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\def\ctln{\centerline}
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\def\ra{\rightarrow}
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\ctln{\bf Math 872 Problem Set 2}

\msk

Starred (*) problems are due Thursday, February 5.


\begin{description}

\item{9.} [Hatcher p.40, \#20] If $H:X\times I\ra X$ is a homotopy
with $H_0=H_1=$ the identity map, show that
$\gamma(t)=H(x_0,t)$ is a loop representing an element
$g=[\gamma]\in\pi_1(X,x_0)$ lying in the center of $\pi_1(X,x_0)$.,
i.e., $gh=hg$ for all $h\in\pi_1(X,x_0)$

\msk

\item{(*)} 10. Show that $\langle a,b\ |\ a^5=b^4,aba=bab\rangle$ is a(nother) presentation
for the trivial group (that is, show that the relations imply that $a=1$ and $b=1$).

\msk

\item{11.} (a) Show that the group $\langle a\ |\ a^n\rangle$ is isomorphic to the cyclic group 
$\bbz_n$ on $n$ elements.

\ssk

\item{\spc} (b) Show that the group $G = \langle a,b,c\ |\ a^2, b^2, c^2, (ab)^3, (ac)^2, (bc)^3\rangle$
is isomorphic to the symmetric group $S_4$ on four letters. [You can/should model this on our demonstration
for $S_3$; find a surjective homomorphism to $S_4$, and show that $G$ has at most 24 elements!]

\msk

\item{(*)} 12. [Hatcher, p.52, \#1] Show that the free product $G*H$ of non-trivial
groups has trivial center; if $x\in G*H$ and $xy=yx$ for every $y\in G*H$, then 
$x=1$. 

\ssk

\item{\spc} [Hint: show that if $x\neq 1$ then there
is an element that it \underbar{doesn't} commute with! 
Think of group elements as products alternately in $G$ and $H$.]

\msk

\item{13.} Show that if $G=\langle A\ |\ R\rangle$, $x\not\in A$, and $w$ is a word in
the alphabet $A$, then $H=\langle A\cup\{x\}\ |\ R\cup\{xw\}\rangle$ is 
isomorphic to $G$. That is, build homomorphisms $G\ra H$ and $H\ra G$ that are
inverses of one another.

\msk

\item{\spc} [This is one of the two {\it Tietze transformations}, that describe how to 
change presentations without changing the group.]

\msk

\item{14.}  Show that $X=\bbr^2\setminus\bbq^2\subseteq \bbr^2$ is path connected,
and $\pi_1(X)$ is uncountable. (I.e., find uncountably many loops no two of which are
homotopic to one another.)

\msk

\item{(*)} 15. Compute (i.e., find a presentation for) the fundamental group of the 
complement $X$ of the coordinate 
axes in $\bbr^3$, i.e., $X=\{(x,y,z)\in \bbr^3\ :\ |xy|+|xz|+|yz|\neq 0\}$. 

\ssk

\item{\spc} [Hint: Find a `simpler' space, homotopy equivalent to $X$, to compute the fundamental group of.]

\msk

\item{16.} Show that if $f:\del D^n\ra X$ is the attaching map of an $n$-cell
$D^n$, with $n\geq 3$, then the inclusion $X\hookrightarrow X\cup_f D^n$ induces
an isomorphism on $\pi_1$. Show that the same is true if we attach any
(finite or infinite) collection of $n\geq 3$ cells. 

\msk

\end{description}
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