Lefschetz duality
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.[1] There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.
Formulations
Let M be an orientable compact manifold of dimension n, with boundary , and let be the fundamental class of the manifold M. Then cap product with z (or its dual class in cohomology) induces a pairing of the (co)homology groups of M and the relative (co)homology of the pair . Furthermore, this gives rise to isomorphisms of with , and of with for all .[2]
Here can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples. Let decompose into subspaces A and B, themselves compact orientable manifolds with common boundary Z, which is the intersection of A and B. Then, for each , there is an isomorphism[3]
Notes
References
- "Lefschetz_duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Lefschetz, Solomon (1926), "Transformations of Manifolds with a Boundary", Proceedings of the National Academy of Sciences of the United States of America, 12 (12), National Academy of Sciences: 737–739, doi:10.1073/pnas.12.12.737, ISSN 0027-8424, JSTOR 84764, PMC 1084792, PMID 16587146