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\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/\#2526}{2025-26}: \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 3}
\hfill\parbox[b]{1.5in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due February 13 at 11:59pm
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{\bf Problem 1}  (30 points). Let $K$ be a knot in $\bbR^3$ presented by a planar diagram $D$. With a massive use of van Kampen's theorem, show that the fundamental group of the complement of $K$ has a presentation (the ``Wirtinger presentation'', as discussed in class) with one generator for each edge of $D$ and two relations for each crossing of $D$, as indicated in the figure below.
\[ \includegraphics[width=0.7\linewidth]{WirtingerExample.png} \]
(There will be a brief discussion of this question in class on Monday Feb 9).

\vskip 2mm
\noindent{\bf Problem 2} (15 points). The trefoil knot above, whose fundamental group is $G_1=\langle \alpha,\beta,\gamma\colon \alpha=\gamma^\beta,\, \beta=\alpha^\gamma,\, \gamma=\beta^\alpha\rangle$ is in fact the torus knot $T_{3/2}$, whose fundamental group, as computed in class, is $G_2=\langle \lambda,\mu\colon \lambda^2=\mu^3\rangle$. Prevent the collapse of mathematics by showing that these two groups are isomorphic.

\parpic[r]{\import{../25-1301-AlgebraicTopology}{figs/OlympicRings.pdf_t}}
\noindent{\bf Problem 3} (10 points). Let $X$ be the ``Olympic Rings'' covering of the figure 8 space, $8^a_b$, whose basepoint is taken to be at the quadrivalent vertex in its centre and whose fundamental group is the free group on two letter $a$ and $b$: $G\coloneqq\pi_1(8^a_b)=F(a,b)$.

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\quad A. Describe the right $G$-set $S$ corresponding to $8^a_b$: it is a set with $\underline{\quad}$ elements, and $a$ and $b$ act on it as the permutations $\underline{\qquad}$ and $\underline{\qquad}$.

\quad B. Taking the basepoint $x_1$ of $X$ to be the point marked as ``1'' on the right, write a set of generators for the image $H$ of $\pi_1(X,x_1)$ within $G\coloneqq\pi_1(8^a_b)$.

\quad C. Is $H$ a normal subgroup of $G$?

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%\noindent{\bf Problem 4} (10 points). Describe all the 2-sheeted and 3-sheeted connected coverings of the figure 8 space, $8^a_b$. (Meanings, all the connected coverings that are 2 to 1 or 3 to 1).

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\noindent{\bf Problem 4} (10 points). Prove Corollary 11 from the Covering Spaces handout: If $X$ is a connected covering of a nice space $B$ (meaning, $B$ is connected, locally connected and semi-locally simply connected) and $H \coloneqq p_\star\pi_1(X) < G \coloneqq \pi_1(B)$, then $\operatorname{Aut}(X)=N_G(H)/H$ where $N_G(H) \coloneqq \{g\in G\colon H=g^{-1}Hg\}$ is the normalizer of $H$ in $G$.

\vskip 2mm
\noindent{\bf Problem 5} (10 points). Describe the universal covering space $U$ of the space $B$ which is the union of a 2-dimensional sphere and one of its diameter lines. (Don't say ``it's the space of spelunkers'' -- you are expected to give a concrete description of $U$ as some familiar space or as a simple subset of some familiar space).

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\noindent{\bf Problem 6} (10 points). If $B$ is a nice space and $U$ its universal cover, show that $U$ is a covering of every connected covering $X$ of $B$.

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