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

  Published:2017-02-21
 Printed: Sep 2017
  • Urtzi Buijs,
    Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59 , 29080-Málaga,, España
  • Yves Félix,
    Institut de Mathématiques et Physique, Université Catholique de Louvain-la-Neuve, Louvain-la-Neuve, Belgique
  • Aniceto Murillo,
    Departamento de Algebra, Geometría y Topología, Universidad de Málaga, Ap. 59, 29080-Málaga,, España
  • Daniel Tanré,
    Département de Mathématiques, UMR 8524, Université de Lille~1, 59655 Villeneuve d'Ascq Cedex, France
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Abstract

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the 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: complete differential graded Lie algebra, Maurer-Cartan element, rational homotopy theory complete differential graded Lie algebra, Maurer-Cartan element, rational homotopy theory
MSC Classifications: 16E45 show english descriptions Differential graded algebras and applications 16E45 - Differential graded algebras and applications
 

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