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\centerline{\bf Algebraic Topology}
\centerline{Homework 4: Due September 22}      % e.g. September 13
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1. Page 53, problem 7.3. Note that ``infinite product'' means ``with the 
product topology''. Except where specifically noted, infinite products
always have the product topology.

2. To see how evil alternatives to the product topology are, 
consider $X = S^1\times S^1 \times \cdots$ 
with the {\it box} topology, where
for definiteness we're taking $S^1$ to be the unit circle in the complex plane
and taking the index set to be the natural numbers. 
(In the box topology, the set $U_1 \times U_2 \times \cdots$ is open if each 
$U_i$ is open. Those infinite boxes form a base for the topology.)
Give a characterization of all the continuous loops in $X$, and 
use this characterization to compute $\pi_1(X,x)$, where $x=(1,1,1,\ldots)$. 

3. Let $Z[1/2]$ denote the set of rational numbers whose denominators
are powers of 2. (This includes integers, since $1=2^0$.)  Show that
$Z[1/2]$ is an Abelian group under addition. Then show, in two ways,
that $Z[1/2]$ is not finitely generated: First show this directly, by
showing that the subgroup generated by any finite set of elements of
$Z[1/2]$ cannot be the entire group. Second, show that $Z[1/2]$ does
not satisfy the conclusions of Theorem 3.6 on page 70 (the
classification theorem for finitely generated Abelian groups).

4. The classification theorem for finitely generated Abelian groups is
stated in different ways in different books. One formulation says that
each such group can be written as $Z^n \oplus Z_{a_1} \oplus \cdots
\oplus Z_{a_k}$, where each $a_i$ is a (positive) power of a prime,
and that the numbers $a_i$ are unique, up to permutation. Another
formulation says that the group can be written as $Z^n \oplus Z_{t_1}
\oplus \cdots \oplus Z_{t_\ell}$, where each $t_i$ divides $t_{i+1}$,
and that the numbers $t_i$ are unique. Give an algorithm for
converting from each of these canonical forms to the other.  Then use
this algorithm to show that each formulation of the theorem implies
the other. (Note: you do not have to prove either formulation from scratch.
Just show that each implies the other.)

5. Let $C$ be a category. Two objects $X,Y$ in $C$ are called ``isomorphic''
if there are morphisms $f: X \to Y$ and $g: Y \to X$ such that $f\circ g$ is
the identity morphism on $Y$ and $g \circ f$ is the identity on $X$. 
An object $Z$ in $C$ is called ``initial'' or
``universal'' if, for every object $Y$ in $C$, there is a unique
morphism $Z \to Y$. For instance, in the category of groups and group 
homomorphisms, the trivial group is universal. In the category of sets and
functions, the empty set is universal. 

Show that any two universal objects are isomorphic, and
show that any object that's isomorphic to a universal object is also universal.
In other words, the universal object is ``unique up to isomorphism''.

[The dual of a universal object is a ``terminal'' or ``final''
object. An object $T$ is final if, for each object $Y$, there is a unique
morphism $Y \to T$.  The proof you just gave, with arrows reversed, shows that
final objects are unique up to isomorphism. 

To every category $C$ there is an {\it opposite category} $C^{op}$, such
that the objects of $C^{op}$ are the same as the objects of $C$, and such
that $Hom_{C^{op}}(X,Y) = Hom_C(Y,X)$. In other words, the morphisms are
the same as in $C$, only with all the arrows reversed.  Compositions
are defined in the opposite order.  Universal objects in $C$ become
final objects in $C^{op}$, and vice-versa.]

6. Find examples of categories with each of the following properties:
(1) a universal object but no final object. (2) no universal object
and no final object, (3) a universal object and a final object, with
those objects not isomorphic, and (4) universal and final objects that
are the same.  (Objects that are both universal and final are called
``zero objects'')

 
7. Let $X$ be a topological space. We build a category out of $X$ whose 
objects are the open sets in $X$. If $U$ and $V$ are open sets, then
$Hom(U,V)$ is empty unless $U \subset V$, in which case it consists of 
the inclusion map of $U$ into $V$.  In this category, what are the universal
object(s) and final object(s)?  (Justify your answers, of course.)



\bye
