Intersection Homology

Jean-Paul Brasselet ()
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Jean-Paul Brasselet: I2M, CNRS and Université d’Aix-Marseille

Chapter Chapter 5 in Handbook of Geometry and Topology of Singularities II, 2021, pp 223-308 from Springer

Abstract: 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. A huge and fantastic step forward was taken by Mark Goresky and Robert MacPherson by the simple but brilliant idea of rediscovering duality by restricting oneself to chains only meeting the singular part of a stratified singular variety in controlled dimensions. Intersection homology was born. In this survey, we recall the first geometric definition as well as the theoretical sheaf definition allowing to describe the main properties of the intersection homology. Fruitful and unexpected developpments have been obtained in the context of singular varieties. For instance de Rham’s theorem and Lefschetz’s fixed point theorem find their place in the theory of intersection homology. The same is true for Morse’s theory (see the Mark Goresky’s survey in this Handbook, Chap. 5, Vol. I). In the last section, we provide some applications of intersection homology, for example concerning toric varieties or asymptotic sets. It must be said that the main application and source, itself, of innumerable applications is the fascinating and fruitful topic of perverse sheaves, which unfortunately it is not possible to develop in such a survey.

Date: 2021
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DOI: 10.1007/978-3-030-78024-1_5

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