DOCTORAL EXAM - TOPOLOGY - JUNE 1999

1. Let X be the 2-complex given by performing the identifications indicated below. Write down a presentation for π_{1}(X). You do not have to give a rigorous Seifert van-Kampen theorem argument to justify your presentation. The group is a familiar group; determine which familiar group it is.

2. Prove that every map f : S^{2} → S^{1} is homotopic to a constant map. (Hint: universal covering space.)

3. Let S be the closed, connected, orientable surface of genus two. Let D_{0}, D_{1}, and D_{2} be disjoint disks in S. Let X be the space obtained by removing the interiors of D_{0}, D_{1}, and D_{2} and then gluing ∂D_{1} to ∂D_{2} by a homeomorphism. Find the Euler characteristic χ(X). What is π_{1}(X)?

4. Let p : X̃ → X be a covering map, where both spaces are path connected. Let x̃_{0} ∈ X̃ and x_{0} = p(x̃_{0}). Let A be a path connected subset of X such that x_{0} ∈ A. Let i : A → X be the inclusion map. Prove that if p^{−1}(A) is path connected, then for every β ∈ π_{1}(X, x_{0}), i_{∗}(π_{1}(A, x_{0})) has non-empty intersection with the right coset p_{∗}(π_{1}(X̃, x̃_{0}))β of p_{∗}(π_{1}(X̃, x̃_{0})) in π_{1}(X, x_{0}).

5. Let ϕ^{p} ∈ C^{p}(X) and ψ^{q} ∈ C^{q}(X). Let ϕ^{p} ⌣ ψ^{q} denote the cup product of these two cochains. State the formula expressing the coboundary δ(ϕ^{p} ⌣ ψ^{q}) in terms of ϕ^{p}, ψ^{q}, δϕ^{p}, and δψ^{q}. (You do not have to prove the formula.) Then use it to prove that {ϕ^{p}} ⌣ {ψ^{q}} = {ϕ^{p} ⌣ ψ^{q}} gives a well defined function H^{p}(X) × H^{q}(X) → H^{p+q}(X).

6. Suppose X is a space such that H_{0}(X) = Z, H_{1}(X) = Z_{4}, and H_{p}(X) = 0 for p > 1. Suppose Y is a space such that H_{0}(Y) = Z, H_{1}(Y) = Z_{6}, and H_{p}(Y) = 0 for p > 1. Calculate H_{p}(X × Y) for all p ≥ 0. (Z_{n} denotes the cyclic group of order n.)

7. Let M be a connected n-manifold (without boundary). Let x ∈ M. Prove that H_{n}(M, M − x) ≅ Z and H_{p}(M, M − x) = 0 for p ≠ n. Do not assume M has a triangulation. (Hints: Excision and homotopy axioms.)

8. Suppose M is a compact, orientable, connected 4-manifold (without boundary). H_{p}(M) = A_{p} ⊕ T_{p}, where A_{p} is a free abelian group of finite rank and T_{p} is a finite abelian group. Prove that if H_{1}(M) = 0, then A_{1} ≅ A_{3} and T_{2} = 0. (Hints: Poincaré Duality and Universal Coefficient Theorem. Recall that every finite abelian group is a direct sum of cyclic groups.)