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UF MTG 6347\\
Topology 2\\
Spring 2025
\begin{center}
{\large \bf Homework 3}\\
Due Friday, Marcy 14, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 206--230.


{\bf Problems.}
\begin{enumerate}

\item \S3.2, Exercise 3.
(a) Using the cup product structure, show there is no map $\RP^n \to \RP^m$ inducing a nontrivial map $H^1(\RP^n;\Z_2)\to H^1(\RP^n;\Z_2)$ if $n>m$.
What is the corresponding result for maps $\CP^n \to \CP^m$?

(b) Prove the Borsuk--Ulam theorem by the following argument.
Suppose on the contrary that $f\colon S^n \to \R^n$ satisfies $f(x)\neq f(-x)$ for all $x$.
The define $g\colon S^n \to S^{n-1}$ by $g(x)=(f(x)-f(-x))/|f(x)-f(-x)|$, so $g(-x)=-g(x)$ and $g$ induces a map $\RP^n\to\RP^{n-1}$.
Show that part (a) applies to this map.

\item \S3.2, Exercise 4.
Apply the Lefschetz fixed point theorem to show that every map $f\colon \CP^n\to \CP^n$ has a fixed point if $n$ is even, using the fact that $f^*\colon H^*(\CP^n;\Z) \to H^*(\CP^n;\Z)$ is a ring homomorphism.
When $n$ is odd show there is a fixed point unless $f^*(\alpha)=-\alpha$, for $\alpha$ a generator of $H^2(\CP^n;\Z)$.
[See Exercise 3 in S2.C for an example of a map without fixed points in this exceptional case.]

\item
Let $X$ be two unlinked circles in $\R^3$ and let $Y$ be two simply linked circles.
Show that the complements $\R^3\setminus X$ and $\R^3\setminus Y$ have isomorphic cohomology groups but different ring structures, and hence are not homotopy equivalent.
[See Example~1.23 in Hatcher for how this problem was solved using the fundamental group.]

\begin{center}
\includegraphics[height=1in]{Unlinked.pdf}
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\end{enumerate}

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