\documentclass[11pt]{amsart}
\textwidth=6.5in
\textheight=8.6in
\oddsidemargin 0.0in
\evensidemargin \oddsidemargin
\topmargin=0in
\RequirePackage{setspace}    
\renewcommand{\onehalfspacing}{\setstretch{1.4}} 
\usepackage[all]{xy}
\usepackage{enumerate}

\title{First assignment}
\author{Due Monday, September 19}

\newcommand{\R}{\mathbb{R}}
\DeclareMathOperator{\Sets}{Sets}
\DeclareMathOperator{\id}{id}


\begin{document}
\maketitle
\onehalfspacing





\begin{enumerate}
\item Let $E \subset X \times X$ be an equivalence relation on a set $X$. Construct the set of equivalence classes as colimit in the category $\Sets$.
\item Let 
\[ \xymatrix{A \ar[r] \ar[d] & B \ar[r] \ar[d] & C  \ar[d] \\
X \ar[r] & Y \ar[r] &  Z   }\]
be a commutative diagram. 
\begin{enumerate}
\item
Prove that if the two inner squares are pushouts, then so is the outer rectangle. That is, suppose that both $Y = B\sqcup_A X$ and $Z = C \sqcup_B Y$. Prove that $Z =  C \sqcup_{A} X$. 
\item What about if $Y = B\sqcup_A X$ and $Z =  C \sqcup_{A} X$, then is $Z = C \sqcup_B Y$? And what if $Z = C \sqcup_B Y$ and $Z =  C \sqcup_{A} X$, is $Y = B\sqcup_A X$?
\end{enumerate}
\item In the following problem, let $S^{n-1} \to D^n$ be the inclusion of the boundary, $X \vee X \to X \times X$ be the map $(\id \times \ast ) \vee (\ast \times \id)$ and $X \vee X \to X$ be the fold map $\id \vee \id$. For a based topological space $X$, let $J_2(X) = (X \times X)/((x,\ast)\sim (\ast, x))$. That is, $J_2(X)$ is the push-out
\[\xymatrix{X \vee X \ar[r]  \ar[d]  & X \ar[d] \\
X \times X \ar[r] & J_2(X).}\] 
Further, you may use the fact that $D^{2n} \cong I^{2n} \cong D^{n} \times D^{n} $ and other such standard homeomorphisms without proof.
\begin{enumerate}
\item Describe $S^n \times S^n$ as a CW-complex obtained from $S^n \vee S^n$ by attaching a single $2n$--cell.
Exhibit this as a pushout.
\item Use your construction in (b) to give $J_2(S^n)$ the structure of a $CW$-complex with one $n$--cell and one $2n$--cell.
\item Show that $S^n$ is an $H$-space if and only if the attaching map of the $2n$-cell of $J_2(S^n)$ is null-homotopic.
\end{enumerate}
\item Let $X$ be $1$--connected (i.e., $\pi_0X = \pi_1 X =0$). Recall that for a covering map $p: E \to B$ of based spaces, given a base point preserving map $f: X \to B$, there exists a unique base point preserving lift $\widetilde{f} : X \to E$ such that $p \widetilde{f} = f$. 
\begin{enumerate}
\item Prove that $\pi_n p : \pi_n E \to \pi_n B$ is an isomorphism for $n\geq 2$.
\item Compute $\pi_k \R P^n$ in terms of $\pi_k S^n$. What about $\pi_k \R P^{\infty}$?
\end{enumerate}
\end{enumerate}


\end{document}