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\fancyhead[R]{\large \textbf{Fall 2022}}
\fancyhead[L]{\large \textbf{Topology Homework 6}}

\paragraph{Instructions:}  Again, I pulled problems from old qualifying exams that you should be able to do.  It would be good to solve them all, but just write up and turn in solutions to 1, 2c, 4, 7, 8, 10, 11, 14 (most of these have short solutions!)  Due on Monday, November 14.

\section{The compact open topology}

\bp \textbf{[Problem 1 from Spring 2015]}
\bigskip

\noindent
Consider $\mathsf{Top}([0,1],\mathbb{R})$ with the compact open topology.  Define a family of functions $\mathcal{F}\subset \mathsf{Top}([0,1],\mathbb{R})$ by $\mathcal{F}=\{f_a:0<a<1\}$ where $f_a(x)=1-\frac{x}{a}.$  Is  $\mathcal{F}$ compact?
\ep



\bp \textbf{[Problem 4 from Fall 2017 and 3 from Spring 2021]}
\bigskip

\noindent Let $X=\mathsf{Top}([0,1],[0,1])$ with the compact open topology.
\begin{enumerate}[label=(\alph*)] 
\item Is the function $ev:X \times [0,1] \to [0,1]$ defined by $(f,x)\mapsto f(x)$ continuous?
\item Is the map $I:X \to [0,1]$ defined by $f\mapsto \ds \int_0^1 f$ continuous?
\item Is the evaluation map $ev:X \times [0,1] \to [0,1]$ defined by $(f,x)\mapsto f(x)$ continuous if $X$ is given the product topology.
\end{enumerate}
\ep

\section{Compactifications}

\bp \textbf{[Problem 7 from Fall 2021]}
\bigskip

\noindent Suppose $X$ and $Y$ are noncompact, locally compact, and Hausdorff.  Prove that $(X\times Y)^* \cong X^* \wedge Y^*$.  Here $(-)^*$ means the one-point compactification and $\wedge$ means the smash product of pointed spaces, with the base point being the point added in the compactification.
\ep


\bp \textbf{[Problem 4 from Fall 2015]}
\bigskip

\noindent
Prove that any compact Hausdorff space is a retract of its Stone-\v Cech compactification.

\ep



\bp \textbf{[Problem 7 from Fall 2015]} 

\bigskip

\noindent Prove that the Stone-\v Cech compactification of the rationals is uncountable.
\ep


\bp \textbf{[Problem 4c and 4d from Spring 2016]}
\bigskip

\noindent Look up what the projective space $\mathbb{RP}^n$ are.

\begin{enumerate}[label=(\alph*)] 
\item Prove or disprove:  the one-point compactification of $\mathbb{R}$ is $\mathbb{RP}^1$.
\item Prove or disprove:  the one-point compactification of $\mathbb{R}^2$ is $\mathbb{RP}^2$.
\end{enumerate}
\ep



\section{CW complexes and Euler characteristic}

\bp \textbf{[Problem 12a from Spring 2015]}
\bigskip

\noindent
Show that $S^1\vee S^1 \vee S^2$ and the torus $T$ both have $CW$-structures with four cells:  one $0$-cell, two $1$-cells, and one $2$-cell.
\ep



\bp \textbf{[Problem 12 from Fall 2016 15c from Spring 2021]}
\bigskip

\noindent The number of cells is a CW structure on a space $X$ is not an invariant of the space.  It turns out, however, that the alternating sum is and invariant called the Euler characteristic of $X$.
\[\chi(X):=\sum (-1)^i d_i \]
where $d_i$ is the number of $i$ cells of a CW structure on $X$.



\begin{enumerate}[label=(\alph*)] 
\item Compute $\chi(T^n)$ where $T^n = (S^1)^{\times n}.$
\item Compute $\chi(S^2 \times T^2)$.
\end{enumerate}
\ep

\section{Suspensions}

\bp  \textbf{[Problems 4 and 7 from Spring 2000]}
\bigskip

\noindent Let $\Sigma X$ be the suspension of $X$.  Prove that
\begin{enumerate}[label=(\alph*)] 
\item If $X$ is connected, then $\Sigma X$ is simply connected.
\item If $X$ is contractible, then $\Sigma X$ is contractible.
\item $X$ is compact if and only if $\Sigma X$ is compact.
\item $X$ is Hausdorff if and only if $\Sigma X$ is Hausdorff.
\end{enumerate}
\ep


\bp \textbf{[Problem 4 from Spring 2017]} Let $\mathbf{Top_*}$ be the category of pointed topological spaces and let $\mathbf{\Sigma}:\mathbf{Top_*} \to \mathbf{Top_*}$ be the functor that sends a pointed space $X$ to its reduced suspension $\Sigma X:= S^1 \wedge X$.  Let $\mathbf{\Omega}:\mathbf{Top_*} \to \mathbf{Top_*}$ be the functor that sends a pointed space $X$ to its based loop space $\Omega X:=\mathbf{Top_*}(S^1,X)$.  The setup
\[\begin{tikzcd}
\mathbf{\Sigma}:\mathbf{Top_*} \arrow[r, shift left]  & \mathbf{Top_*} : \mathbf{\Omega} \arrow[l, shift left]
\end{tikzcd}
\]
defines an adjunction.  Your problem:  describe the unit and counit of this adjunction.
\ep

\section{And a few miscellaneous problems...}

\bp \textbf{[Problem 4 from Fall 2000]}
\bigskip

\noindent Let $1\leq n \leq \infty$ and let $X_n=S^1\vee S^1 \vee \cdots \vee S^1$ be the wedge sum of $n$ copies of $S^1$.  If $n=\infty$, then $X_\infty$ is the colimit of $S^1 \subset S^1 \vee S^1 \subset S^1 \vee S^1 \vee S^1 \subset \cdots $

\begin{enumerate}[label=(\alph*)] 
\item Prove that $X_n$ is compact if and only if $n<\infty$.
\item Prove that $X_n$ is connected for all $n$.
\end{enumerate}
\ep

\bp \textbf{[Problem 5a from Spring 2017]} 

\bigskip

\noindent Let $X$ be the torus and $Y$ be the two-sphere with three points removed.  
\begin{enumerate}[label=(\alph*)] 
\item Are $X$ and $Y$ homotopic?
\item Are $X$ and $Y$ homeomorphic?
\end{enumerate}
\ep


\bp \textbf{[Problem 1 from Part II of Fall 2005]}
\bigskip

\noindent Let $(X,x)$ and $(Y,y)$ be two based spaces.  Prove that $\pi_1((X,x)\times (Y,y)) \simeq \pi_1(X,x) \times \pi_1(Y,y)$.
\ep

\bp \textbf{[Problem 5a from Spring 2002]}
\bigskip

\noindent Let $A$ be a retract of $X$.  This means that $A$ is a subset of $X$ and the inclusion $i:A \to X$ is left-invertible.  Prove that the inclusion $i:A \to X$ induces a one-to-one map in homology.  

\bigskip
\noindent\emph{Note:} For this problem, you only need to know that homology is a functor.

\ep




\bp \textbf{[Problem 7 from Spring 2021 and 8 from Fall 2021]}
\bigskip

\noindent
Does the functor $\pi_1:\mathsf{Top}_* \to \mathsf{Grp}$ have a left or right adjoint $F:\mathsf{Grp} \to \mathsf{Top}_*$?  Is $\pi_1$ representable?
\ep





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