Intersection homology duality and pairings: singular, PL and sheaf-theoretic

2018/12/31 by Greg Friedman, James E. McClure
Mathematics · #Algebraic Geometry and Number Theory #Cohomology #Cup product #De Rham cohomology #Equivariant cohomology #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Intersection (aeronautics) #Intersection homology #Isomorphism (crystallography) #Mathematics #Poincaré duality #Pure mathematics #Sheaf #math.AT #math.GT #msc:55N30 #msc:55N33 #msc:55N45 #msc:57Q99

paper · pdf · doi:10.2140/agt.2021.21.3221

published as Algebr. Geom. Topol. 21 (2021) 3221-3301 · 69 pages; revised from previous version; to appear in Algebraic & Geometric Topology

arxiv created 2020/07/22 · openalex publication_date 2021/12/28 · arxiv updated 2022/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky-MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector, and Saralegi preserves product structures.

Citations

• Sheaves in Topology
• Lectures on Modules and Rings
• Intersection homology theory
• On the chain-level intersection pairing for PL manifolds
• Intersection homology II
• Sheaves on Manifolds
• Blown-up intersection cohomology
• Sheaf Theory
• Differential Analysis on Complex Manifolds