Equivariant cohomology for differentiable stacks

TL;DR

This paper develops models of equivariant cohomology tailored for differentiable stacks with Lie group actions, extending classical manifold results and introducing spectral sequences that generalize known cohomological tools.

Contribution

It introduces new models of equivariant cohomology for differentiable stacks and derives spectral sequences generalizing classical results like Bott's spectral sequence.

Findings

Constructed models extend classical equivariant cohomology to stacks.

Derived spectral sequences that converge to classifying space cohomology.

Generalized Bott's spectral sequence for differentiable stacks.

Abstract

We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.