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{\bf Math 444/539 HW\#2, due Monday 9/9/19} \hspace{.5in} \textbf{NAME:} \hspace{4in}
%outlier malcom
\begin{enumerate}
\item Show that a space $X$ is simply connected if and only if there exists a unique homotopy class of paths connecting any two points in $X$.
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\item Show that for a space $X$, the following three conditions are equivalent:
\begin{enumerate}
\item Every map $S^1\to X$ is homotopic to a constant map, with image a point.
\item Every map $S^1\to X$ extends to a map $D^2\to X$.
\item $\pi_1(X,x_0) = 0 $ for all $x_0 \in X$.
\end{enumerate}
Deduce that a space $X$ is simply-connected if and only if all maps $S^1 \to X$ are homotopic. [In this problem, `homotopic' means `homotopic without regard to basepoints.']
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\item Let $\varphi: X \to Y$ be a continuous map and let $\gamma$ be a class of paths in $X$ from $x_0$ to $x_1$. Prove that the following diagram is commutative:
\[ \begin{tikzcd}
\pi_1(X,x_0) \arrow{r}{\varphi_*} \arrow[swap]{d}{u} & \pi_1(Y,\varphi(x_0)) \arrow{d}{v} \\%
\pi_1(X,x_1)\arrow{r}{\varphi_*}& \pi_1(Y,\varphi(x_1))
\end{tikzcd}
\]
The isomorphism $u$ is defined by $u(\alpha)=\gamma^{-1} \alpha \gamma$ and $v$ is defined similarly using $\varphi_*(\gamma)$ instead of $\gamma$. \emph{Note: A important special case occurs if $\varphi(x_0) =\varphi(x_1)$. Then, $\varphi_*(\gamma)$ is an element of the group $\pi_1(Y,\varphi(x_0))$.}
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\item Show that if $G$ is a topological group (a topological space with a group structure such that inversion and multiplication are continuous), then $\pi_1(G,1)$ is abelian.
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\item Prove that $\R^2$ and $\R^n$ are not homeomorphic if $n \neq 2$. \emph{Hint: Consider the complement of a point in $\R^2$ or $\R^n$. You do not need to prove $\pi_1(S^n) = 0$ for $n \geq 2$.}
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\item[*math:] Let $\{ U_i\}$ be an open covering of the space $X$ having the following properties:
\begin{enumerate}
\item There exists $x_0 \in X$ such that $x_0 \in U_i$ for all $i.$
\item Each $U_i$ is simply connected.
\item If $i \neq j$, then $U_i \cap U_j$ is path connected.
\end{enumerate}
Prove that $X$ is simply connected. \\
\noindent \emph{Hint: To prove that any loop $f: I \to X$ based at $x_0$ is trivial, consider the open covering $\{ f^{-1}(U_i)\}$ of the compact metric space $I$ and use the Lebesgue number\footnote{See page 48 of Massey for more on the Lebesgue number of coverings of intervals.} of this covering. } \\
\noindent Remark: The two most important cases of this exercise are (1) a covering by two open sets and (2) when the sets $U_i$ are linearly ordered by inclusion.
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\item[*math:]
Using the result of the previous exercise in the case that there is a covering consisting of two open sets, prove that the $n$-sphere, $S^n$, for $n\geq 2$ is simply connected. \\
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\item[everyone:] How difficult was this assignment? How many hours did you spend on it?
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\end{enumerate}
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