Poincaré duality

If M is a compactPlanetmathPlanetmath, oriented, n-dimensional manifold, then there is a canonical (though non-natural (http://planetmath.org/NaturalTransformation)) isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


(where Hk(M,) is the kth homology groupMathworldPlanetmath of M with integer coefficients and Hk(M,) the kth cohomologyPlanetmathPlanetmath (http://planetmath.org/DeRhamCohomology) group) for all q, which is given by cap product with a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Hn(M,) (a choice of a generator here corresponds to an orientation). This isomorphism exists with coefficients in /2 regardless of orientation.

This isomorphism gives a nice interpretationMathworldPlanetmathPlanetmath to cup productMathworldPlanetmath. If X,Y are transverse submanifolds of M, then XY is also a submanifold. All of these submanifolds represent homology classes of M in the appropriate dimensionsMathworldPlanetmath, and


where is cup product, and in intersectionMathworldPlanetmathPlanetmath, not cap product.

Title Poincaré duality
Canonical name PoincareDuality
Date of creation 2013-03-22 13:11:36
Last modified on 2013-03-22 13:11:36
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Theorem
Classification msc 55M05
Synonym Poincaré isomorphism
Related topic DualityInMathematics