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\begin{document}
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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 1}
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due Wednesday January 14 at 11:59pm
}~%
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/26-1301}
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\begin{center}\begin{minipage}{6.5in}

\vskip 3mm{\bf Problem 1. } Write explicit formulas for the homotopy in \url{https://drorbn.net/bbs/show?shot=26-1301-260105-110227.jpg} between $e$ and $\gamma\bar{\gamma}$ (and if that picture is wrong, fix it in your mind first). Your solution should take up 3 lines and must be of the form:
\[
  h(t,s)=\begin{cases}
    \text{formula 1} & \text{condition 1} \\
    \text{formula 2} & \text{condition 2} \\
    \text{formula 3} & \text{condition 3} 
  \end{cases}
\]

\vskip 3mm{\bf Problem 2. } Prove the theorem which is implicit in the definition at \url{https://drorbn.net/bbs/show?shot=26-1301-260106-163930.jpg} and \url{https://drorbn.net/bbs/show?shot=26-1301-260106-163954.jpg}. Namely, prove that if $X$ is path-connected then the following are equivalent:
\begin{enumerate}
\item $\pi_1(X,x_0)=0$ for some/any $x_0\in X$.
\item If $\gamma_0$ and $\gamma_1$ are not-necessarily-closed paths that share their endpoints, namely $\gamma_0(0)=\gamma_1(0)=x$ and $\gamma_0(1)=\gamma_1(1)=y$, then they are homotopic via a homotopy that does not move these endpoints.
\item Any two maps $S^1\to X$ are homotopic.
\end{enumerate}

\vskip 3mm{\bf Problem 3. } If $X$ is path-connected, prove that the set of homotopy classes of maps $S^1\to X$ can be put in a bijection with the set of conjugacy classes in the fundamental group of $X$.

\end{minipage}\end{center}

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