Northeastern University Department of Mathematics

Prof. Alex Suciu  MTH 3107 – TOPOLOGY II  Winter 1999

Take-Home Final Exam

Due Monday, March 22

Instructions: Do at least 5 of the following 6 problems. Give complete proofs or justifications for each statement you make. Show all your work.

1. Let X be a space. Show that:

(a) H̃_{n}(X; ℤ) = 0 for all n if and only if H̃_{n}(X; ℚ) = 0 and H̃_{n}(X; ℤ_{p}) = 0 for all n and all primes p.

(b) f : X → Y induces isomorphisms in H_{∗}(−; ℤ) if and only if it induces isomorphisms in H_{∗}(−; ℚ) and H_{∗}(−; ℤ_{p}) for all primes p.

2. Prove the following theorem of Borsuk: If f : S^{n} → S^{n} commutes with the antipodal map, then f has odd degree.

Remarks:

(i) There is a direct (and rather long) proof in Bredon’s book (Theorem 20.6, pp. 244-245). Use instead the Borsuk-Ulam theorem (see hint to Problem 7, p. 245).

(ii) Note that this is not a homotopy-theoretic result. Indeed, fa ≃ af, for all f (show that!).

3. Let X = K × ℝℙ^{3} be the product of the Klein bottle with the 3-dimensional projective space.

(a) Find a CW-decomposition of X.

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

(c) Compute the (cellular) homology groups H_{∗}(X) and H_{∗}(X; ℤ_{2}).

4. Recall that, for a space X, and a short exact sequence 0 → G′ ⟶^{α} G ⟶^{β} G′′ → 0 of abelian groups, there is an associated long exact in homology,

⋯ → H_{i}(X; G′) ⟶^{α_{∗}} H_{i}(X; G) ⟶^{β_{∗}} H_{i}(X; G′′) ⟶^{∂} H_{i−1}(G′) → ⋯

Compute explicitly this homology sequence (the terms and the maps), in case the coefficients sequence is 0 → ℤ ⟶^{2} ℤ → ℤ_{2} → 0, and

(a) X = ℝℙ^{3}.

(b) X = K, the Klein bottle.

5. Let A be an integral 3 × 3 matrix, and let f : T^{3} → T^{3} be the induced self-map of the 3-torus. Compute the Lefschetz number L(f) in terms of the entries of A.

6. Let X be a finite simplicial complex, and let G be a finite group acting simplicially on X.

Show that, for every g ∈ G,

L(g) = χ(X^{g})

where X^{g} = {x ∈ X | gx = x}.