Northeastern University Department of Mathematics

Prof. Alex Suciu  MTH 3107 – TOPOLOGY II  Winter 1998

Take-Home Final Exam

Due Tuesday, March 17

Instructions: Do at least 6 of the following 10 problems. Give complete proofs or justifications for each statement you make. Show all your work.

1. Let X = [0, 1] and A = \{1, \frac{1}{2}, \frac{1}{3}, …, \frac{1}{n}, … \} ∪ \{0\}. Is H_{1}(X, A) isomorphic to H̃_{1}(X/A)? Explain why, or why not.

2. (a) Let X and Y be two finite CW complexes. Show that χ(X × Y) = χ(X)χ(Y).

(b) Let A and B be two subcomplexes of X such that X = A ∪ B. Show that χ(X) = χ(A) + χ(B) − χ(A ∩ B).

3. Let f : (X, A) → (Y, B) be a continuous map. Assume that f : X → Y and f|_{A} : A → B are homotopy equivalences.

(a) Show that f_{∗} : H_{∗}(X, A) → H_{∗}(Y, B) is an isomorphism.

(b) Give an example where H_{∗}(X, A) ≅ H̸_{∗}(Y, B), although X is homotopy equivalent to Y, and A is homotopy equivalent to B.

4. Let X = ℝℙ^{3} × L(4, 1) be the product of the projective space ℝℙ^{3} = S^{3}/ℤ_{2} with the lens space L(4, 1) = S^{3}/ℤ_{4}.

(a) Find a CW-decomposition of X.

(b) Determine the chain complex (C_{•}(X), d) associated to that cell decomposition.

(c) Compute the homology groups H_{∗}(X).

5. Let X = G_{3}(ℝ^{6}) be the Grassmanian of 3-planes in ℝ^{6}.

(a) Find a CW-decomposition of X.

(b) Determine the chain complex (C_{•}(X), d) associated to that cell decomposition.

(c) Compute the homology groups H_{∗}(X).

6. Let T_{g} be the orientable surface of genus g.

(a) Describe a 2-fold cover p : T_{3} → T_{2}.

(b) Compute p_{∗} : H_{1}(T_{3}) → H_{1}(T_{2}).

(c) Compute p_{∗} : H_{2}(T_{3}) → H_{2}(T_{2}).

7. Let X = (ℂ^{2} \ {0})/(z_{1}, z_{2}) ∼ (2z_{1}, 2z_{2}). Compute H_{∗}(X).

8. Show that ℝℙ^{2} is not a retract of ℝℙ^{3}.

9. Problem 1, in Bredon’s book, p. 259.

10. Problem 5, in Bredon’s book, p. 259.