Algebraic Topology · Problem 34

Cohomology — Allen Hatcher

Mathematics — Topology · poincare-duality-boundary-diagram

Problem

For a compact manifold

M

verify that the following diagram relating Poincar\'e duality for

M

and

\partial M

is commutative, up to sign at least:

\begin{array}{ccccccc} H^{k-1}(\partial M; R) & \longrightarrow & H^k(M, \partial M; R) & \longrightarrow & H^k(M; R) & \longrightarrow & H^k(\partial M; R) \\ \downarrow [\partial M] \frown & & \downarrow [M] \frown & & \downarrow [M] \frown & & \downarrow [\partial M] \frown \\ 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) \end{array}