DOCTORAL EXAM - TOPOLOGY - AUGUST 1999

You should attempt all the problems. Partial credit will be given for serious efforts.

1. Let M be the surface obtained by identifying the sides of an octagon according to the pattern acbc^{−1}a^{−1}db^{−1}d^{−1}. Compute the Euler characteristic of this surface and state whether or not the surface is orientable. Then use this information to determine what surface this is.

2. Let M be a closed, connected, orientable surface of genus two. Let C be a simple closed curve on M. Let D be a disk and f : ∂D → C a homeomorphism. Let X be the space obtained by attaching D to M using the map f. List all the possibilities for π_{1}(X). Be sure to identify each group; do not just write down presentations.

3. Let S^{1} be the circle and P^{2} the real projective plane. Prove that every continuous function f : S^{1} → S^{1} which factors through P^{2} must be inessential. (Hint: If g : P^{2} → S^{1} what must the induced map on fundamental groups be?)

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 G = π_{1}(X, x_{0}) and H = p_{∗}(π_{1}(X̃, x̃_{0})). Exhibit a bijection from p^{−1}(x̃_{0}) to the set G/H of all right cosets Hα of H in G. Prove that it is well defined and is a bijection.

5. Let S^{1} be the unit circle in the complex plane. Let D_{1} and D_{2} be two copies of the unit disk. Let f_{1} , f_{2} : S^{1} → S^{1} be given by f_{1}(z) = z and f_{2}(z) = z^{2}. Let X be the space obtained by attaching D_{1} and D_{2} to S^{1} using f_{1} and f_{2}, respectively. Compute H_{p}(X) for all p ≥ 0.

6. Suppose A is a retract of X. Prove that H_{p}(X) is isomorphic to H_{p}(A) ⊕ H_{p}(X, A) for all p ≥ 0.

7. Suppose X is a space such that H_{0}(X) = Z, H_{1}(X) = Z_{5}, H_{2}(X) = Z_{4}, and H_{p}(X) = 0 for p > 2.

(a) Calculate H_{p}(X; Z_{6}) for all p ≥ 0.

(b) Calculate H^{p}(X; Z) for all p ≥ 0.

(Z_{n} denotes the cyclic group of order n.)

8. Let M be a compact, connected, orientable 4-manifold (without boundary). Suppose H_{1}(M) = Z_{2} and H^{2}(M) = Z ⊕ Z. Find H_{p}(M) and H^{p}(M) for all p ≥ 0.