\chapter{Fundamental Group: Capturing Holes}

\begin{thm*}[Fundamental Theorem of Algebra]
A polynomial $p(z) = a_{n} z^{n} + ... + a_{1}z + a_{0}$ with complex coefficients and degree $n>1$ has at least one root.
\end{thm*}

\oneperpage

\begin{exercise}%[special case of the Fundamental Theorem of Algebra]
A polynomial $p(x)= a_{n}x^{n} + ... + a_{1}x + a_{0}$ with real coefficients where $a_{n} \neq 0$ and $n$ is odd has at least one real root.
\end{exercise}

\oneperpage

\begin{thm}
Given topological spaces $X$ and $Y$ with $S\subset X$, 
homotopy relative to $S$ is an equivalence relation on the set of all
continuous functions from $X$ to $Y$.  
In particular, if $S=\varnothing$, homotopy is an equivalence relation on the set of all
continuous functions from $X$ to $Y$.
\end{thm}

\oneperpage

\begin{thm}
If $\alpha$, $\alpha'$, $\beta$, and $\beta'$ are paths in a space $X$ such that $\alpha\sim \alpha'$, $\beta\sim\beta'$, and $\alpha(1)=\beta(0)$,
then $\alpha\cdot\beta\sim\alpha'\cdot\beta'$.
\end{thm}

\oneperpage

\begin{thm}
Given paths $\alpha$, $\beta$, and $\gamma$ where the following products are defined, then $(\alpha\cdot\beta)\cdot
\gamma \sim\alpha\cdot (\beta\cdot \gamma)$ and
$([\alpha]\cdot[\beta])\cdot [\gamma]=[\alpha]\cdot ([\beta]\cdot
[\gamma])$.
\end{thm}

\oneperpage

\begin{thm}
Let $\alpha$ be a path with $\alpha(0)=x_0$.  Then $\alpha\cdot
\alpha^{-1} \sim e_{x_0}$, where $e_{x_0}$ is the
 constant path at $x_{0}$.
\end{thm}

\oneperpage

\begin{thm}
The fundamental group $\pi_1 (X,x_0)$ is a group. The identity element is the class of homotopically trivial loops based at $x_0$.
\end{thm}

\oneperpage

\begin{thm}\label{baseind}
If $X$ is path connected, then $\pi_1(X,p)\cong \pi_1 (X,q)$ for
any points $p,q\in X$.
\end{thm}

\oneperpage

\begin{cor}
Suppose $X$ is a topological space and there is a path between points $p$ and $ q$ in $X$. Then $\pi_1(X,p)$ is isomorphic
to $\pi_1(X,q)$.
\end{cor}

\oneperpage

\begin{exercise}
Let $\alpha$ be a loop into a topological space $X$.  Then $\alpha
=\beta \circ \omega|_{[0,1]}$ where $\omega$ is the standard
wrapping map and $\beta$ is some continuous function from $\Sph^1$ into
$X$.  This relationship gives a correspondence between
loops in $X$ and continuous maps from $\Sph^1$ into $X$.
%\marginpar{I made this exercise}
\end{exercise}

\oneperpage

\begin{thm}
Let $X$ be a topological space and let $p$ be a point in $X$. Then a
loop  $\alpha =\beta \circ \omega|_{[0,1]}$ (where $\omega$ is the
standard wrapping map and $\beta$ is a continuous function from
$\Sph^1$ into $X$) is homotopically trivial if and only if $\beta$
can be extended to a continuous function from the unit disk $\D^2$ to $X$. 
%\marginnote{we never define $\D^n$.}
\end{thm}

\oneperpage

\begin{thm}
Show the following ($1$ denotes the trivial group):
\begin{enumerate}
\item $\pi_1 ([0,1])\cong 1$.
\item $\pi_1$($\R^n$) $\cong 1$ for $n\geq 1$.
\item $\pi_1(X)\cong 1$, if $X$ is a convex set in $\R^n$.
\item $\pi_1(X)\cong 1$, if $X$ is a cone. %(for the definition of cone, see Section \ref{cone}).
\item $\pi_1(X)\cong 1$ if $X$ is a star-like space in $\R^n$ (a
subset $X$ of $\R^n$ is called \textbf{star-like} if there is a
fixed point $x_0\in X$ such that for any $y\in X$, the line segment
between $x_0$ and $y$ lies in $X$; a five pointed `star' is an
example of a star-like space that is not convex).
\end{enumerate}
\end{thm}

\oneperpage

\begin{exercise}
Show the following:
\begin{enumerate}
\item $\pi_1 (\Sph^0, 1)\cong 1$ where $\Sph^0$ is the
zero-dimensional sphere $\{-1,1\}$, the set of points unit distance from the origin in $\R^1$.
\item $\pi_1(\Sph^2) \cong 1$.
\item $\pi_1(\Sph^n)\cong1$ for $n \geq 3$.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise}
Show that the cone over the Hawaiian earring is simply connected. Can you generalize your insight?
\end{exercise}

\oneperpage

\begin{thm}
\begin{enumerate} \vskip.1in
\item Any loop $\alpha :[0,1] \to \Sph^1$ with $\alpha(0)=1$ can be written
$\alpha =\omega\circ\wt{\alpha}$, where $\wt{\alpha}: [0,1] \to \R^1$ satisfies
$\wt{\alpha}(0)=0$ and $\omega$ is the standard wrapping map.
\item If $\alpha: [0,1] \to\Sph^1$ is a loop, then $\wt{\alpha}(1)$ is an integer.
\item Loops $\alpha _1$ and $\alpha _2$ are equivalent in $\Sph^1$ if and only if $\wt{\alpha _1}(1)=\wt{\alpha _2}(1)$.
\item $\pi_1(\Sph^1)\cong\Z$.
\end{enumerate}
\end{thm}

\oneperpage

\begin{thm}
Let $(X, x_0)$, $(Y, y_0)$ be path connected spaces. 
Then
$$
\pi_1(X\times Y,(x_0, y_0))
\cong 
\pi_1(X, x_0)\times \pi_1 (Y, y_0) 
$$ 
via the canonical map that takes a loop $\gamma$ in $X \times Y$ to $(p \circ \gamma, q \circ \gamma)$ where 
$p: X \times Y \rightarrow X$ and $q: X \times Y \rightarrow Y$ are the projection maps.
\end{thm}

\oneperpage

\begin{exercise}
Find:
\begin{enumerate}
\item $\pi_1 (X)$ where $X$ is a solid torus.
\item $\pi_1 (\Sph^2\times\Sph^1)$
\item $\pi_1 (\Sph^2\times\Sph^2\times\Sph^2)$
\item $\pi_1 (X)$, where $X$ is a direct product of $k_n$ copies of $\Sph^n$, with $k_n=0$ for $n$ sufficiently large.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise} 
The fundamental group of the torus $\pi_1(\T^2)$ is $\Z^2$.  
Moreover, if $\mu$ is a meridian and $\lambda$ is a longitude,
%then $\{[\mu],[\lambda]\}$ is a $\Z$-basis for $\pi_1(\T^2)$.
then $\{[\mu],[\lambda]\}$ is a $\Z$-basis for $\pi_1(\T^2)$.
\end{exercise}

\oneperpage

\begin{exercise}
Check that for a continuous function $f:X\to Y$, the induced
homomorphism $f_*$ is well-defined (that
is, the image of an equivalence class is independent of the chosen
representative).  Show that it is indeed a group homomorphism.
\end{exercise}

\oneperpage

\begin{thm}
The following are true:
\begin{enumerate}
\item If $f: (X, x_0)\to (Y, y_0)$ and $g:(Y, y_0)\to (Z, z_0)$ are
continuous maps, then $(g \circ f)_* = g_* \circ f_*$.
\item If $id:(X, x_0)\to (X, x_0)$ is the identity map, then $id_*: \pi_1(X, x_0)\to \pi_1(X, x_0)$ is the identity homomorphism.
\end{enumerate}
\end{thm}

\oneperpage

\begin{thm}
If $h:X\to Y$ is a homeomorphism then $$h_* :\pi_1 (X,x_0)\to \pi_1 (Y,h(x_0))$$ is a group isomorphism. 
Thus homeomorphic path-connected spaces have
isomorphic fundamental groups.
\end{thm}

\oneperpage

\begin{thm} 
Fix a torus with chosen meridian $\mu$ and longitude
$\lambda$.  Suppose $p,q\in\Z$.  Then there is a homeomorphism
of the torus to itself which takes a representative of the class
$q[\mu]+p[\lambda]\in\pi(\T^2)$ to $\mu$ if and only if $p$ and $q$
are relatively prime.
\end{thm}

\oneperpage

\begin{thm}\label{homostar} 
If $f, g:(X, x_0)\to(Y, y_0)$
are continuous functions and $f$ is homotopic to $g$ relative to
$x_0$, then $f_*=g_*$.
\end{thm}

\oneperpage

\begin{lem} 
Homotopy equivalence of spaces is an equivalence
relation.
\end{lem}

\oneperpage

\begin{thm}
If $f:X\to Y$ is a homotopy equivalence and $y_0=f(x_0)$, then
$f_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)$ is an isomorphism.  In particular,
if $X\sim Y$, then $\pi_1(X)\cong\pi_1(Y)$.
\end{thm}

\oneperpage

\begin{exercise} 
Show that for $n\geq0$, $\R^{n+1}-\{0\}$ can be
strong deformation retracted onto $\Sph^n$.
\end{exercise}

\oneperpage

\begin{lem} 
If $A$ is a strong deformation retract of $X$, then $A$
and $X$ are homotopy equivalent.
\end{lem}

\oneperpage

\begin{thm} 
$\R^2$ is not homeomorphic to $\R^{n}$ for any
$n\neq2$.
\end{thm}

\oneperpage

\begin{exercise}  Let $x$ and $y$ be two points in $\R^2$.
Show that $\R^2-\{x,y\}$ strong deformation retracts onto the figure
eight. In addition, show that $\R^2-\{x,y\}$ strong deformation
retracts onto a theta space.
\end{exercise}

\oneperpage

\begin{thm}
If $r:X\to A$ is a strong deformation retraction and $a\in A$, then
$\pi_1(X,a)\cong \pi_1(A,a)$.
\end{thm}

\oneperpage

\begin{exercise} Calculate the fundamental group of the following
spaces.
\begin{enumerate}
\item An annulus.
\item A cylinder.
\item The M\"{o}bius Band.
\item An open 3-ball with a diameter removed.
\end{enumerate}
\end{exercise}

\oneperpage

\begin{exercise} 
Find an example of a space $X$ with a subspace $A$
such that if  $i:A\to X$ is the inclusion map,
$i_*:\pi_1(A)\to\pi_1(X)$ is not injective.
\end{exercise}

\oneperpage

\begin{thm}
Let $A$ be a retract of $X$ via the inclusion $i:A\hookrightarrow X$ and retraction $r:X\hookrightarrow A$. 
Then for $a\in A$, 
 $i_*:\pi_1(A,a)\to \pi_1(X,a)$ is injective 
and $r_*:\pi_1(X,a)\to \pi_1(A,a)$ is surjective.
\end{thm}

\oneperpage

\begin{thm}[No Retraction Theorem for $\D^2$]
\index{No Retraction Theorem!$\D^2$}
There is no retraction from $\D^2$ to its boundary.
\end{thm}

\oneperpage

\begin{thm}[Brouwer Fixed Point Theorem for $\D^2$]
\index{Brouwer Fixed Point Theorem!$\D^2$}
Let $f:\D^2\to\D^2$ be a continuous map.  Then there is some $x\in \D^2$
for which $f(x)=x$.
\end{thm}

\oneperpage

\begin{lem} 
A space is contractible if and only if it is homotopy
equivalent to a point.
\end{lem}

\oneperpage

\begin{thm}
A contractible space is simply connected.
\end{thm}

\oneperpage

\begin{thm}
A retract of a contractible space is contractible.
\end{thm}

\oneperpage

\begin{cor}
The house with two rooms is contractible.
\end{cor}

\oneperpage

\begin{cor}
The Dunce's Hat is contractible.
\end{cor}

\oneperpage

\begin{thm} 
Let $X=U\cup V$, where $U$ and $V$ are open and path connected and $U\cap V$ is path-connected, simply connected, and non-empty. Then $\pi_1(X)$ is isomorphic to the free product of $\pi_1(U)$ and $\pi_1(V)$, that is, $\pi_1(X)\cong\pi_1(U)*\pi_1(V)$.
\end{thm}

\oneperpage

\begin{question}
Let $X$ be the bouquet of $n$ circles. What is $\pi_1(X)$?
\end{question}

\oneperpage

\begin{exercise} 
Find a path-connected space $X$ with open, path-connected subsets $U$ and $V$ of $X$ such that $X=U\cup V$ such that $U$ and $V$ are both simply connected, but $X$ is not simply connected.  Conclude that the hypothesis that $U\cap V$ is path connected is necessary.
\end{exercise}

\oneperpage

\begin{lem}
Let $X=U\cup V$, where $U$ and $V$ are open and $U\cap V$ is path connected,
and let $p\in U\cap V$.  Then any element of $\pi_1(X,p)$ has a representative
$\alpha_1\beta_1\alpha_2\beta_2\cdots\alpha_n\beta_n$, where each $\alpha_i$ is a loop in $U$ based at $p$ and each $\beta_i$ is a loop in $V$ based at $p$.
\end{lem}

\oneperpage

\begin{thm}
Let $X$ be a wedge of two cones over two Hawaiian earrings, where
they are identified at the points of tangency of the circles of each
Hawaiian earring, as in Figure \ref{multicone}.  Then $\pi_1 (X)
\not\cong 1$.
\end{thm}

\oneperpage

\begin{thm}
Let $X=U\cup V$ where  $U,V$ are open, path connected, and simply connected and $U\cap V$ is nonempty and path connected. Then $X$ is simply connected.
\end{thm}

\oneperpage

\begin{thm}
Let $X=U\cup V$ where  $U,V$ are open and path connected and $U\cap V$ is path connected, $x \in U\cap V$, and $\pi_1 (U,x)\cong 1$.
Let $i:U\cap V\to V$ be the inclusion map. Then
\[
\pi_1 (X,x) \cong \frac{\pi_1(V,x)}{N}
\]
where $N$ is the smallest normal subgroup of $\pi_1 (V,x)$ containing the subgroup $i_* (\pi_1(U\cap V, x))$.
\end{thm}

\oneperpage

\begin{thm}[Van Kampen's Theorem]
\index{Van Kampen's Theorem} Let $X=U\cup V$ where  $U,V$ are open
and path connected and $U\cap V$ is path connected and $x\in U\cap V$.
Let $i:U\cap V\to U$ and $j:U\cap V\to V$ be the inclusion maps. Then
\[
\pi_1 (X,x) \cong \frac{\pi_1(U,x) * \pi_1(V,x)}{N}
\]
where $N$ is the smallest normal subgroup containing $\{ i_*
(\alpha)j_* (\alpha^{-1}) \}_{\alpha\in\pi_1 (U\cap V,x)}$ (so
$N$ contains elements created by taking a finite sequence of products and conjugates starting with elements of the form $i_*(\alpha)j_*(\alpha^{-1})$).
\end{thm}

\oneperpage

\begin{thm}[Van Kampen's Theorem; group presentations version]
\index{Van Kampen's Theorem!group presentations} Let $X=U\cup V$
where  $U,V$ are open and path connected and $U\cap V$ is path
connected and $x\in U\cap V$. Let $i:U\cap V\to U$ and $j:U\cap V\to V$
be the inclusion maps.  Suppose $\pi_1 (U,x)=\langle
g_1,\ldots,g_n | r_1, \ldots , r_m \rangle$,
 $\pi_1 (V,x)=\langle h_1,\ldots,h_t | s_1, \ldots , s_u \rangle$ and
$\pi_1 (U\cap V,x)=\langle k_1,\ldots,k_v | t_1, \ldots , t_w
\rangle$ then
\begin{eqnarray*}
\pi_1 (X,x) & =  \left\langle g_1,\ldots,g_n , h_1,\ldots,h_t \right.|
                & r_1, \ldots , r_m, s_1, \ldots , s_u,\\
            &   & \left. i_*(k_1)=j_*(k_1),\ldots
            ,i_*(k_v)=j_*(k_v)\right\rangle.
\end{eqnarray*}
\end{thm}

\oneperpage

\begin{exercise} 
Let $P$ be a polygonal representation of a compact, connected $2$-manifold such that the
all the vertices of $P$ are identified in the corresponding
quotient.  Give a presentation for $\pi_1(P)$.
\end{exercise}

\oneperpage

\begin{exercise} 
Give presentations of the fundamental groups for
our canonical polygonal presentations of $\num_{i=1}^n\T^2$ and
$\num_{i=1}^n\PP^2.
$\end{exercise}

\oneperpage

\begin{thm} 
Each 2-manifold in the following infinite list is topologically different from all the others on the list: $\Sph^2$, $\num_{i=1}^n\PP^2$, and $\num_{i=1}^n\T^2$.
\end{thm}

\oneperpage

\begin{thm}  
Suppose $G$ is a finitely presented group.  Then there
exists a 2-complex $(K,T)$ such that $\pi_1(K)\cong G$.
\end{thm}

\oneperpage

\begin{thm}
The fundamental group of the Hawaiian earring is not finitely
generated.  In fact, it is not countably generated.
\end{thm}

\oneperpage

\begin{lem} If $p,q\in\N$ are relatively prime, the line described
in the above process will eventually intersect the upper-right
vertex of the square.  Moreover, the line will not intersect itself
until it does.\end{lem}

\oneperpage

\begin{lem} Let $p$ and $q$ be relatively prime integers and let
$\rho_{p,q}$ be the simple closed curve constructed above.  Then
there is a homeomorphism of the square (with the standard
identifications made for the torus) that takes $\rho_{p,q}$ to our
canonical meridian.\end{lem}

\oneperpage

\begin{thm} 
For $p,q\in\Z$ relatively prime, the lens space $L(p,q)$
is triangulable.
\end{thm}

\oneperpage

\begin{exercise} 
Let $p,q\in\Z$ be relatively prime.
Calculate the fundamental group of the Lens space
$L(p,q)$.
\end{exercise}

\oneperpage

\begin{lem}
Every loop in $M_K$ is homotopic in $M_K$ to a product of $a_i$'s.
In other words, the loops $\{a_i\}$ generate $\pi_1(M_K)$.
\end{lem}

\oneperpage

\begin{lem}
At every crossing, such as that illustrated in Figure
\ref{crossings}, the following relation holds: $acb^{-1}=c$ or
$acb^{-1}c^{-1}=1$.
\end{lem}

\oneperpage

\begin{thm}\index{knot!fundamental group}
Let $K$ be a knot in $\Sph^3$ and let $\{a_i\}$ be the set of loops
consisting of one loop for each arc in a knot projection of $K$ as
described above. Then $\pi_1(M_K)=\{a_1,a_2,\ldots,a_n|
a_ia_ja_k^{-1}a_j^{-1}$ where there is one relation of the form
$a_ia_ja_k^{-1}a_j^{-1}$ for each crossing in the knot
projection$\}$.
\end{thm}

\oneperpage

\begin{exercise}\index{knot!unknot}
Find the fundamental group of the complement of the unknot (See
Figure \ref{unknot}).


\begin{figure}
\begin{center}
%\includegraphics{Figures/unknot}
\caption{\label{unknot} The unknot.}
\end{center}
\end{figure}
\end{exercise}

\oneperpage

\begin{exercise}
Find the fundamental group of the complement of the trefoil knot.
\end{exercise}

\oneperpage

\begin{exercise}\index{knot!figure-8}
Find the fundamental group of the complement of the figure-8 knot,
shown in Figure \ref{figeight}.
\end{exercise}

\oneperpage

\begin{exercise}
The collection of homotopy classes of continuous maps of the type
$f:(\D^n,\partial\D^n)\to(X,x_0)$, with the product defined above,
forms a group.
\end{exercise}

\oneperpage

\begin{thm}
Homotopy equivalent spaces have the same homotopy groups.
\end{thm}

\oneperpage

