Northeastern University Department of Mathematics

Prof. A. Suciu  MTH 3481 — TOPOLOGY 3  Spring 1998

FINAL EXAM

This is a take-home exam, due Monday, June 15, at 9AM. Do 6 of the following 7 problems. Give complete proofs or justifications for each statement you make. Show all your work.

1. Let Y = \{(x, y) ∈ ℝ^{2} | x > 0, y = sin \frac{1}{x}\} ∪ \{(x, y) ∈ ℝ^{2} | x = 0, −1 ≤ y ≤ 1\} and X = \{0, 1\}. Let f : X → Y, given by f(0) = (0, 0) and f(1) = (\frac{1}{π}, 0).

(a) Show that f_{∗} : π_{n}(X) → π_{n}(Y) is an isomorphism, for all n ≥ 0.

(b) Show that f is not a homotopy-equivalence.

(c) Does this contradict Whitehead’s theorem? Why, or why not?

2. Let X be a connected, finite CW-complex, with π_{1}(X) having a non-trivial element of finite order. Let Y = X̃ × K(π_{1}(X), 1), where X̃ is the universal cover of X.

(a) Show that π_{n}(X) ≅ π_{n}(Y), for all n ≥ 0.

(b) Show that X is not homotopy-equivalent to Y.

(c) Does this contradict Whitehead’s theorem? Why, or why not?

3. Let f : T^{3} → S^{2} be the composite of the Hopf bundle map p : S^{3} → S^{2} and the quotient map q : T^{3} → S^{3}, which collapses the 2-skeleton of the 3-torus to a point.

(a) Show that f_{∗} = 0 : π_{n}(T^{3}) → π_{n}(S^{2}), for all n ≥ 0.

(b) Show that f_{∗} = 0 : H_{n}(T^{3}) → H_{n}(S^{2}), for all n > 0.

(c) And yet f is not homotopic to a constant map.

4. Let X be a connected CW-complex, with π_{i}(X) = 0 for 1 < i < n, for some n ≥ 2. Let h : π_{n}(X) → H_{n}(X) be the Hurewicz homomorphism. Show that H_{n}(X)/h(π_{n}(X)) ≅ H_{n}(K(π_{1}(X), 1).

5. Let G be an abelian group.

(a) Show that H_{n+1}(K(G, n)) = 0, for n > 1.

(b) Show that there is a Moore space M(G, 1) if and only if H_{2}(K(G, 1)) = 0.

(c) For what values of n does there exist a Moore space of type M(ℤ^{n}, 1)?

6. Let X = ℂℙ^{2}∪e^{3}, with attaching map S^{2} ⟶^{×p} S^{2} = ℂℙ^{1} ⊂ ℂℙ^{2}, and Y = M(ℤ_{p}, 2)∨S^{4}. (a) Show that M(ℤ_{p}, 2) can be chosen so that X and Y have the same 3-skeleta.

(b) Show that H^{∗}(X; ℤ) ≅ H^{∗}(Y; ℤ) (as graded rings).

(c) Show that H^{∗}(X; ℤ_{p}) ≅ H̸^{∗}(Y; ℤ_{p}) (as graded rings).

7. Let G be a group, and let {M_{n}}_{n=1}^{∞} be a sequence of ℤG-modules.

(a) Construct a CW-complex X with π_{1}(X) = G, and π_{n}(X) = M_{n} (as ℤG-modules).

(b) If X = K(G, 1) × Y, where π_{1}(Y) = 0, show that π_{n}(X) is trivial as a ℤG-module, for all n > 1.

(c) If X = ℝℙ^{n}, show that π_{n}(X) = ℤ is trivial as a ℤℤ_{2}-module if and only if n is odd.