Algebraic Topology · Problem 35

Cohomology — Allen Hatcher

Mathematics — Topology · poincare-duality-noncompact-boundary

Problem

M

is a noncompact

R

-orientable

n

-manifold with boundary

\partial M

having a collar neighborhood in

M

, show that there are Poincar\'e duality isomorphisms

H^k_c(M; R) \approx H_{n-k}(M, \partial M; R)

for all

k

, using the five-lemma and the following diagram:

\begin{array}{ccccccc} \cdots & \longrightarrow & H^{k-1}_c(\partial M; R) & \longrightarrow & H^k_c(M, \partial M; R) & \longrightarrow & H^k_c(M; R) & \longrightarrow & H^k_c(\partial M; R) & \longrightarrow & \cdots \\ & & \downarrow D_{\partial M} & & \downarrow D_M & & \downarrow D_M & & \downarrow D_{\partial M} & \\ \cdots & \longrightarrow & H_{n-k}(\partial M; R) & \longrightarrow & H_{n-k}(M; R) & \longrightarrow & H_{n-k}(M, \partial M; R) & \longrightarrow & H_{n-k-1}(\partial M; R) & \longrightarrow & \cdots \end{array}