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\begin{document}

\title{Homework 1 Part 1: Due Wednesday September 4}

\author{Instructor: Paul Apisa}

\maketitle 

\noindent All of the subsequent problems are either self-contained or from Chapter 1.1 of Hatcher. $X$ will always denote a topological space unless otherwise stated. All arrows denote continuous maps unless otherwise stated. 

\begin{description}
\item[Exercises 6 - Conjugacy Classes in $\pi_1$]
\end{description}

\begin{description}
\item[Exercises 9 - Ham Sandwich Theorem]
\end{description}

\noindent \textbf{Two Problems on Constructing Explicit Homotopies}

\begin{description}
\item[Exercise 10]
\end{description}

\begin{description}
\item[Exercise: The fundamental group of a topological group]
A topological group is a topological space $G$ that is also a group so that the map $m: G \times G \ra G$ that sends $(g, h)$ to $gh$ is continuous and so that the map $i: G \ra G$ that sends $g$ to $g^{-1}$ is continuous. Examples include $S^1$ and closed subgroups of $\mathrm{GL}(n, \mathbb{R})$ - viewed as an open subset of $\R^{n^2}$. Show that the fundamental group of any topological group is abelian by using the continuity of multiplication to exhibit a homotopy between $\alpha \cdot \beta$ and $\beta \cdot \alpha$ for any two loops $\alpha$ and $\beta$ based at the identity element $e$. 
\end{description}

\noindent \textbf{Three (Hopefully) Very Short Proofs}

\begin{description}
\item[Exercises 11, 12, 13]
\end{description}

\noindent \textbf{Two More Perspectives on the Fundamental Group}

\begin{description}
\item[Exercise: The fundamental group corresponds to path components of the loop space]
Given a topological space $X$ with a point $x_0 \in X$, let $\Omega X$ be the set of loops in $X$ based at $x_0$. Give $\Omega X$ the compact-open topology. Show that path components of $\Omega X$ are in bijective correspondence with elements of $\pi_1(X, x_0)$. 
\end{description}

\begin{description}
\item[Exercise: Separate continuity vs. Continuity]
Say that a map $F: I \times I \ra X$ is separately continuous if for all $t \in I$ the maps $f_i : I \ra X$ given by $f_1(s) := F(s, t)$ and $f_2(s) := F(t, s)$ for $i \in \{1, 2\}$ are continuous. Prove or disprove the following: if $F$ is separately continuous then it is continuous. 
\end{description}

\noindent \textbf{Fibrations}

\begin{description}
\item[Exercise: Product spaces are fibrations]
Let $X_1$ and $X_2$ be two topological spaces. Let $p_1: X_1 \times X_2 \ra X_1$ be the projection onto the first factor. Show that this is a fibration. 
\end{description}

\begin{description}
\item[Exercise: The fundamental group of $\mathbb{R} P^n$]
Let $\pi: S^n \ra \mathbb{R} P^n$ be the quotient of $S^n$ by the antipodal map. Show that $\pi$ is a covering map and imitate the proof that $\pi_1(S^1) \cong \Z$ to show that $\pi_1(\RP^n) \cong \Z/2\Z$ for $n > 1$. Notice that a path in $\RP^n$ corresponds to a continuously varying family of lines in $\R^{n+1}$ - using this description of paths, describe in words what distinguishes the nontrivial element of the fundamental group of $\RP^n$ from the trivial element when $n > 1$ (Hint: ``orient" the lines). 
\end{description}

\noindent \textbf{Teeing up van Kampen}

\begin{description}
\item[Exercises 18 and 19]
\end{description}

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\end{document}