Poincaré duality

Poincaré duality identifies the cohomology group in degree k of a closed oriented n-manifold with its homology group in degree n−k, via cap product with the fundamental class.

Cap product: An operation combining a cohomology class and a homology class to produce a homology class of lower degree. Capping with the fundamental class converts degree-k cohomology into degree n−k homology.
Singular homology: A homology theory built from formal sums of continuous maps from simplices into a topological space. It provides the homology groups appearing on the theorem's target side.
Henri Poincaré: A French mathematician whose work helped establish topology and the qualitative study of differential equations. His work on analysis situs established the ideas behind manifold duality.
Poincaré–Lefschetz duality: A duality theorem relating relative homology and cohomology for compact oriented manifolds with boundary. It replaces the closed-manifold statement when a boundary is present.
Fundamental class: The distinguished top-dimensional homology class representing the orientation of a connected closed manifold. It supplies the geometric class used in the duality isomorphism.
Singular cohomology: A cohomology theory formed from algebraic duals of singular chain groups, with a natural cup product. It supplies the cohomology groups identified with complementary homology.
Analysis situs: The historical name for early topological study of properties preserved under continuous deformation. Poincaré duality grew from this foundational program in topology.
Alexander duality: A theorem relating the reduced cohomology of a subspace to the homology of its complement in a sphere. It is a complementary duality theorem for subspaces and their complements, not manifold self-duality.
Orientation: A consistent choice of local orientation across a manifold. The theorem uses orientation to define an integral fundamental class.
Closed manifold: A compact manifold without boundary. Compactness and absence of boundary are central hypotheses of the classical theorem.

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