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\def\square{\hfill${\vcenter{\vbox{\hrule height.4pt \hbox{\vrule width.4pt
height7pt \kern7pt \vrule width.4pt} \hrule height.4pt}}}$}

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\def\O{\mathcal O}
\def\N{\mathbb N}
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\def\T{\mathcal T}
\def\U{\mathcal U}
\def\V{\mathcal V}
\def\RP{\mathbb{RP}}
\def\G{\mathcal G}
\def\Aut{\text{Aut}}
\def\Inn{\text{Inn}}
\def\Out{\text{Out}}
\def\Isom{\text{Isom}}
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\def\id{\text{id}}

\newenvironment{pf}{{\it Proof:}\quad}{\square \vskip 12pt}
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\title{Homework 5 --- Fundamental groups and covering spaces}

\author{Danny Calegari}

\maketitle
\begin{prob}
Let $X$ be path connected. That is, given any two points $x,y \in X$
there is a path $\alpha:I \to X$ with $\alpha(0) = x$ and $\alpha(1) = y$.
Show that the following are equivalent:
\begin{enumerate}
\item{$\pi_1(X,x)$ is trivial for all $x$
(we also say in this case that $X$ is {\em simply
connected})}
\item{Every map $\beta:S^1 \to X$ extends to a map of the closed unit
disk $D$ into $X$.}
\item{If $\sigma,\tau$ are paths in $X$ with the same initial points and
the same terminal points then $\sigma \simeq \tau$ rel.
endpoints.}
\end{enumerate}
\end{prob}

\begin{prob}
Let $f,g:\S^n \to \S^n$ be maps such that for all $x \in \S^n$, the
points $f(x)$ and $g(x)$ are not antipodal. Show $f \simeq g$. If in
addition there is $x_0 \in X$ such that $f(x_0) = g(x_0)$, show
$f \simeq g$ rel. $x_0$.
\end{prob}

\begin{prob}[Brouwer's fixed point theorem]
In this problem we prove the $2$--dimensional version of Brouwer's theorem,
and derive a theorem of linear algebra as a consequence.
\begin{itemize}
\item{Show that the circle is not a retract of the closed unit disk.
That is, show there is no map $r:D^2 \to S^1$ where $S^1 = \partial D^2$
such that $r|_{S^1} = \id$. In general for $A \subset X$ a map
$X \to A$ whose restriction to $A$ is the identity is called a {\em retraction}
from $X$ to $A$.}
\item{As a consequence, prove Brouwer's theorem: a continuous map
$f:D^2 \to D^2$ has a fixed point. Hint: suppose a fixed
point free map $f$ exists. Then we can join $x$ to $f(x)$ by a
line, and move along this line from $f(x)$ to $x$ and beyond
until we reach a point
$r(x) \in S^1$, thus defining a retraction $D^2 \to S^1$.}
\item{Show that every $3\times 3$ matrix with positive real entries
has an eigenvector with positive eigenvalues. Hint: consider the
triangle $T = \lbrace x + y + z = 1; x,y,z \ge 0 \rbrace$ and
the self--map of $T$ obtained by composing the matrix transformation with
radial projection.}
\end{itemize}
\end{prob}

\begin{prob}
Suppose $\phi:F \to E$ is a covering space, and suppose $E$ is
path--connected. Show that all the points in $E$ have the same number
of preimages.
\end{prob}

\begin{prob}
There is a natural projection $SO(3,\R) \to \S^2$ which is the projection
to the coset space of a subgroup $\text{stab}(p)$ for any $p \in S^2$ under the
standard action of the group. The preimage of any point in $S^2$ is a circle.
Show that $SO(3,\R)$ is not homotopy equivalent to $S^2 \times S^1$.  
\end{prob}

\begin{prob}
Let $G$ be the topological space obtained from two circles by gluing them
together at a point (so that $G$ is homeomorphic to the figure ``8''). Draw
all the connected coverings of $G$ of order $3$. Which coverings are regular?
\end{prob}

\begin{prob}
Find a presentation for $\pi_1(\S^3-K)$ for $K$ the figure $8$ knot.
\end{prob}

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