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The Poincaré–Lefschetz duality theorem is a generalisation for manifolds with boundary. In the non-orientable case, taking into account the sheaf of local orientations, one can give a statement that is independent of orientability: see twisted Poincaré 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 , at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem.
2. Nov. 2021 · Jean-Paul Brasselet. 1398 Accesses. 2 Citations. Abstract. The famous duality theorems for compact oriented manifolds: Poincaré duality between cohomology and homology, and Poincaré-Lefschetz duality, intersection between cycles, are no longer true for a singular variety.
- Jean-Paul Brasselet
- jean-paul.brasselet@univ-amu.fr
4. Dez. 2019 · The Poincare-Alexander-Lefschetz duality. Ask Question. Asked 4 years, 5 months ago. Modified 4 years, 4 months ago. Viewed 2k times. 6. I came across the Poincare-Alexander-Lefschetz duality here: If M M is a closed (compact and no boundary) manifold, B ⊂ A ⊂ M B ⊂ A ⊂ M are closed subsets, then.
10. Mai 2024 · Traditionally Poincaré duality is stated as a duality of chain homology groups. The passage from chain complexes to their homology groups, hence the passage from full homotopy theory to just some invariants, however forgets a lot of information. But it turns out that this can always be lifted: Theorem 0.5.
4. Apr. 2008 · In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space.
their intersections as in Lefschetz. Our proof of generalized Poincare duality is entirely geometric and somewhat similar to the cell-dual cell proof of Poincare: it uses certain ‘basic sets” (43) constructed from a triangulation of X. For convenience, we work in the piecewise linear category. The subanalytic category[8] would work as
