EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 4, November 10th 2005

Solutions due on Thursday, 17th of November.

Exercise 4.1. Let π_{n} be the permutation of the set {0, 1, …, n} given by π_{n}(i) = n − 1, and let ε_{n} ∈ {±1} be its signature. Let π : Δ^{n} → Δ^{n} be the affine map that sends the vertex e_{i} to e_{π_{n}(i)}. Consider the homomorphism θ_{n} : S_{n}(X; R) → S_{n}(X; R) defined on a simplex σ : Δ^{n} → X by σ ↦ ε_{n}σ ∘π_{n}, and extended linearly.

(a) Show that {θ_{n}}_{n} is a morphism of chain complexes.

(b) Construct an explicit chain homotopy h_{∗} : S_{∗}(X; R) → S_{∗+1}(X; R) between θ_{∗} and the identity.

(c) Use the Theorem of Acyclic Models to show that θ_{∗} is chain homotopic to the identity.

Exercise 4.2. Determine, by explicit computation with the definition of the cupproduct, the cohomology algebra H^{∗}(S^{1} × S^{1}; ℤ) of the torus.

Here are some suggestions. First find explicit cycles that generate the homology of the torus. Consider the cycles σ = AB and τ = AD in S_{1}(S^{1} × S^{1}; ℤ) and the cycle z = ABC − ADC in S_{2}(S^{1} × S^{1}; ℤ), as pictured below.

[figure: B; C; A; D]

(The torus is the quotient of this square obtained by identifying AB with DC and AD with BC.) Show that

H_{n}(S^{1} × S^{1}; ℤ) ≅ \{ \begin{matrix}ℤ\{σ̄\} ⊕ ℤ\{τ̄\} & if n = 1, \\ ℤ\{z̄\} & if n = 2, \\ 0 & if n ⩾ 3. \end{matrix}

Dualize, and compute the cup product σ̄^{∗} ∪ τ̄^{∗}, where {σ̄^{∗}, τ̄^{∗}} is the basis of H^{1}(S^{1} × S^{1}; ℤ) dual to {σ̄, τ̄}.

Exercise 4.3. Let m ⩾ 2 be an integer. Compute the cohomology algebra

H^{∗}(M(ℤ/m, 1); ℤ/m)

of the Moore space M(ℤ/m, 1) by a method similar to the one proposed in the previous exercise.

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