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UF MTG 6346\\
Topology 1\\
Fall 2024
\begin{center}
{\large \bf Homework 1}\\
Due Friday, August 30, anytime, on Canvas
\end{center}

{\bf Reading.}
Hatcher pages 1--17.


{\bf Problems.}
\begin{enumerate}

\item Chapter 0, Exercise 1.
Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian circles of the torus.

\emph{Remark: ``Explicit'' means give formulas.}

\emph{Hint: I recommend you parameterize your torus as a square in the plane with opposite sides identified.}

\item Chapter 0, Exercise 3.

(a) Show that the composition of homotopy equivalences $X\to Y$ and $Y\to Z$ is a homotopy equivalence $X\to Z$.
Deduce that homotopy equivalence is an equivalence relation.

(b) Show that the relation of homotopy among maps $X\to Y$ is an equivalence relation.

(c) Show that a map homotopic to a homotopy equivalence is a homotopy equivalence.

\item Chapter 0, Exercise 16.
Show that $S^\infty$ is contractible.

\item Chapter 0, Exercise 19.
Show that the space otained from $S^2$ by attaching $n$ 2-cells along any collection of $n$ circles in $S^2$ is homotopy equivalent to the wedge sum of $n+1$ $2$-spheres.

\end{enumerate}

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

\item Chapter 0, Exercise 2.

\item Chapter 0, Exercise 6.

\item Chapter 0, Exercise 9.

\end{itemize}

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