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{\bf Math 177 HW 8 Due ??}
\bigskip

\begin{enumerate}

\setcounter{enumi}{49}

\item Suppose that $X$ and $Y$ are spaces (that is, each is a subset of some Euclidean space) and that $f$ and $g$ are continuous functions from $X$ to $Y$. We say that $f$ is {\it homotopic} to $g$ if there exists a continuous function $H:X\times I\to Y$ such that $H(x,0)=f(x)$ for all $x\in X$ and $H(x,1)=g(x)$ for all $x\in X$. We write $f\sim g$ and call $H$ a {\it homotopy} between $f$ and $g$. We can jazz this up a little bit as follows: Suppose $A$ is a subset of $X$ and furthermore that $H(a, s)=f(a)=g(a)$ for all $a\in A$ and for all $s\in I$. In this case we say that $f$ is homotopic to $g$ {\it relative to $A$} and we write $f\sim g \mbox{ rel } A$. This definition extends our definition of homotopic loops. In the case of loops, we have $X=I$, $A=\{0,1\}$, and we require that $f$ and $g$ are loops, that is $f(0)=f(1)=g(0)=g(1)$.

Show that $\sim$ is an equivalence relation on the set of continuous functions from $X$ to $Y$.

\item Suppose that $f_1$ and $f_2$ are homotopic maps from $X$ to $Y$ and that $g_1$ and $g_2$ are homotopic maps from $Y$ to $Z$. Show that the composition $g_1\circ f_1$ is homotopic to the composition $g_2 \circ f_2$.

\item Suppose that $f:X\to Y$ is a continuous map from the space $X$ to the space $Y$ taking the point $x_0\in X$ to the point $y_0=f(x_0)\in Y$.  We can use $f$ to obtain a map  $f_*:\pi_1(X, x_0)\to \pi_1(Y,y_0)$ by
defining $f_*$ as $f_*([\alpha])=[f\circ \alpha]$. Show that:
\begin{enumerate}
\item $f_*$ is well-defined.
\item $f_*$ is a homomorphism.
\item If $f, g:X\to Y$ are homotopic maps relative to $x_0$, then $f_*=g_*$.
\item If $f:X\to Y$ and $g:Y\to Z$ and $z_0=g(y_0)$ and $y_0=f(x_0)$, then $(g\circ f)_*=g_*\circ f_*$ (where these are maps from $\pi_1(X, x_0)$ to $\pi_1(Z,z_0)$).
\end{enumerate}


\item Suppose that $X$ and $Y$ are spaces. A function $f:X \to Y$ is a {\it homeomorphism} if $f$ is a continuous bijection and $f^{-1}$ is also continuous. A cool way to say this is that $f$ is a ``bicontinuous bijection.'' Notice that $f\circ f^{-1}:Y\to Y$ is equal to the identity function $1_Y$ on $Y$ and $f^{-1}\circ f:X\to X$ is equal to the identity function $1_X$ on $X$ (by definition of the inverse of a function). We can relax this and consider  functions $f:X\to Y$ and $g:Y \to X$ such that the compositions $f\circ g$ and $g\circ f$ are not necessarily the identity, but just homotopic to the identity. That is, we want  $f\circ g\sim 1_Y$ and $g\circ f\sim 1_X$. In this case we say that  both $f$ and $g$ are {\it homotopy equivalences} and we say that $X$ and $Y$ are {\it homotopy equivalent}.
\begin{enumerate}
\item Show that homotopy equivalence is an equivalence relation on the set of spaces.
\item Show that if $f:X\to Y$ is a homotopy equivalence then $f_*:\pi_1(X,x_0)\to \pi_1(Y, f(x_0))$ is an isomorphism.
\end{enumerate}

\item Suppose that $X$ is a space and that $A$ is a subset of $X$. A {\it retraction} from $X$ to $A$ is a continuous map $r:X\to A$ such that  $r(a)=a$ for all $a \in X$. We say that $A$ is a {\it retract} of $X$. 
\begin{enumerate}
\item Show that   the following are retractions:
\begin{enumerate}
\item Let $X$ be any space and $x_0\in X$ any point in $X$. Let $r:X\to x_0$ be the map that sends every point of $X$ to $x_0$.
\item Let $X$ and $Y$ be any spaces and let $r:X \times Y\to X$ be defined as $r((x,y))=x$.
\end{enumerate}
\item If $r:X\to A$ is a retraction sending $x_0$ to $a_0$, show that $r_*:\pi_1(X, x_0)\to \pi_1(A, a_0)$ is a surjection.
\item Let $B^2=\{(x,y)\in \mathbb R^2\,|\,x^2+y^2\le1\}$ and let $S^1$ be the subset of $B^2$ defined as $S^1=\{(x,y)\in \mathbb R^2\,|\,x^2+y^2=1\}$. Show that there cannot be a retraction from $B^2$ to $S^1$.
\end{enumerate}

\item Let $X$ be a space, $A$ a subset, and $i:A\to X$ the inclusion map $i(a)=a$. We say that $A$ is a {\it deformation retract} of $X$ and that $X$ {\it deformation retracts} to $A$ if there is a retraction $r:X \to A$ and a homotopy between the identity function $1_X$ on $X$ and $i\circ r$. 
\begin{enumerate}
\item Suppose that $X$ is a convex subset of $\mathbb R^n$ and that $x_0$ is any point in $X$. Show that $X$ deformation retracts to $x_0$. A space that deformation retracts to a point is called {\it contractible}.
\item  Suppose that $X$ and $Y$ are spaces, $Y$ is contractible, and $y_0$ is a point in $Y$. Show that $X\times Y$ deformation retracts to $X\times \{y_0\}$.
\item Suppose that $T$ is a finite connected tree contained in $\mathbb R^2$. That is, $T$ is a connected graph with a finite number of vertices and edges that contains no loops. Show that $T$ is contractible.
\end{enumerate}
 
 \item Suppose that  $X$ deformation retracts to $A$, with retraction $r:X\to A$. Show that $r$ is a homotopy equivalence. 
 
 \item A {\it bouquet of $n$ circles} is a subset of $\mathbb R^2$ consisting of $n$ ``circles'' such that each circle contains the origin and the intersection of all the circles is the origin. A bouquet of six circles is shown below:
 \begin{center}
 \includegraphics[scale=.5]{bouquet.pdf}
 \end{center}
 Alternatively, if you really want to use geometrically round circles, we can picture the bouquet as seen below:
  \begin{center}
 \includegraphics[scale=.5]{bouquet2.pdf}
 \end{center}
 Another way to think about a bouquet of $n$ circles is that it is a graph with one vertex and $n$ edges, each of which is a loop with both of its endpoints at the single vertex.
 
Let $\Gamma$ be any connected graph with $v$ vertices and $e$ edges. Show that $\Gamma$ is homotopy equivalent to a bouquet of $e-v+1$ circles.
\end{enumerate}
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