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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 2}
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  \null\hfill\sheeturl
  \newline\null\hfill Due January 21 at 11:59pm
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\begin{center}\begin{minipage}{6.5in}

\parpic[r]{\includegraphics[width=2in]{../25-1301-AlgebraicTopology/ConnectedButNotLocally.png}}
{\bf Problem 1. } A topological space $X$ is called ``locally path connected'' if every point in it has arbitrarily small neighborhoods that are path connected. Namely, if $x\in X$ and if $U$ is a neighborhood of $x$, then there is a path connected open set $V$ such that $x\in V\subset U$.

{\em On the subject of liftings:} Prove that if $X$ is path connected, locally path connected, and simply connected, and if $p\colon(E,e_0)\to(B,b_0)$ be a covering map, then every $f\colon(X,x_0)\to(B,b_0)$ has a unique lift to a map $\tilde{f}\colon(X,x_0)\to(E,e_0)$ such that $\tilde{f}\act p=f$.

{\em Hint.} Lift paths, lift endpoints of paths, worry about well-definedness, worry about continuity.

{\em On the right.} A space that is path-connected but not locally path-connected.

\picskip{0}\vskip -1.9in
\vskip 3mm{\bf Problem 2. } With the obvious assumptions and definitions, prove that $\pi_1((X,x_0)\times(Y,y_0))\cong\pi_1(X,x_0)\times\pi_1(Y,y_0)$.

\vskip 3mm{\bf Problem 3. } Prove that if $\gamma\colon S^1\to S^1$ is even (meaning, $\forall z\in S^1\ \gamma(-z)=\gamma(z)$) then $[\gamma]$, an integer, is even. Likewise, prove that if $\gamma\colon S^1\to S^1$ is odd (meaning, $\forall z\in S^1\ \gamma(-z)=-\gamma(z)$) then $[\gamma]$ is odd.

{\em Hint.} Re-interpret $\gamma$ as a map with domain $[0,1]$ with some symmetry property. In both cases, $\gamma$ is determined by its values on $[0,\frac12]$, and if the lift $\bar\gamma$ of $\gamma|_{[0,\frac12]}$ is known then the full lift $\tilde{\gamma}$ of $\gamma|_{[0,1]}$ is determined by it. What can you say about $\bar\gamma(\frac12)$? What does it say about $\tilde{\gamma}(1)$?

\parpic[r]{\includegraphics[width=2.5in]{../../2014-08/17Worlds/EscherMobius_800.jpg}}
\vskip 3mm{\bf Problem 4. } Let $M$ denote the M\"obius band.
\begin{enumerate}
\item Show that $\pi_1(M)\cong\bbZ$.
\item Is there a retraction from $M$ to its boundary?
\end{enumerate}

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