
Let M_{g} denote a surface of genus
g. Let S ⊆M be a subset such that
MS is connected. Prove that
H_{1}(S;Q) has dimension at most 2g.
 Let M be an oriented manifold and L,K ⊆
M compact, locally contractible subsets with L
⊆K. Prove that there is an isomorphism
H^{l}(K,L) → H_{nl}(ML,MK).
[Hint: Use Proposition 3.46 in Hatcher twice. Then use long
exact sequences and the 5lemma.]
 Bonus problem. Find a manifold M
with boundary, such that the boundary does not have a collar
neighborhood. ("Don't try this at home". Tell me if you can do
it, but don't write it up.)