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\centerline{\bf Algebraic Topology}
\centerline{Homework 10: Due Wednesday, November 10}      % e.g. September 13
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{\bf Problem 1.} As we defined in class, 
a short exact sequence $0 \to A \to B \to C \to 0$,
with maps $i: A \to B$ and $j:B \to C$ {\it splits} if there is an
isomorphism $\alpha: B \to A \oplus C$ with $\alpha \circ i$ being
inclusion in the first factor and $j \circ \alpha^{-1}$ being
projection onto the second. Prove that the sequence splits if (and
only if) there exists a homomorphism $\ell: B \to A$ such that $\ell \circ i$
is the identity on $A$. 

{\bf Problem 2.} Prove the 5-lemma. Yes, we did this in class, but 
writing out the details of this sort of diagram-chasing will give you a better
feel for the technique.  Of course you can just copy the notes you (possibly)
took in class, and you're free to do so, but please try to prove it on your
own first. 

{\bf Problem 3.} Likewise, prove the snake lemma. 
You don't need to construct the maps in the long exact sequence, 
but you do need to show that the sequence is exact. I'll do most, if not
all, of this in class, but the only way to get a feel for the $\partial_*$
operator is to do a bunch of calculations and proofs with it.  

The last three problems are exercises in using the Mayer-Vietoris sequence
to compute homology. In each case you can assume that the preliminary
version of Mayer-Vietoris is true, even though we haven't proven it yet. 
(I expect to cover this preliminary version on November 3 or 5.)

{\bf Problem 4.} Compute the homology of $\R^n$ with $p$ points removed. [Hint:
use induction on $p$.]

{\bf Problem 5.} Compute the homology of $\R^3$ with $p$ non-intersecting
lines removed. Note that the lines may not be parallel, so this doesn't 
instantly reduce to $\R^2$ with $p$ points removed. (Yes, if you're clever you
can find a way to show that a plane with $p$ points removed is a deformation
retract, but that probably requires techniques from the second semester 
prelim class.  Tackling the problem directly, in analogy to what you did
in problem 4, is easier.)


{\bf Problem 6.} Now consider $\R^3$ with $n_0$ points removed and
$n_1$ lines removed, with none of the lines intersecting, and none of
the points on any of the lines. What is the homology of this space?
[Hint: Mayer-Vietoris doesn't just compute the homology of $U \cup V$
from that of $U$, $V$ and $U \cap V$, but it can also be used to compute
the homology of $U \cap V$ from that of $U$, $V$ and $U \cup V$.]

\bye
