Due Thursday, April 25, in class.

A. Prove the splitting lemma.

p. 176, Hatcher

# 1, 2, 8

Extra hint for 2: it is easy to get confused about the subscripts. Recall that a connected graph has (reduced) nonzero homology only in dimension one. This observation will save you a lot of trouble when considering Mayer-Vietoris sequences. Once you have figured out the case where \(X\) is a tree, build up to arbitrary graphs by adding edges one at a time.