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UF MTG 6347\\
Topology 2\\
Spring 2025
\begin{center}
{\large \bf Homework 4}\\
Due Friday, April 4, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 230--260.


{\bf Problems.}
\begin{enumerate}

\item \S3.3, Exercise 7.
For a map $f\colon M\to N$ between connected closed orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, the degree of $f$ is defined to be the integer $d$ such that $f_*([M]) = d[N]$, so the sign of the degree depends on the choice of fundamental classes.
Show that for any connected closed orientable $n$-manifold $M$ there is a degree $1$ map $M\to S^n$.

\item \S3.3, Exercise 6.
Given two disjoint connected $n$-manifolds $M_1$ and $M_2$, a connected $n$-manifold $M_1\#M_2$, their \emph{connected sum}, can be constructed by deleting the interiors of closed $n$-balls $B_1 \subset M_1$ and $B_2 \subset M_2$ and identifying the resulting boundary spheres $\partial B_1$ and $\partial B_2$ via some homeomorphism between them.
(Assume that each $B_i$ embeds nicely in a larger ball in $M_i$.)

(a) Show that if $M_1$ and $M_2$ are closed then there are isomorphisms $H_i(M_1\# M_2;\Z) \cong H_i(M_1;\Z)\oplus H_i(M_2;\Z)$ for $0 < i < n$, with one exception:
If both $M_1$ and $M_2$ are nonorientable, then $H_{n-1}(M_1\# M_2;\Z)$ is obtained from $H_{n-1}(M_1;\Z)\oplus H_{n-1}(M_2;\Z)$ by replacing one of the two $\Z_2$ summands by a $\Z$ summand.
[Euler characteristics may help in the exceptional case.]

(b) Show that $\chi(M_1\#M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)$ if $M_1$ and $M_2$ are closed.

\item \S3.3, Exercise 26.
Compute the cup product structure in $H^*(S^2\times S^8\#S^4\times S^6;\Z)$, and in particular show that the only nontrivial cup products are those dictated by Poincar\'{e} duality.
[See Exercise 6.
The result has an evident generalization to connected sums of $S^i \times S^{n-i}$'s for fixed $n$ and varying $i$.]

\end{enumerate}

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