Talk:Poincaré duality/Comments
From Wikipedia, the free encyclopedia
One of the very important results in algebraic topology, and one with major influence elsewhere too. Should aim to get this to high level of completeness. Things to add:
- Duality in cohomology in terms of cup product and evaluation against the fundamental class; the de Rham picture with differential forms and integration is particularly good as an introduction;
- More details the cap-product-based pairing currently discussed;
- Discussion on relaxing the conditions of the "easy" case: orientability (use orientation bundle or Z/2Z coefficients), compactness (use cohomology with supports or homology/cohomology pairing) and smoothness (intersection cohomology);
- Brief introduction to Verdier duality and how it generalises the previous models;
- Discuss restrictions PD implies for, e.g., Betti numbers of smooth compact manifolds;
- Add applications in differential geometry / topology, elsewhere;
- Give pointers to Poincaré-type duality theorems in other fields (Serre duality, the more general coherent duality in algebraic / analytic geometry, duality in étale cohomology)
Stca74 13:22, 15 May 2007 (UTC)
