A spectral sequence for stratified spaces and configuration spaces of points

TL;DR

This paper introduces a new spectral sequence framework for stratified spaces and configuration spaces, enabling computation of cohomology groups and establishing a broad representation stability theorem applicable to algebraic and topological contexts.

Contribution

It develops a sheaf-theoretic spectral sequence for stratified spaces that generalizes existing sequences and applies to both topological and algebraic varieties, leading to a general stability result.

Findings

Derived a spectral sequence for stratified spaces and configuration spaces.

Unified various known spectral sequences as special cases.

Proved a broad representation stability theorem for point configuration spaces.

Abstract

We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology groups of the poset of strata. Several familiar spectral sequences arise as special cases. The construction is sheaf-theoretic and works both for topological spaces and for the \'etale cohomology of algebraic varieties. As an application we prove a very general representation stability theorem for configuration spaces of points.