EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 5, November 17th 2005

Solutions due on Thursday, 24th of November.

Exercise 5.1. Let R be a commutative and unital ring, and M, N be R-modules. Describe how the algebras T (M ⊕ N), P (M ⊕ N) and E(M ⊕ N) are related to T (M) and T (N), P (M) and P (N), and E(M) and E(N), respectively.

Exercise 5.2. Consider the polynomial algebra P (x) over ℤ. Define on P (x) a coproduct ψ : P (x) → P (x) ⊗ P (x) by setting ψ(x) = 1 ⊗ x + x ⊗ 1, and extending it to be a map of ℤ-algebras. Describe the algebra Γ(x), which as ℤ-module is the dual (P (x))^{#} of P (x), with product induced by the coproduct of P (x). Also describe the coproduct of Γ(x) induced by the product of P (x).

Exercise 5.3. Consider a product of spheres X = S^{n_{1}} × ⋯ × S^{n_{k}} with n_{i} ⩾ 1. Describe the algebra H^{∗}(X; ℤ).

Exercise 5.4. Let (X, x_{0}) and (Z, y_{0}) be based spaces, and let X ∨ Y be their wedge, i.e. X ∨ Y = (X ⊔ Y)/(x_{0} = y_{0}). Show that there is an isomorphism of (non-unital) R-algebras

H̃^{∗}(X; R) ⊕ H̃^{∗}(Y; R) ≅ H̃^{∗}(X ∨ Y; R),

where the product on the left is component-wise.

Exercise 5.5. Show that if m, n ⩾ 1, then the spaces S^{m}×S^{n} and S^{m}∨S^{n}∨S^{m+n} are not homotopy equivalent.

Exercise 5.6. Let M_{g} be an orientable surface of genus g. Compute the cohomology algebra H^{∗}(M_{g}; ℤ).

Hint: M_{g} is the connected sum of S^{2} with g tori T_{i} = S^{1} × S^{1}, i = 1, …, g, i. e.

M_{g} = [(S^{2} ∖ (D_{1}^{∘} ⊔ ⋯ ⊔ D_{g}^{∘})) ⊔ (T_{1} ∖ D̄_{1}^{∘}) ⊔ ⋯ ⊔ (T_{g} ∖ D̄_{g}^{∘})]/ ∼

Here D_{1}, …, D_{g} are g two-dimensional disks disjointly embedded in S^{2}, D̄_{i} is a disk embedded in T_{i}, and the superscript (−)^{∘} denotes the interior. The relation ∼ identifies ∂D_{i} with ∂D̄_{i} for 1 ⩽ i ⩽ g. Assume g ⩾ 1. The space X = S^{2} ∖ (D_{1}^{∘} ⊔ ⋯ ⊔ D_{g}^{∘}) ⊂ M_{g} is homotopy equivalent to ⋁_{g−1} S^{1}. The quotient of M_{g} by X is homeomorphic to ⋁_{g} T. Use the cofibration up to homotopy

⋁_{g−1} S^{1} → M_{g} → ⋁_{g} T

and the induced long exact sequence in cohomology to solve the exercise.

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