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UF MTG 6346\\
Topology 1\\
Fall 2024
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{\large \bf Homework 4}\\
Due Wednesday, October 30, anytime, on Canvas
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{\bf Reading.}
Hatcher pages 97--133.


{\bf Problems.}
\begin{enumerate}

\item \S2.1, Exercise 4.
Compute the simplicial homology groups of the triangular parachute obtained from $\Delta^2$ by identifying its three vertices to a single point.

\item \S2.1, Exercise 5.
Compute the simplicial homology groups of the Klein bottle using the $\Delta$-complex structure described at the beginning of this section.
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\item \S2.1, Exercise 8.
Construct a 3-dimensional $\Delta$-complex $X$ from $n$ tetrahedra $T_1,\ldots,T_n$ by the following steps.
First arrange the tetrahedra in a cyclic pattern as in the figure, so that each $T_i$ shares a common vertical face with its two neighbors $T_{i-1}$ and $T_{i+1}$, subscripts being taken mod $n$.
Then identify the bottom face of $T_i$ with the top face of $T_{i+1}$ for each $i$.
Show the simplicial homology groups of $X$ in dimensions $0, 1, 2, 3$ are $\Z, \Z_n, 0, \Z$, respectively.
[The space $X$ is an example of a \emph{lens space}; see Example 2.43 for the general case.]
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\item \S2.1, Exercise 11.
Show that if $A$ is a retract of $X$ then the map $H_n(A)\to H_n(X)$ induced by the inclusion $A\subset X$ is injective.

\end{enumerate}

\vfill

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

\item
Compute the simplicial homology of the Klein bottle with $\Z/2\Z$ coefficients (defined on pages 153--154 of Hatcher).

\item \S2.1, Exercise 3.

\end{itemize}

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