On the homotopy fixed points of Maurer-Cartan spaces with finite group actions

TL;DR

This paper develops a theory connecting Maurer-Cartan spaces with finite group actions, showing fixed points and homotopy fixed points are homotopy equivalent under certain conditions, and provides rational models for related mapping spaces.

Contribution

It establishes a homotopy equivalence between fixed points and homotopy fixed points of Maurer-Cartan spaces with finite group actions, extending the understanding of equivariant $L_$ algebraic structures.

Findings

Fixed points inclusion is a homotopy equivalence under certain conditions.

Provides rational algebraic models for fixed and homotopy fixed points.

Applicable to mapping spaces with finite group actions.

Abstract

We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) $L_\infty$ algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the $L_\infty$ algebra is concentrated in non-negative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite group.