Mathematics 730 Final Examination December 15, 1998

Instructions. Answer each question in your exam booklet. The point value of each problem is indicated. The exam is worth a total of 200 points. In problems with multiple parts, whether the parts are related or not, the parts are graded independently of one another. Be sure to go on to subsequent parts even if there is some part you cannot do.

You are allowed to appeal to “standard theorems” proved in class or in the textbook, but if you do so, it’s your responsibility to state clearly exactly what you’re using and why it applies.

1. (40 points, divided as indicated) Let X and Y be topological spaces and let f, g: X → Y be continuous maps.

(a) (25) Prove that if Y is Hausdorff, then A = {x ∈ X : f(x) = g(x)} is closed in X.

(b) (15) Show by example that if X is Hausdorff but Y is not, then A may fail to be closed in X.

2. (40 points, divided as indicated) (i) (30) Let X be a topological space and let A ⊆ X be a closed subset. Prove that the following two conditions are equivalent:

(a) There is a retraction r from X × [0, 1] onto (X × {0}) ∪ (A × [0, 1]).

(b) Given any topological space Y and a map h: A × [0, 1] → Y, if h|_{A×{0}} has an extension g: X × {0} → Y, then h has an extension H: X × [0, 1] → Y with H|_{X×{0}} = g.

(ii) (10) Show that the conditions of (i) are satisfied if X = [0, 1] and A = {0}.

3. (50 points, divided as indicated) Define an identification space X = (T_{1} ⨿ T_{2}) / ∼, where T_{1} = D^{2} × S^{1} × S^{1}, T_{2} = S^{1} × D^{2} × S^{1}, and ∼ identifies x ∈ ∂T_{1} = S^{1} × S^{1} × S^{1} with f(x) ∈ ∂T_{2} = S^{1} × S^{1} × S^{1} via the map f: T^{3} → T^{3}, T^{3} = ℝ^{3}/ℤ^{3}, induced by the matrix

A = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) .

(We think of ℝ^{3} as consisting of column matrices and let A act by left multiplication.)

(a) (20) Show that X has the natural structure of a connected smooth 4-dimensional manifold (without boundary).

(b) (30) Use Van Kampen’s Theorem to compute the fundamental group of X.

4. (70 points, divided as indicated) Short Answer Problems. Give brief explanations but not complete proofs.

(a) (10) Describe the one-point compactification of ℝ^{2} ⨿ ℝ.

(b) (20) Give an example of a space which is path-connected but not locally path-connected.

(c) (20) What are the critical points and critical values of the map f: ℝ^{3} → ℝ defined by f(x, y, z) = x^{2}y + 2xz? What is the topology of the inverse image of a regular value?

(d) (20) Classify all 2-to-1 connected coverings of the Klein bottle.