G$ is
compact.  Then $K$ is contained in a finite subgraph of
$G$.\end{lem}

\oneperpage

\begin{thm} Every tree is simply connected.\end{thm}

\oneperpage

\begin{thm} Let $G$ be a graph, then $\pi_1(G)$ is free.
\end{thm}

\oneperpage

\begin{exercise} Show that the free group of rank {2} has finite index subgroups that are isomorphic to free groups of arbitrarily large rank.
\end{exercise}

\oneperpage

\begin{lem} Let $X$ be the bouquet of $n$ circles.  Then every cover
of $X$ is homeomorphic to a graph.\end{lem}

\oneperpage

\begin{cor}[Nielsen-Schreier Theorem]
Let $F_n$ be the free group on $n$ letters.  A subgroup of $F_n$ is
always free.
\end{cor}

\oneperpage

\begin{exercise}
Describe a regular $k$-fold cover $\wt X$ of a bouquet of
$n$-circles. What (in terms of $k$ and $n$) is the rank of the free
group $\pi_1(\wt{X})$? What does this insight tell us about the normal
subgroups of finite index of the free group on $n$ letters?
\end{exercise}

\oneperpage

\begin{exercise}{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\begin{enumerate}
\item Let $F$ be a free group on $n$ letters. Let $G< F$ be of finite index $k$
and contain $7$ free generators. What can the value of $n$ be?
\item Let $F$ be a free group on $n$ letters. Let $G< F$ be of finite index $k$
and contain $4$ free generators. What can the value of $n$ be?
\item Let $F$ be a free group on $n$ letters. Let $G<F$ be of finite index $k$
and contain $24$ free generators. What can the value of $n$ be?
\end{enumerate}\end{exercise}

\oneperpage

\begin{thm}
Let $F$ be a 2-manifold and $(\wt{F}, p)$ be a
covering space of $F$. Then $\wt{F}$ is a 2-manifold. 
\end{thm}

\oneperpage

\begin{thm}
Let $F$ be a compact connected surface and $p_n:\wt{F}\to F$ be an
$n$-fold covering of $F$ (for $n<\infty$). Then $\wt{F}$ is a
compact surface and $\chi(\wt{F})=n\chi(F)$.  Moreover, if $F$ is
orientable, then $\wt{F}$ is as well.
\end{thm}

\oneperpage

\begin{exercise}{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\begin{enumerate}
\item Describe all non-homeomorphic 3-fold covers of the Klein bottle.
\item Describe all non-homeomorphic 2-fold covers of $\T^2\num\T^2$.
\item Describe all non-homeomorphic 3-fold covers of $\T^2\num\T^2\num\T^2$.
\item Describe all non-homeomorphic 3-fold covers of $\PP^2$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
Given a compact, connected 2-manifold and a natural number $n$, describe all non-homeomorphic $n$-fold covers of that surface.
\end{exercise}

\oneperpage

