The equivariant de Rham complex on a simplicial G_*-manifold

TL;DR

This paper constructs a bisimplicial manifold and de Rham complex for a simplicial Lie group acting on a simplicial manifold, establishing a quasi-isomorphism and cohomology isomorphism with the equivariant complex.

Contribution

It introduces a new bisimplicial manifold framework that links the equivariant de Rham complex to the cohomology of the fat realization.

Findings

The constructed complex is quasi-isomorphic to the equivariant simplicial de Rham complex.

The cohomology of the complex matches that of the fat realization.

A specific cocycle in the equivariant complex is exhibited.

Abstract

We show that when a simplicial Lie group acts on a simplicial manifold $\{X_*\}$, we can construct a bisimplicial manifold and the de Rham complex on it. This complex is quasi-isomorphic to the equivariant simplicial de Rham complex on $\{X_*\}$ and its cohomology group is isomorphic to the cohomology group of the fat realization of the bisimplicial manifold. We also exhibit a cocycle in the equivariant simplicial de Rham complex.