DOCTORAL EXAM - TOPOLOGY - JUNE 1998

1. Carefully state the Seifert-van Kampen Theorem for the case X = U ∪ V. Be sure to state all the hypotheses on the spaces involved. Then give a rigorous proof that for any path connected 3-manifold M (without boundary) and any point x ∈ M, one has π_{1}(M − x) ≅ π_{1}(M).

2. Give an example of a retract which is not a deformation retract. Prove your answer.

3. Let X be the space obtained by removing the interiors of n disjoint disks from a closed, connected, orientable surface of genus g. Suppose π_{1}(X) is free of rank 4. Compute the Euler characteristic χ(X). List all the possibilities for the pair (g, n).

4. Find the universal covering spaces of the following spaces. (X ∨ Y denotes the space obtained by gluing together X and Y at a single point. Draw a picture in (d).)

(a) S^{1}

(b) S^{1} × S^{1}

(c) S^{2}

(d) S^{1} ∨ S^{2}

5. Let C = {C_{p}, ∂_{p}} and C′ = {C′_{p}, ∂′_{p}} be chain complexes. Define what it means for f : C → C′ to be a chain map. Let g : C → C′ be another chain map. Define what it means for f and g to be chain homotopic. Then prove that in this case f_{∗} = g_{∗} : H_{p}(C) → H_{p}(C′).

6. Suppose X is a space such that H_{0}(X) = Z, H_{1}(X) = Z_{6}, and H_{p}(X) = 0 for p > 1. Let P^{2} be the real projective plane. Calculate H_{p}(X × P^{2}) for all p ≥ 0. (Z_{n} denotes the cyclic group of order n.)

7. Let M and N be connected n-manifolds (without boundary). Let M ∨ N be the union of M and N at a single point. For p > 0 express H_{p}(M ∨ N) in terms of H_{p}(M) and H_{p}(N), carefully justifying your answer. Do not assume that M and N have triangulations. What is H_{0}(M ∨ N)?

8. Suppose M is a compact, orientable, connected 3-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 A_{1} ≅ A_{2} and T_{2} = 0. (Hints: Poincaré Duality and Universal Coefficient Theorem. Recall that every finite abelian group is a direct sum of cyclic groups.)