Hatcher-Algebraic-Topology · Problem 35

Cohomology

Mathematics — Graph Theory · poincare-duality-noncompact-boundary

Problem

If MM is a noncompact RR-orientable nn-manifold with boundary ∂M\partial M having a collar neighborhood in MM, show that there are Poincar\'e duality isomorphisms Hck(M;R)≈Hn−k(M,∂M;R)H^k_c(M;R)\approx H_{n-k}(M,\partial M;R) for all kk, using the five-lemma and the following diagram: ⋯→Hck−1(∂M;R)→Hck(M,∂M;R)→Hck(M;R)→Hck(∂M;R)→⋯\cdots \to H^{k-1}_c(\partial M;R) \to H^k_c(M,\partial M;R) \to H^k_c(M;R) \to H^k_c(\partial M;R) \to \cdots ⋯→Hn−k(∂M;R)→Hn−k(M;R)→Hn−k(M,∂M;R)→Hn−k−1(∂M;R)→⋯\cdots \to H_{n-k}(\partial M;R) \to H_{n-k}(M;R) \to H_{n-k}(M,\partial M;R) \to H_{n-k-1}(\partial M;R) \to \cdots

Related theory

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