nLab
Poincaré duality space

Contents

Context

Duality

Integration theory

Contents

Definition

Definition

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

()[X]:H (X)H d(X) (-) \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 XX is moreover a CW-complex then this it is sometimes called a Poincaré complex or even Poincaré manifold.

See at Poincaré duality for more.

References

In the general concext of spectral geometry (spectral triples):

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

Last revised on March 20, 2014 at 07:57:05. See the history of this page for a list of all contributions to it.