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\centerline{\bf Algebraic Topology}
\centerline{Homework 13: Due Friday, December 3}      % e.g. September 13
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{\bf Problem 1.} We saw in class how $RP^2$ is a CW complex with one 2-cell,
one 1-cell, and one 0-cell, with $\partial e^{(2)}=2 e^{(1)}$ and $\partial
e^{(1)}=0$, which implies that $H_2(RP^2)=0$, $H_1=\Z_2$ and $H_0=\Z$. Find a
CW decomposition for $RP^2 \times RP^2$ and use it to compute the homology
of $RP^2 \times RP^2$.  This example provides a counterexample to an
obvious guess of how the homology of $X \times Y$ is related to the homology
of $X$ and $Y$. The obvious formula works for the free part of the homology,
but the torsion part behaves differently. 

%Let $X$ be a topological space. If $H_k(X)$ is finitely generated,
%then $H_k(X) = F \oplus T$ for some finitely generated free Abelian
%group $F$ and a torsion subgroup $T$. If $F = \Z^{b_k}$, we call
%$b_k$ the {\it $k$-th Betti number of $X$}. The {\it Euler
%Characteristic} of $X$ is $\sum_k (-1)^k b_k$, assuming that each
%$b_k$ is finite and that only a finite number of $b_k$'s are nonzero.

{\bf Problem 2.} Let $X$ be the torus of revolution 
$(\sqrt{x^2+z^2}-2)^2+y^2=1$. Let $Y$ be the union of $X$ with the two disks
$(x-2)^2+y^2 \le 1, z=0$ and $(x+2)^2 + y^2 \le 1, z=0$. 
Exhibit a CW structure for this torus-with-two-disks space
(which appeared on last year's midterm), and use this structure 
to compute the homology of this space. 

{\bf Problem 3.} Construct a 3-dimensional CW complex (that is, at least one
3-cell and no cells of dimension greater than 3) with the following
homology groups: $H_0=\Z$, $H_1 = \Z_2 \oplus \Z_3$, $H_2=\Z$, $H_3=0$. There
are many, many possible answers. 

{\bf Problem 4.} Let $X$ be a simplicial complex with a finite number
of simplices (aka a ``finite simplicial complex''). Prove that the
simplicial homology of $X$ is isomorphic to the singular homology of
$X$.  [Hint: writing $X$ as a CW complex is easy, and this gives a
correspondence between simplicial chains and cellular chains. What
takes work is showing that the boundary maps in the simplicial complex
are the same as the maps in the cellular computation of the singular
homology.]
 

Let $G_1$, $G_2, \ldots$ be Abelian groups, and suppose that we have
maps $\rho_n:G_n \to G_{n+1}$. The {\it direct limit} of the groups
$G_n$ is the quotient of the product $G_1 \times G_2 \times \cdots$ by
the identification $x \sim y$ if $x \in G_n$, $y\in G_{n+1}$, and
$y=\rho_n(x)$. That is, $G_1$ is identified with a subgroup of $G_2$, 
$G_2$ is identified with a subgroup of $G_3$, etc. The groups $G_n$ are called
{\it approximants} to the direct limit. 

The direct limit of spaces is similar. Let $X_1, \ldots,$ be spaces,
and consider continuous maps $f_n: X_n \to X_{n+1}$. As a set, 
the direct limit of the $X_n$'s
is the union $\coprod X_n$, 
modulo the identifications $x_n \sim f_n(x_n)$ if $x_n \in X_n$. A set 
is considered closed if its intersection with (the image of) 
each $X_n$ is closed (in $X_n$). 

The most common example is when $X_1 \subset X_2 \subset \cdots$, so the
maps $f_n$ are just inclusions. The direct limit is then the union, albeit
with a particular topology. For instance, $X_n$ might be the $n$-skeleton
of a CW complex, or $X_n$ might be a finite sub-complex of an infinite
(but finite-dimensional) CW complex. 

CW complexes have the following useful property that you are free to use
below: Every compact subspace 
is contained in a finite sub-complex, and in particular in 
a finite skeleton $X_n$.  A proof may be found on p520 of Hatcher,
but here's a sketch.  If a compact set $C$ hits infinitely many cells, then
we could find a subset $S=\{x_1, x_2, \ldots\}$ of $C$ 
with each $x_i$ in a different cell. 
Since $S$ hits the interior of each cell at most once, it hits the closure of
each cell only a finite number of times (I'm skipping some details here), 
so the intersection of $S$ with each cell is closed, so $S$ is closed. 
Being a closed subset of the compact $C$, $S$ must be compact. 
But $S$ has the discrete topology, and so is not compact. Contradiction. 

{\bf Problem 5.} Let $X$ be the direct limit of a collection of spaces 
$X_1 \subset X_2 \subset \cdots$, such that any compact subset of $X$ lies
in some $X_n$.  Prove that the homology of $X$ is the direct limit of the
homology of $X_n$. 

{\bf Problem 6.} As a corollary to Problem 5, prove the following
theorem from last week's lectures: Let $X$ be a CW complex, and let
$F^n$ be a free abelian group whose generators are the $n$-cells of
$X$.  Then there exist maps $\partial_n: F^n \to F^{n-1}$ with
$\partial^2=0$ such that the $n$-th singular homology of $X$ is
isomorphic to $ker(\partial_n) / Im(\partial_{n+1})$. (We can take
$\partial_0: F^0 \to 0$ to be the zero map.)  When $X$ is
finite-dimensional, this was proved in class, and you are free to use
those results without reproving them. The trick is extending
this to infinite dimensions.

{\bf Problem 7.} The inclusion $\R^n \to \R^{n+1}$, where $(x_1,\ldots,x_n)$
goes to $(x_1,\ldots,x_n,0)$, induces an inclusion of $S^{n-1}$ into
$S^n$. Let $S^\infty$ be the direct limit of $S^1 \subset S^2 \subset
\cdots$, and let $RP^\infty$ be the quotient of $S^\infty$ by the antipodal
map. Let $CP^\infty$ be the direct limit of the spaces $CP^n$ induced by the
obvious inclusion of each $C^n$ in $C^{n+1}$.  Compute the homologies of 
$S^\infty$, $RP^\infty$, and $CP^\infty$. 
  
\bye
