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UF MTG 4303/5317\\
Introduction to Topology 2\\
Spring 2024
\begin{center}
{\large \bf Homework 3}\\
Due Wednesday, February 28, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S37, 56--59.


{\bf Problems.}
\begin{enumerate}

\item \S52 \#3.
Let $x_0$ and $x_1$ be points of the path-connected space $X$.
Show that $\pi_1(X,x_0)$ is abelian if and only if for every pair $\alpha$ and $\beta$ of paths from $x_0$ to $x_1$, we have $\hat{\alpha}=\hat{\beta}$.

\item \S55 \#4(d,e).
Suppose that you are given the fact that for each $n$, there is no retraction $r\colon B^{n+1}\to S^n$.
(This result can be proved using more advanced techniques of algebraic topology.)
Prove the following:

%(a) The identity map $i \colon S^n \to S^n$ is not nullhomotopic.

(d) Every continuous map $f\colon B^{n+1}\to B^{n+1}$ has a fixed point.

(e) Every $n+1$ by $n+1$ matrix with positive real entries has a positive eigenvalue.

\item \S57 \#4(a,b,c).
Suppose you are given the fact that for each $n$, no continuous antipode-preserving map $h\colon S^n \to S^n$ is nullhomotopic.
(This result can be proved using more advanced techniques of algebraic topology.)
Prove the following:

(a) There is no retraction $r\colon B^{n+1}\to S^n$.

(b) There is no continuous antipode-preserving map $g\colon S^{n+1}\to S^n$. 

(c) (Borsuk--Ulam theorem) Given a continuous map $f\colon S^{n+1}\to \mathbb{R}^{n+1}$, there is a point $x$ of $S^{n+1}$ such that $f(x)=f(-x)$.

\end{enumerate}



\vfill

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

\item \S55 \#4(a)

\item \S57 \#1

\item \S57 \#2

\end{itemize}

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