Algebraic Topology II – Lecture 18

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Nathan Broaddus. February 27, 2015. 1 Poincaré Duality. 1.1 Orientation in manifolds. Lemma 1.1. M an n-manifold and K ⊂ M compact. Then. 1. Hk(M|K; R )=0 for k>n. 2. αK ∈ Hk(M|K; R) is 0 if and only if ix. ∗(αK)=0 for all x ∈ K. 3. If α : M → ∐x∈M Hn(M|x; R) is locally consistent then there is unique. αK ∈ Hn(M|K; R ).

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