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UF MTG 6346\\
Topology 1\\
Fall 2024
\begin{center}
{\large \bf Homework 6}\\
Due Friday, November 22, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 134--159.


{\bf Problems.}
\begin{enumerate}

\item
\S2.2, Exercise 1.
Prove the Brouwer fixed point theorem for maps $f\colon D^n \to D^n$ by applying degree theory to the map $S^n\to S^n$ that sends both the northern and southern hemispheres of $S^n$ to the southern hemisphere via $f$.
[This was Brouwer's original proof.]

\item
\S2.2, Exercise 9. Compute the homology groups of the following $2$-complexes:
\begin{enumerate}
\item[(a)] The quotient of $S^2$ obtained by identifying the north and south poles to a point.

\emph{Remark: } For pedagogical purposes, I recommend using the long exact sequence for the singular homology of the pair $(S^2, S^0)$.

\item[(c)] The space obtained from $D^2$ by first deleting the interiors of two disjoint subdisks in the interior of $D^2$ and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise orientations of these circles.

\end{enumerate}

\item
\S2.2, Exercise 32.
For $SX$ the suspension of $X$, show by a Mayer--Vietoris sequence that there are isomorphisms $\tilde{H}_n(SX)\cong \tilde{H}_{n-1}(X)$ for all $n$.

\item
Consider the CW-complex structure on real projective space $\RP^n$ that has a single $k$-cell for each $0\le k\le n$.
\begin{enumerate}
\item Using this CW structure, compute the cellular homology of $\RP^n$.

\emph{Remark: Some steps look different for the case of $n$ odd versus $n$ even.}
\item Compute the homology of $\RP^n$ with $\Z_2$ coefficients (compute $H_i(\RP^n;\Z_2)$).
\item Compute the homology of $\RP^n$ with $\Z_3$ coefficients (compute $H_i(\RP^n;\Z_3)$).
\end{enumerate}

\end{enumerate}

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

\item
Prove that $S^n$ has a continuous nonzero vector field iff $n$ is odd.

\item
Prove that $\Z_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even.

\item
\S2.2, Exercise 12.

\item
\S2.2, Exercise 29.

\end{itemize}

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