The homotopy fixed point set of Lie group actions on elliptic spaces

Buijs, Urtzi; Félix, Yves; Huerta, Sergio; Murillo, Aniceto

doi:10.1112/plms/pdv015

Cited by 4 publications

(2 citation statements)

References 27 publications

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“…Some classical references on the rational homotopy type of G-spaces include [31,18,1,28]. The relation between fixed and homotopy fixed points for non-discrete group actions on rational (not necessarily simply-connected) spaces has been studied in [8,7].…”

Section: Equivariant Rational Homotopy Theorymentioning

confidence: 99%

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

Moreno-Fernández¹,

Wierstra²

2022

Preprint

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We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) L ∞ 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 ∞ 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.There are other alternative constructions: Getzler's γ • (L), Robert-Nicoud-Vallette's R(L), and Buijs-Félix-Murillo-Tanré's 〈L〉 (see [16,26,9], respectively). We explain in Section 5 that our main results also hold for these smaller alternative simplicial sets. Conventions:In this paper, all the algebraic structures are considered over the field of the rational numbers, but our results hold over any field of characteristic zero. We use a homological grading for L ∞ -algebras. All topological spaces are assumed to be CW-complexes. If V is a graded vector space, and G is a group that acts linearly on V , then the graded vector space of fixed points isand the graded vector space of coinvariants isThe linear dual of a graded vector space V is denoted V ∨ . Acknowledgments:The authors are very grateful to Marc Stephan for teaching them about equivariant homotopy theory, and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.

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Section: Equivariant Rational Homotopy Theorymentioning

confidence: 99%

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

Moreno-Fernández¹,

Wierstra²

2022

Preprint

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“…Our approach is different and conceptually simpler in the sense that we work simplicially and do not rely on the Federer-Schultz spectral sequence. The non-discrete rational case for not necessarily simply-connected spaces has been studied in [7,6].…”

Section: The Sullivan Conjecture For the Maurer-cartan G-simplicial Setmentioning

confidence: 99%

Iterated suspensions are coalgebras over the little disks operad

Moreno-Fernández,

Wierstra

2019

Preprint

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We study the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every n-fold suspension is a coalgebra over the little n-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an n-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an n-fold suspension to be an (n + 1)-fold suspension, i.e. if this cooperation is non-zero then the space is not an (n + 1)-fold suspension. We prove several results in equivariant rational homotopy theory that play an essential role in our results. Namely, we prove a version of the Sullivan conjecture for the Maurer-Cartan simplicial set of certain L ∞ -algebras equipped with a finite group action, and we provide rational models for fixed and homotopy fixed points of (mapping) spaces under some connectivity assumptions in the context of finite groups. We further show that by using the Eckmann-Hilton dual of the Browder operation we can use the rational homotopy groups to detect the difference between certain spaces that are rationally homotopy equivalent, but not homotopy equivalent.

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