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

Moreno-Fernández, José M.; Wierstra, Felix

doi:10.1215/21562261-2024-0004

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

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Moreno-Fernández, José M.
Max Planck Institute for Mathematics, Max Planck Society;

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Wierstra, Felix
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.48550/arXiv.2203.03200
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https://doi.org/10.1215/21562261-2024-0004
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Citation

Moreno-Fernández, J. M., & Wierstra, F. (2024). On the homotopy fixed points of Maurer-Cartan spaces with finite group actions. Kyoto Journal of Mathematics, 64(4), 759-787. doi:10.1215/21562261-2024-0004.

Cite as: https://hdl.handle.net/21.11116/0000-000F-F2AF-B

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.