\documentclass[12pt]{article}
\usepackage{amsmath, amssymb}
\usepackage{graphicx, stmaryrd, datetime, amscd, multicol, txfonts, mathabx, tensor, needspace, mathtools, ../picins}
\usepackage[margin=0.5in,foot=0.1in]{geometry}
\usepackage[setpagesize=false]{hyperref}
%\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
\usepackage{soul}\setstcolor{red}

\def\myurl{http://www.math.toronto.edu/~drorbn/}

\def\red{\color{red}}

\def\bbe{\mathbb{e}}
\def\bbi{\mathbb{i}}
\def\bbC{{\mathbb C}}
\def\bbF{{\mathbb F}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbV{{\mathbb V}}
\def\bbZ{{\mathbb Z}}
\def\calS{{\mathcal S}}
\def\Aut{\operatorname{Aut}}

\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#2425}{2024-25}:
  \hfill\url{http://drorbn.net/25-1301}
}

\vskip 6mm
\noindent{\bf Do not turn this page over until instructed.}
\vskip 6mm
\begin{center}
  \href{http://drorbn.net/25-1301}{\large MAT 1301 Algebraic Topology}
  \vskip 6mm
  {\LARGE Final Exam}
  \vskip 6mm
  University of Toronto, April 9, 2025
  \vskip 5mm{\bf Respond to all 4 tasks on the other side of this page. }\\
  The tasks are of equal weight.\\You have three hours to write this test.
\end{center}

\vfill

\noindent{\bf Notes}
\begin{itemize}
\item No outside material allowed other than stationery, minimal hydration and snacks, and stuffed animals.
\item Do not write on this form.
\item Three of the four tasks on this form are not-fully-specified ``write an essay'' tasks. This is a graduate class, and one of the greatest challenges you will be facing later on will be to free-style write rather than demonstrate to a TA that you've memorized some material. So the essay tasks do not specify precisely what you need to do -- it is up to you to decide what is important and what isn't, which figures to draw and when are they superior to formulas, which formulas to write and when are they superior to figures, etc. Think that you really are writing a paper and you really are trying to transmit information that will be read and understood.
\item {\bf Neatness counts! Language counts!} You may want to write ``draft essays'' and only then write the final versions of your solutions. Messy essays cannot be read.
\item {\red {\bf In Red.} Comments added after the test took place.}
\end{itemize}

\vfill\begin{center} {\bf Good Luck!} \end{center}\vfill\eject

\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#2425}{2024-25}:
  \href{http://drorbn.net/25-1301}{MAT 1301 Algebraic Topology}:
  \hfill\url{http://drorbn.net/25-1301}
}

\vfill\parpic[r]{\input{figs/OlympicRings.pdf_t}}
\noindent{\bf Task 1. } 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 $b$: $G\coloneqq\pi_1(8^a_b)=F(a,b)$.

\picskip{3}
\quad A. In your examination notebook (not here), describe the right $G$-set $S$ corresponding to \st{$8^a_b$} {\red$X$}: $S$ 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 $G$?

\vfill\noindent{\bf Task 2. } Write an essay on ``the prism construction''. What is it? How is it used? Who cares?

\vfill\noindent{\bf Task 3. } Write an essay on ``the transfer sequence for the $\bbZ/2$ homology of a double cover and the homology of projective spaces''. What is it? How is it used? Who cares?

\vfill\noindent{\bf Task 4. } Write an essay on ``the degrees of maps $S^n\to S^n$''. What are they? What are their properties? Who cares?

%\vfill\noindent{\bf Task 4. } Suppose $n$ is even.
%
%\quad A. Show that for any continuous map $f\colon S^n\to S^n$ there is a point $x$ such that $f(x)=\pm x$.
%
%\quad B. Show that any continuous map $g\colon\bbR P^n\to\bbR P^n$ has a fixed point.

\vfill\begin{center} {\bf Good Luck!} \end{center}\vfill

\end{document}

