Ph.D. COMPREHENSIVE EXAM - TOPOLOGY - JANUARY 1995

1. (12 points) Prove the “tube lemma”: Let X and Y be topological spaces. Give X × Y the product topology. Assume Y is compact. Let x_{0} ∈ X. Let N be an open subset of X × Y which contains the “slice” {x_{0}} × Y. Then there is an open neighborhood W of x_{0} in X such that the “tube” W × Y is contained in N.

2. (12 points) Let X be a topological space which satisfies the second axiom of countability. Let A be an uncountable subset of X. Prove that A has a limit point in X.

3. (12 points) Let (X, d) be a metric space. Let x_{0} ∈ X. Define f : X → R by f(x) = d(x, x_{0}). Prove that f is continuous.

4. (12 points)

(a) State the Urysohn Lemma.

(b) Prove that every connected normal space having at least two points is uncountable.

5. (12 points) Let x_{0} ∈ A ⊆ X. Suppose r : X → A is a retraction.

(a) Prove that r_{∗} : π_{1}(X, x_{0}) → π_{1}(A, x_{0}) is onto.

(b) Prove that if r is a deformation retraction, then r_{∗} is one-to-one.

(c) Give an example of a retraction that is not a deformation retraction.

6. (12 points) Let G = G_{1} ∗ G_{2}.

(a) Prove that if G_{1} and G_{2} are free groups, then G is a free group.

(b) Is the converse true? (No proof required.)

7. (12 points) Let p : X̃ → X be a covering space. X̃ and X are assumed to be path connected and locally path connected. Let x̃_{0} ∈ X̃. Let x_{0} = p(x̃_{0}).

(a) Prove that p_{∗} : π_{1}(X̃, x̃_{0}) → π_{1}(X, x_{0}) is one-to-one.

(b) Let x̃_{1} ∈ p^{−1}(x_{0}). Prove that p_{∗}(π_{1}(X̃, x̃_{0})) and p_{∗}(π_{1}(X̃, x̃_{1})) are conjugate subgroups

of π_{1}(X, x_{0}).

8. (12 points) Give presentations for the fundamental groups of the following spaces. P denotes a projective plane.

(a) P × P

(b) P #P, where “#” denotes connected sum.

(c) P ∨ P, where “∨” denotes the gluing of the two spaces together at a single point.

(d) P − {x}, where x ∈ P.

9. (4 points) State the classification theorem for compact surfaces.