<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt,notitlepage]{article}
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\begin{document}
\parindent 0in

\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2425}{2024-25}: \href{http://drorbn.net/25-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 8}
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due April 2 at 11:59pm
}~%
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/25-1301}
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\vskip -3mm
\rule{\linewidth}{1pt}
\vskip 5mm

\begin{center}\begin{minipage}{6.5in}\parindent 12pt

\noindent{\bf Problem 1.}
\begin{enumerate}
\item Formulate and prove a naturality property for the Mayer-Vietoris sequence. Your property must be at least strong enough to answer part 2 of this question.
\item Use part 1 of this question to prove that if $f\colon S^n\to S^n$ then $\deg(f)=\deg(\Sigma f)$ where $\Sigma$ is the suspension functor, mentioned previously both in class and in HW7. 
\end{enumerate}

\vskip 2mm\noindent{\bf Problem 2.} Suppose $n$ is even.
\begin{enumerate}
\item Show that for any continuous map $f\colon S^n\to S^n$ there is a point $x$ such that $f(x)=\pm x$.
\item Show that any continuous map $f\colon\bbR P^n\to\bbR P^n$ has a fixed point.
\end{enumerate}

\vskip 2mm\noindent{\bf Problem 3.} 
\begin{enumerate}
\item Compute the homology over $\bbZ$ of a the space $X$ obtained from the 2D disk $D^2$ by identifying each of its boundary points with the point you get from it by applying a $1/3$ rotation counterclockwise.
\item Same question, but over $\bbZ/3$.
\end{enumerate}

\vskip 2mm\noindent{\bf Problem 4.} The statement ``all reasonable everyday spaces are at least homotopy equivalent to CW complexes'' sounds completely reasonable. At least until you hit the first example where it's hard.

Show that the complement $X$ of the trefoil knot {\huge$\righttrefoil$} in $\bbR^3$ is homotopy equivalent to a 3-dimensional CW complex.

\end{minipage}\end{center}

\end{document}


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