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All known realizations
of complete Lie algebras coincide
Yves Félix, Mario Fuentes and Aniceto Murillo |
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Algebraic & Geometric Topology 25 (2025) 1155–1167
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Abstract |
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We prove that for any reduced differential graded Lie algebra , the classical Quillen geometrical realization is homotopy equivalent to the realization constructed via the cosimplicial free complete differential graded Lie algebra . As the latter is a deformation retract of the Deligne–Getzler–Hinich realization we deduce that, up to homotopy, all known topological realization functors of complete differential graded Lie algebras coincide. Immediate consequences of our main result include an elementary proof of the Baues–Lemaire conjecture and the description of the Quillen realization as a representable functor. |
Keywords
CDGAs, Baues–Lemaire conjecture, Quillen realization
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Mathematical Subject Classification
Primary: 17B55, 55P62
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References |
Publication
Received: 20 July 2023
Revised: 24 February 2024
Accepted: 23 March 2024
Published: 16 May 2025
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Authors |
| Yves Félix | |
| Département de Mathématiques Université Catholique de Louvain Louvain la-Neuve Belgium |
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| Mario Fuentes | |
| CIMI, Insitut de Mathématiques Université Paul Sabatier Toulouse France |
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| Aniceto Murillo | |
| Departamento de Álgebra, Geometría y
Topología Universidad de Málaga Málaga Spain |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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