Ph.D Comprehensive Exam: TOPOLOGY January 1997

Part I: The first part is true or false problems or statements. Support your answer by some short argument or explicit example. Typical answer should look like: true, since every closed bounded subset of R^{n} is compact.

1. Every covering space of a compact surface is compact.

2. The free product of two Abelian groups is Abelian.

3. The torus T^{2} is a covering space of S^{2}.

4. There is a continuous function r : S^{2} → S^{1} such that r(x, y, 0) = (x, y) for all (x, y) ∈ S^{1}.

5. Any continuous function f : D^{2} → D^{2} has the property: x = f(x) for some x ∈ D^{2}.

Part II: Solve five of the following problems. Give detailed solutions for credits.

1. If the quotient space X/R is Hausdorff, then R is closed in the product space X × X.

2. (a) Identify the surface whose triangulation is given by the following.

124 235 156 236 134 245 135 126 346 456

(b) For any triangulation of a compact surface, show that 2v ≥ 7 + \sqrt{49 − 24χ},

where v is the total number of vertices and χ is the Euler number of the surface.

3. If Y_{1} and Y_{2} are connected closed 3-manifolds, then the fundamental group of connected sum π_{1}(Y_{1}#Y_{2}) is the free product of π_{1}(Y_{1}) and π_{1}(Y_{2}).

4. (a) List all connected covering spaces of S^{1}.

(b) If X is simply connected and (X̃, p) is a covering space of X, then p : X̃ → X is a homeomorphism.

5. Prove that there is no continuous map f : S^{2} → S^{1} such that f(−x) = −f(x).

6. Let X be a path-connected and Hausdorff space with a fixed point free involution i : X → X (i.e. an homeomorphism with i∘i = id_{X} and i(x) ≠ x for each point x ∈ X). Let π : X → Y be the projection from X onto the quotient space Y = X/ ∼, where x_{1} ∼ x_{2} if and only if x_{1} = x_{2} or x_{1} = i(x_{2}). Show that π : X → Y is a covering map.