EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 9, December 15th 2005

Solutions due on Thursday, 22th of December.

Exercise 9.1. Show that the ℤ-orientable manifold ℂP^{2n} does not admit any self homotopy equivalence f : ℂP^{2n} → ℂP^{2n} which reverses a given ℤ-orientation.

Exercise 9.2. Let M be a compact manifold of dimension n, such that M is homotopy equivalent to a suspension ΣY for some path-connected space Y. Show that H^{∗}(M; ℤ) ≅ H^{∗}(S^{n}; ℤ). Can you find a continuous map that realizes this isomorphism ?

Exercise 9.3. Prove the following result on colimits (which we used in the proof of the Poincaré Duality Theorem). Let I and J be directed sets, and assume that J has a filtration {J_{i}}_{i∈I} with J_{i} ⊆ J_{i′} whenever i ⩽ i′ and J = ⋃_{i∈I} J_{i}. Given a functor F : J → R−modules, show that the natural map

colim_{i∈I} colim_{j∈J_{i}} F (j) → colim_{j∈J} F (j)

is an isomorphism.

Exercise 9.4. Let A be the “Hawaiian earrings”, viewed as the compact subset of S^{2} = ℝ^{2} ∪ {∞} defined by A = ⋃_{n⩾1} A_{n}, where A_{n} is the circle in ℝ^{2} with center (1/n, 0) and radius 1/n.

(a) Compute the first Čech-cohomology group of A from the definition, and verify that your result is compatible Alexander Duality.

(b) Show that the natural map Ȟ^{1}(A; ℤ) → H^{1}(A; ℤ) is not surjective.

Exercise 9.5. Let Y ⊂ ℝ^{2} be the subspace Y = {(x, sin(1/x)) ∈ ℝ^{2}|x ≠ 0}, and let A be its closure in S^{2} = ℝ^{2} ∪ {∞}.

(a) Some question as (a) in the previous exercise.

(b) Show that the natural map Ȟ^{1}(A; ℤ) → H^{1}(A; ℤ) is not injective.

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