\section{Fundamental group}

The \emph{fundamental group}\index{fundamental group} of a space
is the first homotopy group. In this section we describe two
basic methods how to compute it.

\begin{cislo}\label{FGcs}{\bf Covering space.} A \emph{covering
space}\index{covering space} of a space $X$ is a space $\widetilde X$
together with a map $p:\widetilde X\to X$ such that $(\widetilde X,X,p)$
is a fibre bundle with a discrete fibre.

In the previous section we have proved that every fibre bundle
has homotopy lifting property with respect to CW-complexes. In
the case of covering spaces the lifts of homotopies are unique:

\begin{prop*} Let $p:\widetilde X\to X$ be a covering space and let
$Y$ be a space. Given a homotopy $F:Y\times I\to X$ and a map
$\widetilde f:Y\times\{0\}\to \widetilde X$ such that $F(-,0)=p\widetilde f$,
there is a unique homotopy $\widetilde F:Y\times
I\to\widetilde X$ making the following diagram commutative:
$$
\xymatrix{
  Y\times\{0\}\ar[d]\ar[r]^-{\widetilde f}
& \widetilde X \ar[d]^p\\
  Y\times I \ar[r]_-{F}\ar[ru]_-{\widetilde F}
& X
}
$$
\end{prop*}

\begin{proof} Since the proof follows the same lines as the proof
of the analogous proposition in 10.5, %\ref{HOMfib} 
we outline only the main steps.
\begin{enumerate}
\item Using compactness of $I$ we show that for each $y\in Y$
there is a neighbourhood $U$ such that
$\widetilde F$ can be defined on $U\times I$.
\item $\widetilde F$ is uniquely determined on $\{y\}\times I$
for each $y\in Y$.
\item The lifts of $F$ defined on $U_1\times I$ and $U_2\times I$
concide on $(U_1\cap U_2)\times I$.
\end{enumerate}
\end{proof}

From the uniquiness of lifts of loops and their homotopies
starting at a fixed point we get immediately the following

\begin{cor*} The group homomorphism $p_*:\pi_1(\widetilde X,\widetilde
x_0)\to \pi_1(X,x_0)$ induced by a covering space $(\widetilde X,X,p)$
is injective. The image subgroup $p_*(\pi_1(\widetilde X,\widetilde x_0))$
in $\pi_1(X,x_0)$ consists of loops in $X$ based at $x_0$ whose
lifts in $\widetilde X$ starting at $\widetilde x_0$ are loops.
\end{cor*}
\end{cislo}

\begin{cislo}\label{FGact}{\bf Group actions.} A \emph{left action}
\index{left group action} of a discrete group $G$ on a space $Y$ is a map
$$G\times Y\to Y,\quad (g,y)\mapsto g\cdot y$$
such that $1\cdot y=y$ and $(g_1g_2)\cdot y=g_1\cdot(g_2\cdot
y)$. We will call this action \emph{properly discontinuous}
\index{properly discontinuous action} if each point $y\in Y$
has an open neighbourhood $U$ such that
$g_1U\cap g_2U\ne \emptyset$ implies $g_1=g_2$.

An action of a group $G$ on a space $Y$ induces the equivalence
$x\sim y$ if $y=g\cdot x$ for some $g\in G$. The \emph{orbit
space}\index{orbit space} $Y/G$ is the factor space $Y/\sim$.

A space $Y$ is called  \emph{simply connected}\index{simply
connected space} if it is path connected
and $\pi_1(Y,y_0)$ is trivial for some (and hence all) base point
$y_0$.

The following theorem provides a useful method for computation of
fundamental groups.

\begin{thm*} Let $Y$ be a path connected space with a properly
discontinuous action of a group $G$. Then
\begin{enumerate}
\item The natural projection $p:Y\to Y/G$ is a covering space.
\item $G\cong \pi_1(Y/G,p(y_0))/p_*\pi_1(Y,y_0)$. Particularly,
if $Y$ is simply connected, then $\pi_1(Y/G)\cong G$.
\end{enumerate}
\end{thm*}

\begin{proof} Let $y\in Y$ and let $U$ be a neighbourhood of $y$
from the definition of properly discontinuous action. Then
$p^{-1}(p(U))$ is a disjoint union of $gU$, $g\in G$. Hence
$(Y,Y/G,p)$ is a fibre bundle with the fibre $G$.

Applying the long exact sequence of homotopy groups of this
fibration we obtain
$$0=\pi_1(G,1)\to \pi_1(Y,y_0)\xrightarrow{p_*}
\pi_1(Y/G;p(y_0))\xrightarrow{\delta}\pi_0(G)=G\to \pi_0(Y)=0.$$
In general $\pi_0$ of a fibre is only the set with distinguished
point. However, here it has the group structure given by $G$. Using the
definition of $\delta$ from 10.3 %\ref{HOMles} 
one can check that $\delta$ is a group homomorphism. Consequently, the exact
sequence implies that
$$G\cong\pi_1(Y/G,p(y_0))/p_*\pi_1(Y,y_0).$$
\end{proof}

\begin{example} $\Z$ acts on real numbers $\R$ by addition.
The orbit space is $\R/\Z=S^1$. According to the previous theorem
$$\pi_1(S^1,s)=\Z.$$
The fundamental group of the sphere $S^n$ with $n\ge 2$ is
trivial. The reason is that any loop $\gamma:S^1\to S^n$ is
homotopic to a loop which is not a map onto $S^n$ and $S^n$
without a point is contractible.

Next, the group $\Z_2=\{1,-1\}$ has an action on $S^n$, $n\ge 2$
given by $(-1)\cdot x=-x$. Hence
$$\pi_1(\mathbb{RP}^n)=\Z_2.$$
\end{example}

\begin{example} The abelian group $\Z\oplus \Z$ acts on $\R^2$
$$(m,n)\cdot(x,y)=(x+m,y+n).$$
The factor $\R^2/(\Z\oplus\Z)$ is two dimensional torus
$S^1\times S^1$. Its fundamental group is $\Z\oplus \Z$.
\end{example}

\begin{example} The group $G$ given by two generators
$\alpha$, $\beta$ and the relation
$\beta^{-1}\alpha\beta=\alpha^{-1}$
acts on $\R^2$ by
$$\alpha\cdot(x,y)=(x+1,y),\quad \beta\cdot(x,y)=(1-x,y+1).$$
The factor $\R^2/G$ is the Klein bottle. Hence its fundamental
group is $G$.
\end{example}
\end{cislo}

\begin{cislo}\label{FGfp}{\bf Free product of groups.} As a set the
\emph{free product $*_{\alpha} G_{\alpha}$ of groups} \index{free
product of groups} $G_\alpha$, $\alpha\in I$ is the set of finite
sequences $g_1g_2\dots g_m$ such that $1\ne g_i\in G_{\alpha_i}$,
$\alpha_i\ne\alpha_{i+1}$, called words. The elements $g_i$ are
called letters. The group operation is
given by
$$(g_1g_2\dots g_m)\cdot (h_1h_2\dots h_n)=
(g_1g_2\dots g_mh_1h_2\dots h_n)$$
where we take $g_mh_1$ as a single letter $g_m\cdot h_1$ if both
elements belong to the same group $G_{\alpha}$. It is easy to
show that $*_{\alpha}G_{\alpha}$ is a group with the empty word
as the identity element. Moreover, for each $\beta\in I$
there is the natural inclusion
$i_{\beta}:G_{\beta}\hookrightarrow *_{\alpha}G_{\alpha}$.


Up to isomorhism the free product of groups is characterized by
the following universal property: Having a system of group
homomorphism $h_{\alpha}:G_{\alpha}\to G$ there is just one group
homomorphism $h:*_{\alpha}G_{\alpha}\to G$ such that $h_{\alpha}=
hi_{\alpha}$.

\begin{ex*} Describe $\Z_2*\Z_2$.
\end{ex*}

\end{cislo}

\begin{cislo}\label{FGvkt}{\bf Van Kampen Theorem.} Suppose that
a space $X$ is a union of path connected open subsets
$U_{\alpha}$ each of which contains a base point $x_0\in X$.
The inclusions $U_{\alpha}\hookrightarrow X$ induce homomorphisms
$j_{\alpha}:\pi_1(U_{\alpha})\to \pi_{1}(X)$ which determine
a unique homomorphism $\p:*_{\alpha}\pi_1(U_{\alpha})\to
\pi_1(X)$.

Next, the inclusions $U_{\alpha}\cap
U_{\beta}\hookrightarrow U_{\alpha}$ induce
the homomorphisms $i_{\alpha\beta}:\pi_1(U_{\alpha}\cap
U_{\beta})\to \pi_1(U_{\alpha})$. We have
$j_{\alpha}i_{\alpha\beta}=j_{\beta}i_{\beta\alpha}$.
Consequently, the kernel of $\p$ contains elements
of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega^{-1})$
for any $\omega\in \pi_{1}(U_{\alpha}\cap U_{\beta})$.

Van Kampen Theorem provides the full description of the homomorphism
$\p$ which enables us to compute $\pi_1(X)$ using
groups $\pi_1(U_{\alpha})$
and $\pi_1(U_{\alpha}\cap U_{\beta})$.

\begin{thm*} [Van Kampen Theorem] If $X$ is a union of path connected open sets
$U_{\alpha}$ each containing a base point $x_0\in X$ and if each
intersection $U_{\alpha}\cap U_{\beta}$ is path connected, then
the homomorhism $\p:*_{\alpha}\pi_1(U_{\alpha})\to \pi_1(X)$
is surjective. If in addition each intersection $U_{\alpha}\cap
U_{\beta}\cap U_{\gamma}$ is path connected, then the kernel of
$\p$ is the normal subgroup $N$ in $*_{\alpha}\pi_1(U_{\alpha})$
generated by elements $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega^{-1})$
for any $\omega\in \pi_{1}(U_{\alpha}\cap U_{\beta})$. So $\p$
induces an isomorphism
$$\pi_1(X)\cong *_{\alpha}\pi_1(U_{\alpha})/N.$$
\end{thm*}

\begin{example*} If $X_{\alpha}$ are path connected spaces, then
$$\pi_1(\bigvee X_{\alpha})=*_{\alpha}\pi_1(X_{\alpha}).$$
\end{example*}

\begin{proof}[Outline of the proof of Van Kampen Theorem] For simplicity we suppose that
$X$ is a union of only two open subsets $U_1$ and $U_2$.

\emph{Surjectivity of $\p$.} Let $f:I\to X$ be a loop starting at
$x_0\in U_1\cup U_2$. This loop is up to homotopy a composition
of several paths,
for simplicity suppose there are three such that
$f_1:I\to U_1$, $f_2:I\to U_2$ and $f_3:I\to U_1$ with end points
succesively $x_0,x_1,x_2,x_0\in U_1\cap U_2$. Since $U_1\cap U_2$
is path connected there are paths $g_1:I\to U_1\cap U_2$ and
$g_2:I\to U_1\cap U_2$ from $x_0$ to $x_1$ and $x_2$,
respectively. Then the loop $f$ is up to homotopy the composition of loops
$f_1-g_1:I\to U_1$, $g_1+f_2-g_2:I\to U_2$ and $g_2+f_3:I\to
U_1$. Consequently, $[f]\in \pi_1(X)$ lies in the image of $\p$.

\begin{figure}[htb]
  \centering
  \def\svgwidth{7cm}
  \input{img/fig-11_1.pdf_tex}
  \caption{$[f]=[f_1+f_2+f_3]=[f_1-g_1]+[g_1+f_2-g_2]+[g_2+f_3]$}
\end{figure}

\emph{Kernel of $\p$.} Suppose that the image under $\p$ of a word
with $m$ letters $[f_1][g_1][f_2]\dots$, where
$[f_i]\in \pi_1(U_1)$, $[g_i]\in \pi_1(U_2)$,
is zero in $\pi_1(X)$. Then there is a homotopy
$F:I\times I\to X$ such that
$$F(s,0)=f_1+g_1+f_2+\dots,\quad F(s,1)=x_0,\quad
F(0,t)=F(1,t)=x_0$$
where we suppose that $f_i$ is defined on
$[\frac{2i-2}{m},\frac{2i-1}{m}]$ and $g_i$ is defined on
$[\frac{2i-1}{m},\frac{2i}{m}]$.
Since $I\times I$ is compact, there is an integer $n$, a multiple
of $m$, such that
$$F\left(\left[\frac{i}{n},\frac{i+1}{n}\right]\times \left[\frac{j}{n},\frac{j+1}{n}\right]\right)$$
is a subset in $U_1$ or $U_2$. Using homotopy extension property,
we can construct a homotopy
from $F$ to $\widetilde F$ rel
$J^1$ such that again
$$\widetilde F\left(\left[\frac{i}{n},\frac{i+1}{n}\right]\times \left[\frac{j}{n},\frac{j+1}{n}\right]\right)$$
is a subset in $U_1$ or $U_2$, and moreover,
$$\widetilde F\left(\frac{i}{n},\frac{j}{n}\right)=x_0.$$
Further,
$\widetilde F(s,0)= {f'}_1+{g'}_1+{f'}_2+\dots$
where ${f'}_i\sim f_i$, ${g'}_i\sim g_i$ in $U_1$ and
$U_2$, respectively, rel the boundary of the domain of definition.
We want to show that the word
$[{f'}_1]_1[{g'}_1]_2[{f'}_2]_1\dots$
belongs to $N$. Here $[\ ]_i$ stands for an element in
$\pi_1(U_i)$.

We can decompose
$$I\times I=\bigcup_{i} M_i$$
where $M_i$ is a maximal subset with the properties:
\begin{enumerate}
\item $M_i$ is a union of several squares
$[\frac{i}{n},\frac{i+1}{n}]\times [\frac{j}{n},\frac{j+1}{n}]$.
\item $\inte M_i$ is path connected.
\item $\widetilde F(M_i)$ is a subset in $U_1$ or $U_2$.
\end{enumerate}

For simplicity suppose that we have four sets $M_i$ as indicated
in the picture.

%\begin{figure}[H]
%\centering
%\includegraphics[height=7.4cm,width=7.2cm]{fg6.eps}%
%\caption{}\label{fg6}
%\end{figure}

\begin{figure}[htb]
  \centering
  \def\svgwidth{7.5cm}
  \input{img/fig-11_2.pdf_tex}
  \caption{$[f'_1]_1 [g'_1]_2 [f'_2]_1 \in \text{Ker} \varphi$}
\end{figure}

In this situation there are three loops $k$, $l$ and $p$ starting
at $x_0$ and lying in $U_1\cap U_2$. They are defined by $\widetilde
F$ on common boundary of $M_1$ and $M_2$, $M_2$ and $M_3$, $M_3$ and
$M_4$, respectively. Now, we get
\begin{align*}
[{f'}_1]_1 [{g'}_1]_2 [{f'}_2]_1&=[k]_1 [-k+l]_2
[-l+p]_1=[k]_1[-k]_2[l]_2[-l]_1[p]_1\\
&=[k]_1[-k]_2[l]_2[-l]_1 \in N.
\end{align*}
\end{proof}

\begin{cor*} Let $X$ be a union of two open subsets $U$ and $V$ where
$V$ is simply connected and $U\cap V$ is path connected.
Then
$$\pi_1(X)=\pi_1(U)/N$$
where $N$ is the normal subgroup in $\pi_1(U)$ generated by the
image of $\pi_1(U\cap V)$.
\end{cor*}
\end{cislo}

\begin{ex*} Use the previous statement to compute the fundamental
group of the Klein bottle and other 2-dimensional closed
surfaces.
\end{ex*}

\begin{cislo}\label{FGfgh}{\bf Fundamental group and homology.}
Here we compare the fundamental group of a space with the first
homology group. We obtain a special case of Hurewitz theorem,
see 13.6. %\ref{EHht}.

\begin{thm*} By regarding loops as 1-cycles, we obtain a
homomorphism $h:\pi_1(X,x_0)\to H_1(X)$. If $X$ is path
connected, then $h$ is surjective and its kernel is the commutator
subgroup of $\pi_1(X)$. So $h$ induces isomorphism from
the abelization of $\pi_1(X,x_0)$ to $H_1(X)$.
\end{thm*}

For the proof we refer to [Hatcher], %\cite{Ha}, 
Theorem 2A.1, pages 166--167.
\end{cislo}