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In this exercise sheet, we work out some of the details in the proof of the Poincar´e Duality Theorem

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EXERCISES FOR TOPOLOGY III, WS05/06 Sheet 8, December 8th 2005

Solutions due on Thursday, 15th of December.

In this exercise sheet, we work out some of the details in the proof of the Poincar´e Duality Theorem.

Exercise 8.1. Show that if M is R-oriented of dimension n, and if U ⊂ V ⊂ M are open subsets, then the diagram

Hcq(U) D //

i∗

Hn−q(U)

i∗

Hcq(V) D //Hn−q(V) is commutative, with i∗ induced by the inclusion U ⊂V. Exercise 8.2. Prove the Poincar´e Duality Theorem for Rn.

Exercise 8.3. Consider an open triad (X, U, V), i.e. X is a space and U and V are open subsets. Construct and prove the exactness of the relative Mayer-Vietoris sequence

−→∂ Hq(X, U ∩V)→Hq(X, U)⊕Hq(X, V)→Hq(X, U ∪V)−→∂ Hq−1(X, U ∩V)→ Similarly, prove the existence of the relative Mayer-Vietoris sequence in cohomology

←δ−Hq(X, U∩V)←−Hq(X, U)⊕Hq(X, V)←−Hq(X, U∪V)←−δ Hq−1(X, U∩V)←− Exercise 8.4. LetU andV be open subsets ofX, and consider the relative Mayer- Vietoris sequences of the triad (X, U, V), as constructed in the previous exercise.

Show that if a∈Hn+1(X, U ∪V) and if β ∈Hq(X, U ∩V) then the equality

∂(a)∩β =±a∩δ(β) holds in Hn−q(X).

Exercise 8.5. Prove that taking the colimit over a directed set of indices is an exact functor. Find an example where the colimit functor (over a non-directed indexing category) is not exact.

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