EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 6, November 24th 2005

Solutions due on Thursday, 1st of November.

Exercise 6.1. Compute the cohomology algebra H^{∗}(ℂP^{n}; ℤ) for n ⩾ 1.

Exercise 6.2. A division R-algebra structure on an R-module A is an R-bilinear homomorphism μ : A × A → A (not necessarily unital, associative, nor commutative), such that for any a ∈ A\{0} the homomorphisms μ(a, −) and μ(−, a) : A → A are surjective.

Show that if ℝ^{n} is a division ℝ-algebra, then n is a power of 2. Hint: assume that n ⩾ 2 and that we have a division ℝ-algebra structure on ℝ^{n} given by μ : ℝ^{n} × ℝ^{n} → ℝ^{n}. We then have a map S^{n−1} × S^{n−1} → S^{n−1} given by (x, y) ↦ μ(x, y)/|μ(x, y)|. This induces a map f : ℝP^{n−1} × ℝP^{n−1} → ℝP^{n−1}. Show that the homomorphism of 𝔽_{2}-algebras

(×)^{−1} ∘ f^{∗} : H^{∗}(ℝP^{n−1}; 𝔽_{2}) → H^{∗}(ℝP^{n−1}; 𝔽_{2}) ⊗ H^{∗}(ℝP^{n−1}; 𝔽_{2})

satisfies α ↦ α ⊗ 1 + 1 ⊗ α, where α is the generator of H^{1}(ℝP^{n−1}; 𝔽_{2}). Deduce from α^{n} = 0 that n is a power of 2.

Exercise 6.3. Show that if a space X is the union of n open contractible subspaces, then all n-fold cup products α_{1} ∪ ⋯ ∪ α_{n} with |α_{i}| ⩾ 1 are 0 in H^{∗}(X; R). Hint: Induction on n, using the relative cup product.

Exercise 6.4. Using the cup product, show that if m, n ⩾ 1 then any map S^{m+n} → S^{m} × S^{n} induces the zero homomorphism H_{m+n}(S^{m+n}; ℤ) → H_{m+n}(S^{m} × S^{n}; ℤ).

Exercise 6.5. Find a connected space X so that for any field F, the cross product

H^{∗}(X; F) ⊗ H^{∗}(X; F) ⟶^{×} H^{∗}(X × X; F)

is not an isomorphism

Typeset by AMS-TEX