nLab Poincaré duality space

Contents

analytic integration	cohomological integration
measurable space	Poincaré duality
measure	orientation in generalized cohomology
volume form	(virtual) fundamental class
Riemann/Lebesgue integration of differential forms	push-forward in generalized cohomology/in differential cohomology

Analytic integration

A topological space for which there is $d \in \mathbb{N}$ and a class $[X] \in H_d(X)$ such that the cap product induces isomorphisms

(-) \cap [X] \;\colon\; H^\bullet(X) \stackrel{\simeq}{\to} H_{d-\bullet}(X)

between ordinary cohomology and ordinary homology groups as indicated, is called a Poincaré duality space.

If $X$ is moreover a CW-complex then this it is sometimes called a Poincaré complex or even Poincaré manifold.

See at Poincaré duality for more.

James Munkres, Duality in Manifolds, Chapter 8 in: Elements of Algebraic Topology, Addison-Wesley (1984) [pdf]

Alain Connes, page 10 of Noncommutative geometry and reality, J. Math. Phys. 36 (11), 1995 (pdf)

Last revised on August 14, 2023 at 18:37:23. See the history of this page for a list of all contributions to it.