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\begin{document}

\begin{center}
\textbf{Spring 2021, Math 621:\ Problem Set 4} \\
\textbf{Due:\ Thursday, February 25th, 2021} \\
\textbf{Homotopy Equivalence (Week 2)}
\end{center}


\bigskip
\noindent
\textbf{Homework problems.}
You must submit \emph{all} homework problems in order to receive full credit.  

\begin{enumerate}[(H1)]
\item 
Prove that homotopy equivalence of topological spaces is an equivalence relation.  

\begin{proof}

\end{proof}


\item 
Prove that if $X \simeq X'$ and $Y \simeq Y'$, then $X \times Y \simeq X' \times Y'$.  

\begin{proof}

\end{proof}


\item 
Prove if $A$ is contractible, the quotient map $q:X \to X/A$ is a homotopy equivalence.  Use the following steps as a guide.  

\begin{enumerate}[(i)]
\item 
Fix a contraction $f_t:A \to A$.  Use the fact that $(X,A)$ has the homotopy extension property to obtain $f_t:X \to X$ with $f_0 = \Id_X$.  

\item 
Define $\ol f_t:X/A \to X/A$ so the left diagram below commutes (i.e., $qf_t = \ol f_tq$).  
\vspace{-0.3em}
$$
\begin{tikzcd}[row sep=small]
X \arrow[dd, "q"] \arrow[r, "f_t"] & X \arrow[dd, "q"] \\ \\
X/A \arrow[r, "\ol f_t"] & X/A
\end{tikzcd}
\qquad
\begin{tikzcd}[row sep=small]
X \arrow[dd, "q"] \arrow[r, "f_1"] & X \arrow[dd, "q"] \\ \\
X/A \arrow[r, "\ol f_1"] \arrow[ruu, "g"] & X/A
\end{tikzcd}
$$
\vspace{-0.3em}

\item 
Define a map $g:X/A \to X$ so that the right diagram above commutes.  

\item 
Verify $qg$ and $gq$ are each \emph{nullhomotopic} (i.e., homotopic to the identity map).  

\end{enumerate}

\begin{proof}

\end{proof}


\item 
Given a continuous map $f:X \to Y$, the \emph{mapping cylinder} is the space
$$M_f = ((X \times I) \amalg Y)/\til$$
where $(x,1) \sim f(x)$ for each $x \in X$.  It turns out that if $f$ is a homotopy equivalence, then~$M_f$ deformation retracts onto both $Y$ and $X \times \{0\} \cong X$.  

Prove that $M_f$ deformation retracts onto $Y$.  

\begin{proof}

\end{proof}


\item 
If $X$ and $Y$ are cell complexes, under what conditions (if any) on $f:X \to Y$ does the mapping cylinder $M_f$ inherit a ``natural'' a cell structure from $X$ and $Y$?  

\begin{proof}

\end{proof}


\item 
Determine whether each of the following statements is true or false.  Prove each true statement, and give a counterexample for each false statement.  

\begin{enumerate}[(a)]
\item 
Any 1-dimensional cell complex with finitely many cells is homotopy equivalent to a wedge sum of circles.  

\begin{proof}

\end{proof}


\item 
If $X$ is a cell complex equal to the union of two contractible subcomplexes whose intersection is also contractible, then $X$ is contractible.  

\begin{proof}

\end{proof}

\end{enumerate}

\end{enumerate}



\bigskip
\noindent
\textbf{Challenge problems.}
Challenge problems are not required for submission, but bonus points will be awarded for submitting a partial attempt or a complete solution.  

\begin{enumerate}[(C1)]
\item 
Suppose $(X,A)$ is a CW-pair and that $A$ is contractible in $X$ (that is, the inclusion map $A \hookrightarrow X$ is homotopic to a constant map).  Determine the homotopy type of $X/A$.  

Hint:\ first find an example where $A$ is not a contractible space but \textbf{is} constractible in $X$.  

\begin{proof}

\end{proof}

\end{enumerate}

\end{document}

