• Let M_g 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 H^l(K,L) → H_n-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.)