Lattice Spectral Sequences and Cohomology of Configuration Spaces

Abstract

For a topological space $X$, we introduce a criterion for the $\mathrm{FI}$ module $H^i({\mathrm{Conf}}_n(X))$ to be finitely generated and give several applications. For instance, if $C$ is a finite connected $CW$ complex, then $X=C\times {\mathbb{R}}^2$ satisfies the criterion. Our main tool is a spectral sequence that we derive from the homological algebra of the partition lattice.

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References in corpus (4)

• FI-modules and the cohomology of modular representations of symmetric groups
• A spectral sequence for stratified spaces and configuration spaces of points
• Comparison of sheaf cohomology and singular cohomology
• Representation stability for the cohomology of arrangements associated to root systems

Cited by in corpus (3)

• A generating function approach to new representation stability phenomena in orbit configuration spaces
• FI-hyperhomology and ordered configuration spaces
• Cohomology of configuration spaces on punctured varieties