Let Mg denote a surface of genus
g. Let S ⊆M be a subset such that
M-S is connected. Prove that
H1(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
Hl(K,L) → Hn-l(M-L,M-K).
[Hint: Use Proposition 3.46 in Hatcher twice. Then use long
exact sequences and the 5-lemma.]
- 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.)