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\centerline{\bf Algebraic Topology}
\centerline{Homework 12: Due Wednesday, November 24}      % e.g. September 13
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(If you're going to miss Wednesday's class because of Thanksgiving, 
please drop off your homework before you go.)
 
Let $X_1, X_2, \ldots$ be a sequence of path-connected
topological spaces, and suppose that $x_i \in X_i$ for each $i$. The {\it wedge}
of the $X$'s, denoted $\vee_i X_i$, is the quotient space of the disjoint
union $\coprod_i X_i$
by the identification $x_i \sim x_j$ for all $i,j$. 
For instance, the wedge of two circles is a figure 8. The topology of 
$\vee_i X_i$ is that a set is open if (and only if) its intersection with
each $X_i$ is open. For the wedge of a finite number of spaces, this definition
is obvious, but it is more subtle for infinite wedges. (The Hawaiian earring
is {\it not} an infinite wedge of circles!)

{\bf Problem 1} Prove the following: If $Y=\vee_i X_i$ for a 
collection of spaces $X_i$, and if each $x_i$ has a neighborhood that
(strongly) deformation retracts to $x_i$, then $\tilde H_k(Y) =
\oplus_i \tilde H_k(X_i)$. [Warning: as with Van Kampen's theorem,
which says that the fundamental group of $Y$ is the free product of
the fundamental groups of the $X_i$'s, the condition of having an
appropriate neighborhood is crucial. For finite collections, this problem could
be solved by induction and Mayer-Vietoris, but there is a better way. For
infinite collections, don't even think about induction.]
 

{\bf Problem 2.} Suppose that the topological spaces $X$ and $Y$ are
path-connected. Show that $H_1(X\times Y)$ is isomorphic to $H_1(X)
\oplus H_1(Y)$. [More generally, there is a simple formula for the free
part of $H_n(X\times Y)$ in terms of the homology of $X$ and $Y$, but
the torsion part is more complicated. If $H_k^f$ denotes the free part
of $H_k$, then $H_n^f(X\times Y) = \oplus_k (H_k^f(X)\otimes
H_{n-k}^f(Y))$. No, you are {\it not} expected to prove this!]

Since $H_n(S^n)$ is infinite cyclic, there are two choices of
generators for $H_n(S^n)$. Each is called an {\it orientiation} of
$S^n$. Given a choice of orientation, we can think of the top homology
as the integers.  If $f: S^n \to S^n$ is a continuous map between
oriented spheres, then $f_*: \Z=H_n(S^n) \to \Z=H_n(S^n)$ is
multiplication by an integer, which we call the {\it degree} of
$f$. Note that, for maps from an $n$-sphere to itself, the degree
doesn't depend on the orientation (since switching the orientation
would involve multiplying by $-1$ twice), but for maps between distinct
spheres, you have to pick orientations before you can define a degree.
Note that $deg(f\circ g) = deg(f) deg(g)$ if both $f$ and $g$ are maps
$S^n \to S^n$, and that $deg(f_0)=deg(f_1)$ if $f_0$ and $f_1$ are homotopic. 


{\bf Problem 3.} Consider the reflection map $R(x_1,x_2)=
(-x_1,x_2)$ that maps $\R^2$ to itself, and also 
maps $S^1$ to itself. Show that $R$ has degree $-1$. [Hint: Pick an explicit
generator $\alpha$ of $H_1(S^1)$ such that $R_\#(\alpha)=-\alpha$.]
Now consider reflection on the $n$-sphere, where 
$R(x_1,x_2,\ldots,x_{n+1})=(-x_1,x_2,\ldots,x_{n+1})$. Show that $R$ has 
degree $-1$. 
 
{\bf Problem 4.} Show that the antipodal map of $S^n$ is homotopic to the
identity if, and only if, $n$ is odd. 

The van Kampen computation of $\pi_1$ of
a surface, when Abelianized, essentially gives the Mayer-Vietoris computation
of $H_1$ of the surface. The following problem generalizes that idea. 


{\bf Problem 5.}  Let $U$, $V$, and $X = U \cup V$ be as in the setup
for Van Kampen's theorem. That is, each is open and 
path-connected and contains
the base point $x_0$. Without using homology, write down (and
prove) an ``Abelianized van Kampen's theorem'' that relates $\pi^A(X)$
to $\pi^A(U)$, $\pi^A(V)$, $\pi^A(U \cap V)$ and the way these groups map
to one another, where $\pi^A$ denotes the Abelianization of
$\pi_1$. (You can take the standard van Kampen's theorem as a given --
there's no need to reprove that!)  Then write down the (reduced) Mayer-Vietoris
sequence that computes $H_1(X)$ from $H_1(U)$, $H_1(V)$ and $H_1(U
\cap V)$, and show how the maps in the M-V sequence relate to the maps
in your Abelianized van Kampen's theorem.

{\bf Book problems:} Page 191, problem 2.5 and
page 201, problem 3.7. 

\bye
