EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 7, December 1st 2005

Solutions due on Thursday, 8th of December.

Exercise 7.1.

(a) Let ∩ : S_{∗}(X)⊗_{R}S^{∗}(X) → S_{∗}(X) be defined as ∩ = (1⊗⟨−, −⟩)∘((AWΔ_{∗})⊗1), and assume that the cup product on the cochains level is also defined using AW. Prove that if σ ∈ S_{p+q}(X), α ∈ S^{p}(X) and β ∈ S^{q}(X), then the formula

⟨β, σ ∩ α⟩ = ⟨β ∪ α, σ⟩

holds. Deduce that ∩ makes S_{∗}(X) into a left S^{∗}(X)-module and H_{∗}(X) into a left H^{∗}(X)-module. State and prove the corresponding results for relative homology and cohomology.

(b) Show that for any z ∈ H_{p+q}(X, A) the diagram

\xymatrix{H^{p}(X, A) \ar[r] \ar[d]_{z∩−} & H^{p}(X) \ar[d]^{z∩−} \\ H_{q}(X) \ar[r] & H_{q}(X, A)}

is commutative

Exercise 7.2. Let X = S^{n_{1}} × ⋯ × S^{n_{k}} for k ⩾ 1 and n_{j} ⩾ 1. Let M_{g} be the orientable surface of genus g. Compute

(a) H_{∗}(X; ℤ) as a module over H^{∗}(X; ℤ),

(b) H_{∗}(M_{g}; ℤ) as a module over H^{∗}(M_{g}; ℤ),

(c) H_{∗}(ℝP^{n}; 𝔽_{2}) as a module over H^{∗}(ℝP^{n}; 𝔽_{2}), and

(d) H_{∗}(ℂP^{n}; ℤ) as a module over H^{∗}(ℂP^{n}; ℤ).

For all these cases, deduce that if m is maximal such that H_{m} is non-zero, then H_{m} is free of rank one on a generator z and z ∩ − : H^{k} → H_{m−k} is an isomorphism for any k. Show that the space in question is a compact topological manifold of dimension m.

Exercise 7.3. Let n ⩾ 1. Show that ℝP^{n} is ℤ-orientable if and only if n is odd.

Exercise 7.4. Show that R-orientations of a topological manifold M are in 1-1 correspondence with sections s ∈ Γ(M) such that s(M) ⊂ ε^{−1}(1).

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