Hatcher-Algebraic-Topology · Problem 34

Poincaré duality says that cap product with the fundamental class $[M]$ (or $[\partial M]$) turns cohomology into homology, reversing dimensions. The key insight is that this duality must respect the long exact sequences of the pair $(M, \partial M)$—the boundary maps and inclusion maps that relate the manifold to its boundary should commute with cap product up to sign. You're checking that duality is a natural transformation: the diagram's commutativity ensures that the algebraic structure of the pair survives the duality isomorphism, so homological and cohomological information about $M$ and $\partial M$ remain compatible.

Poincaré duality says that cap product with the fundamental class $[M]$ (or $[\partial M]$) turns cohomology into homology, reversing dimensions. The key insight is that this duality must respect the long exact sequences of the pair $(M, \partial M)$—the boundary maps and inclusion maps that relate the manifold to its boundary should commute with cap product up to sign. You're checking that duality is a natural transformation: the diagram's commutativity ensures that the algebraic structure of the pair survives the duality isomorphism, so homological and cohomological information about $M$ and $\partial M$ remain compatible.