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\begin{center}
{\Large Intro to Topology -- HW 9/10.}
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\bigskip\noindent
\begin{enumerate}

\item Using the examples of various fundamental groups we talked about (some intuitively) so far, find topological spaces $X$, $Y$ and $f:X\to Y$  a continuous map to show that
even if $f$ is surjective, the induced homomorphism
$f_*:\Pi(X, x_0)\to \Pi(Y,f(x_0))$ need not be surjective.

Also, even if  $f$ is injective,  $f_*$ need not be injective.





\item  I.a.) Is there a retraction $r:\R\to [0,1]$? \newline
1.b.) Is there a  retraction $r:\R\to (0,1)$? \newline
1.c.) Is there a  retraction $\R^2\to \overline{D}^2$? \newline
1.d.) Is there a  retraction $\R^2\to S^1$?\newline


\smallskip\noindent
II. For the spaces below show explicitly (i.e. providing formulae for $G$)
 that there is a deformation retraction deforming $X$ onto some $A\sim S^1$ of your choice\newline
a.) $X=S^1\cup \{ (x,0)\, |\,x>0\}\ \subset\ \R^2$\newline
b.) $X=\{ (x,y,u,w)\, |\, x^2+y^2\leq 1, u^2+w^2=1 \}$ in $\R^4$. 
Also, what is $X$?

\item Consider $X=S^1$, the unit circle centered at the origin in $\R^2$ and fix $x_0=(1,0)$.
Choose an integer $m\in\Z$. For that integer provide a formula for a loop (of your choice)
based at $x_0$ which is homotopic to the "standard loop" $\beta_m(s)=(\cos 2\pi ms, \sin 2\pi ms)$, 
but is not equal to it.

Prove that your loop is indeed homotopic to $\beta_m$.





\item We will define covering maps on Tuesday.
Which of the following are covering maps? Explain briefly.\newline
a.) The quotient map $q:\R^2\to \R^2/\sim$ where $(x,y)\sim (x',y')$ if $x-x'\in\Z$ and $y=y'$.\newline
b.) The quotient map $q:\R^2\to\R^2/\sim$ where $(x,y)\sim (x',y')$ if
$xx'>0$ and $y=y'$.\newline
c.) The $n$-th power mapping of $\C\to \C$ given by
$z\to z^n$. (Here $z$ denotes a complex number.)\newline
d.) The $n$-th power mapping of $S^1\to S^1$ given by
$z\to z^n$. (Here $S^1=\{ z\in\mathbb{C}\ |\ ||z||=1\}$.)\newline



 \smallskip
{\it In part d.) and e.) I use complex numbers since then the maps
have a simple description. But you can use if you wish $\R^2$ instead of $\C$,
the unit circle in it and the maps
$z=(x,y)=(r\cos\theta, r\sin\theta)\to (r^n\cos\, n\theta, r^n\sin\, n\theta)=z^n$.}\newline


\item  Show that if $p: Y\rightarrow X$ is a covering map
and $X$ is connected, then the sets $p^{-1}(x)$ have the same
cardinality for every $x\in X$.


\smallskip
{\it (Hint:
Fix $x_0\in X$, let $|p^{-1}(x_0)|=\lambda$
 and consider the set\newline $A=\{ x\in X\, |\, |p^{-1}(x)|=\lambda\}$.
Show that $A$ is both open and closed.)}




\item  The next formula is very useful to know:
$$\pi_1(X\times Y, (x_0,y_0))\simeq\pi_1(X,x_0)\times \pi_1(Y,y_0)$$

a.) What is the isomorphism between the two groups? No proof is necessary, but
write a formula for the candidate and its inverse.

\noindent{\it With a good/proper notation, this is a straightforward problem. Make sure you state clearly what your functions/maps are.} 


b.) 
Show that the following two spaces, $Z$ and $W$, are not homeomorphic by finding their fundamental groups.

Let $Z$ be the quotient space of $[0,1]\times [0,1]\times [0,1] $,
where we set $(x,0,z)\sim (x, 1,z)$, $(0,y,z)\sim (1,y,z)$
and $(x,y,0)\sim (x,y,1)$.

On the other hand $W$ is the quotient space of 
$[0,1]\times [0,1]\times [0,1] $ where we set
$(x,0,z)\sim (0,x,z)$, $(x,1,z)\sim (1,x,z)$
and $(x,y,0)\sim (x,y,1)$.

{\it (No precise proofs are necessary.)}


\item[Extra Credit]   When proving that the fundamental group $\pi_1(X,x_0)$ is a group indeed, one has to show (among other facts) that  $<x_0>$ is a
left unit of the fundamental group $\Pi_1 (X, x_0)$, that is  $<x_o>*<\alpha>=<\alpha>$ for all $\alpha$ loops
of a topological space $X$ based at $x_0$. 
I posted a podcast proving this.
Have a look at it, then prove
 that  $<x_0>$ is also a
right unit of the fundamental group, that is \newline $<\alpha>*<x_0>=<\alpha>$ for all $\alpha$ loops based at $x_0$.


\end{enumerate}

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