DOCTORAL EXAM (NEW OPTION) - TOPOLOGY - JANUARY 1998

1. Give examples of the following. Support your answers.

(a) A space with trivial fundamental group.

(b) A space with a finite non-trivial fundamental group.

(c) A space with an infinite cyclic fundamental group.

(d) A space with a non-abelian fundamental group.

For i = 1, 2, let F_{i} be the free group on the set S_{i} with respect to the function φ_{i} : S_{i} → F_{i}. Assume that S_{1} and S_{2} are disjoint. Let S = S_{1} ∪ S_{2}. Let F be the free product of F_{1} and F_{2} with respect to the homomorphisms ψ_{i} : F_{i} → F. Define φ : S → F by φ|_{S_{i}} = ψ_{i} ∘ φ_{i}. Prove that F is the free group on S with respect to φ.

2. Let T be a triangulation of S^{2} such that each vertex has valence either 5 or 6. (Recall that the valence is the number of edges meeting the vertex.) Prove that T has exactly 12 vertices of valence 5.

3. Let T be the torus S^{1} × S^{1}.

(a) Give an example of a covering map p : X → T such that π_{1}(X) is isomorphic to Z.

(b) Give an example of a 3-sheeted covering map p : Y → T with Y connected.

(c) Up to isomorphism how many 3-sheeted covering maps p : Y → T with Y connected are there? Justify your answer.

4. (a) Compute the integral homology groups of S^{2} × S^{7}.

(b) Suppose X is a space such that H_{0}(X; Z) = Z, H_{1}(X; Z) = Z_{3}, H_{2}(X; Z) = 0, H_{3}(X; Z) = Z, and H_{n}(X; Z) = 0 for n > 3. Calculate the homology and the cohomology groups of X with Z_{3} coefficients.

5. Let T be the torus S^{1} × S^{1}. Let z_{0} be a point on S^{1}, and let c be the curve S^{1} × {z_{0}}. Let X be the space obtained by attaching a disk D to T so that ∂D is homeomorphically identified with c and D ∩ T = c. Compute the integral homology groups of X.

6. Suppose M is a closed, orientable, connected 3-manifold. (Recall that “closed” in this context means “compact and without boundary.”) Prove that χ(M) = 0. Hint: Duality.