Natural operations in Intersection Cohomology

TL;DR

This paper constructs simplicial sets for intersection cohomology of stratified spaces, enabling natural operations and classifying spaces, thus extending classical cohomology concepts to the stratified setting.

Contribution

It introduces a new simplicial set construction for intersection cohomology, providing a framework for natural operations and classifying spaces in stratified spaces.

Findings

Constructed simplicial sets respecting stratification

Defined functors from perversities to cochain complexes

Identified classifying spaces as Joyal's projective cones

Abstract

Eilenberg-MacLane spaces, that classify the singular cohomology groups of topological spaces, admit natural constructions in the framework of simplicial sets. The existence of similar spaces for the intersection cohomology groups of a stratified space is a long-standing open problem asked by M. Goresky and R. MacPherson. One feature of this work is a construction of such simplicial sets. From works of R. MacPherson, J. Lurie and others, it is now commonly accepted that the simplicial set of singular simplices associated to a topological space has to be replaced by the simplicial set of singular simplices that respect the stratification. This is encoded in the category of simplicial sets over the nerve of the poset of strata. For each perversity, we define a functor from it, with values in the category of cochain complexes over a commutative ring. This construction is based upon a simplicial blow up and the associated cohomology is the intersection cohomology as it was defined by M. Goresky and R. MacPherson. This functor admits an adjoint and we use it to get classifying spaces for intersection cohomology. Natural intersection cohomology operations are understood in terms of intersection cohomology of these classifying spaces. As in the classical case, they form infinite loop spaces. In the last section, we examine the depth one case of stratified spaces with only one singular stratum. We observe that the classifying spaces are Joyal's projective cones over classical Eilenberg-MacLane spaces. We establish some of their properties and conjecture that, for Goresky and MacPherson perversities, all intersection cohomology operations are induced by classical ones.