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UF MTG 6346\\
Topology 1\\
Fall 2024
\begin{center}
{\large \bf Homework 5}\\
Due Friday, November 8, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 113--137.


{\bf Problems.}
\begin{enumerate}

\item
\begin{enumerate}
\item Pick a $\Delta$-complex structure on the $n$-dimensional ball $D^n$ for $n\ge 1$.
This induces a $\Delta$-complex structure on its boundary $(n-1)$-sphere $S^{n-1}$.
Compute the simplicial relative homology $H^\Delta_n(D^n,S^{n-1})$ from its definition (i.e., the homology of the chain complex $\ldots\to\Delta_{i+1}(D^n,S^{n-1})\to\Delta_i(D^n,S^{n-1})\to\Delta_{i-1}(D^n,S^{n-1})\to\ldots)$.
\item Compute the singular relative homology $H_n(D^n,S^{n-1})$ for $n\ge 1$ by using the LES for a pair of spaces (on pages 115 and 117 of Hatcher).
\end{enumerate}

\item The proof of Theorem~2.16 in Hatcher contains 6 verifications of inclusions.
Prove them.
For one of the last three such inclusions ($\ker j_*\subseteq\im i_*$ or $\ker\partial\subseteq\im j_*$ or $\ker i_*\subseteq\im\partial$, your choice of which one), draw a diagram of the SES of chain complexes $0\to A_\bullet\to B_\bullet\to C_\bullet\to 0$, and show where all the elements you consider live in this diagram.

\item \S2.1, Exercise 16.
\begin{enumerate}
\item Show that $H_0(X, A) = 0$ iff $A$ meets each path-component of $X$.
\item Show that $H_1(X, A) = 0$ iff $H_1(A)\to H_1(X)$ is surjective and each path-component of $X$ contains at most one path-component of $A$.

\emph{Remark: For (b), Exercise 15 of Hatcher is useful. It says that if $A\to B\to C\to D\to E$ is an exact sequence, then $C = 0$ iff the map $A\to B$ is surjective and $D\to E$ is injective.}
\end{enumerate}

\item \S2.1, Exercise 20.
Show that $\tilde{H}_n(X)\cong \tilde{H}_{n+1}(SX)$ for all $n$, where $SX$ is the suspension of $X$.
More generally, thinking of $SX$ as the union of two cones $CX$ with their bases identified, compute the reduced homology groups of the union of any finite number of cones $CX$ with their bases identified.

\end{enumerate}

\vfill

{\bf Recommend Problems (not to turn in).}
\begin{itemize}

\item \S2.1, Exercise 17.

\item \S2.1, Exercise 22.

\end{itemize}

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