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\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}:
  \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}:
  \newline \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 8}
\hfill\parbox[b]{1.5in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due April 3 at 11:59pm
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/26-1301}
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\noindent{\bf Problem 1} (5 points). For $n>0$, construct a surjective map $f\colon S^n\to S^n$ whose degree is 0.

\vskip 2mm\noindent{\bf Problem 2} (10 points). Let $n>0$.
\begin{enumerate}
\item Use degree theory to show that any non-surjective map $g\colon S^n\to S^n$ must have a fixed point.
\item Use the above to re-prove the Brouwer fixed point theorem: Any map $f\colon D^n\to D^n$ has a fixed point. (Hint: Make a $g$ out of $f$ by mapping both the upper hemisphere of $S^n$ and the lower hemisphere of $S^n$ to the lower hemisphere using $f$.)
\end{enumerate}

\vskip 2mm\noindent{\bf Problem 3} (10 points). 
\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} (10 points). 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.

\vskip 2mm\noindent{\bf Problem 5} (20 points). For the first 3 parts of this problem, sketches are sufficient.
\begin{enumerate}
\item Using the same ideas as in the case of $S^2\to S^2$, define the degree of a map $T^2\to S^2$, where $T^2=S^1\times S^1$ is the two dimensional torus. Similarly define the local degree $\deg_x(f)$ when $f\colon T^2\to S^2$ when $x\in T^2$ is isolated in $f^{-1}(f(x))$.
\item Show that the degree of maps $T^2\to S^2$ is invariant under homotopy of such maps.
\item Show that the ``local formula'' for the degree, $\deg(f)=\sum_{x\in f^{-1}(y)} \deg_x(f)$, if $y\in S^2$ and $F^{-1}(y)$ is finite, holds for $f\colon T^2\to S^2$.
\item A pair of curves $\gamma_1,\gamma_2\colon S^1\to \bbR^3$ whose images are disjoint induces a ``direction of sight'' map $\lambda\colon T^2\to S^2$ by $(s_1,s_2)\mapsto\frac{\gamma_1(s_1)-\gamma_2(s_2)}{|\ \cdot\ |}$. Show that $\ell(\gamma_1,\gamma_2)\coloneqq\deg(\lambda)$, the so-called ``linking number of $\gamma_1$ and $\gamma_2$'', is invariant under homotopies of the pair $(\gamma_1,\gamma_2)$ that preserve the disjointness of their images.
\item Show that $\ell(\gamma_1,\gamma_2)=\ell(\gamma_2,\gamma_1)$.
\item Compute $\ell$ for the following pairs (orient each loop counterclockwise yet pick whatever orientation you want for the $\infty$-like loop):
\[
  \adjustbox{height=0.5in,valign=m}{$\blue\bigcirc\purple\bigcirc$}
  \qquad\includegraphics[valign=m,height=0.5in]{../../Projects/Theta/LinkFigs/L_2a_1.pdf}
  \qquad\includegraphics[valign=m,height=0.5in]{../../Projects/Theta/LinkFigs/L_4a_1.pdf}
  \qquad\includegraphics[valign=m,height=0.5in]{../../Projects/Theta/LinkFigs/L_5a_1.pdf}
\]
\end{enumerate}

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