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Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

Published online by Cambridge University Press:  20 November 2018

Urtzi Buijs ,
Yves Félix ,
Aniceto Murillo  and
Daniel Tanré
Urtzi Buijs
Affiliation:
Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080-Málaga, España. e-mail: ubuijs@uma.esaniceto@uma.es
Yves Félix
Affiliation:
Institut de Mathématiques et Physique, Université Catholique de Louvain-la-Neuve, Louvainla-Neuve, Belgique. e-mail: Yves.felix@uclouvain.be
Aniceto Murillo
Affiliation:
Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080-Málaga, España. e-mail: ubuijs@uma.esaniceto@uma.es
Daniel Tanré
Affiliation:
Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France. e-mail: Daniel.Tanre@univ-lille1.fr
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Abstract

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In a previous work, we associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Keywords

55P6216E45complete diÒerential graded Lie algebraMaurer–Cartan elementsrational homotopy theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 470 - 477
Copyright
Copyright © Canadian Mathematical Society 2017

References

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