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\centerline{\bf Algebraic Topology}
\centerline{Homework 6: Due October 8}      % e.g. September 13
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1. Page 94, problems 3.1 and 3.2.  Both of these were essentially done in
class, but spell them out anyway. 


2. As noted in class, the {\it Hawaiian Earring} is the union of
infinitely many circles of decreasing diameter, all meeting at a
point. For instance, for each positive integer $n$ let $S_{n}$ be a
circle of radius $1/n$ in the plane, centered at $(1/n,0)$, and let $X
= \cup_{n} S_n$. (It helps to DRAW A PICTURE!)  Let $x_0=(0,0)$. Show
that this example does not meet the hypotheses of Exercise 3.2.  (Again,
already more or less done in class.)

3. Not only does the Hawaiian Earring not meet the hypotheses of
Exercise 3.2, it also doesn't meet the conclusions. Show that the
inclusions $i_n: S_n \hookrightarrow X$ induce injections $\pi_1(S_n,
x_0) \to \pi_1(X,x_0)$, and hence an injection $f: \prod^*
\pi_1(S_n,x_0) \hookrightarrow \pi_1(X,x_0)$, but that $f$ is not
onto. In other words, $\pi_1(X,x_0)$ is {\it not} the free product of
all of the cyclic groups $\pi_1(S_n,x_0)$.  [Hint: Can every loop in
$X$ be expressed by a finite concatenation of loops in the $S_n$'s?]

4. Page 94, problem 3.4.  Hint: with the correct choice of open sets, 
this is an easy corollary of problem 3.2. 

5. Page 95, problem 3.7. 


5. Page 103, problem 5.4.

6. Let $X_2$ be a path-connected Hausdorff space, 
let $k>2$, and let $f:S^{k-1}\to X_2$
be a continuous map. Let $B$ be the closed unit ball in $\R^k$, and for
each $x \in S^{k-1} \subset B$, identify $x$ with $f(x) \in X_2$. Let $X$
be the union of $X_2$ with $B$, modulo these identifications. This is called
{\it adjoining a $k$-cell to $X_2$.} Let $x_0 \in X_2$.
Show that $\pi_1(X,x_0)$ is isomorphic to $\pi_1(X_2,x_0)$. 

7. Let $X_1$ be another path-connected Hausdorff space, and let $X_2$ be
obtained by adjoining a 2-cell to $X_1$. Let $x_0 \in X_1$.
Show that the inclusion $X_1 \hookrightarrow X_2$ induces a surjection
$\pi_1(X_1,x_0) \to \pi_1(X_2,x_0)$. Give an example where this surjection is 
not an isomorphism. 

The importance of exercises 6 and 7 is that it is possible to build
any manifold, and a lot of other spaces, recursively. To get a space
$X$, start with a set $X_0$ of disconnected points, called the
0-skeleton of $X$.  Then add 1-cells to get a graph $X_1$ that is
called the 1-skeleton.  Then add 2-cells to $X_1$ to get $X_2$, add
3-cells to $X_2$ to get $X_3$, and in general add $k$-cells to the
$k-1$ skeleton $X_{k-1}$ to get the $k$-skeleton $X_k$. Finally, take
$X = \cup_k X_k$. The upshot is that $\pi_1(X)$ is isomorphic to
$\pi_1(X_2)$, and is a quotient of $\pi_1(X_1)$ by some relations that
come from the 2-cells.  This is both a blessing and a curse.  The
blessing is that we don't have to worry about higher dimensional
structures when computing a fundamental group. The curse is that we
can't use the fundamental group to keep track of higher dimensional
structures. To do that we need other tools, such as homology.
 
\bye
