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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

[1] Buijs, U., Felix, Y., Murillo, A., and Tanre, D., Lie models of simplicial sets and representability of the Quillen functor. arxiv:1 508.01442Google Scholar
[2] Buijs, U., Felix, Y., Murillo, A., and Tanre, D., The Deligne groupoid of the Lawrence-Sullivan interval. Topology Appl. 204(2016), 1–7. http://dx.doi.Org/10.101 6/j.topol.201 6.02.004 Google Scholar
[3] Buijs, U., Felix, Y., Murillo, A., and Tanre, D., Rational Lie models for non-simply connected spaces and Bousfield-Kan completion. arxiv:1 601.05331Google Scholar
[4] Buijs, U. and Murillo, A., The Lawrence-Sullivan interval is the right model ofl+. Algebr. Geom. Topol. 13(2013), no. 1, 577–588. http://dx.doi.Org/10.2140/agt.2013.13.577 Google Scholar
[5] Felix, Y., Halperin, S., and Thomas, J.-C., Rational homotopy theory. Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001. http://dx.doi.Org/10.1007/978-1 -4613-0105-9 Google Scholar
[6] Felix, Y., Halperin, S., and Thomas, J.-C., Rational homotopy theory. II. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. http://dx.doi.Org/10.1142/9473 Google Scholar
[7] Gomez-Tato, A., Halperin, S., and Tanre, D., Rational homotopy theory for non-simply connected spaces. Trans. Amer. Math. Soc. 352(2000), no. 4,1493-1525. http://dx.doi.Org/10.1090/S0002-9947-99-02463-0 Google Scholar
[8] Lawrence, R. and Sullivan, D., A formula for topology/deformations and its significance. Fund. Math. 225(2014), no. 1, 229–242. http://dx.doi.Org/10.4064/fm225-1-10 Google Scholar
[9] Parent, P.-E. and Tanre, D., Lawrence-Sullivan models for the interval. Topology Appl. 159(2012), no. 1, 371–378. http://dx.doi.Org/10.101 6/j.topol.2011.1 0.006 Google Scholar
[10] Quillen, D., Rational homotopy theory. Ann. of Math. (2) 90(1969), 205–295. http://dx.doi.Org/10.2307/1 970725 Google Scholar
[11] Sullivan, D., Infinitesimal computations in topology. Inst. Hautes Etudes Sci. Publ. Math. 47(1977), 269–331. Google Scholar
[12] Tanre, D., Homotopie rationnelle: modeles de Chen, Quillen, Sullivan. Lecture Notes in Mathematics, 1025, Springer-Verlag, Berlin, 1983. http://dx.doi.Org/!0.1007/BFb0071482 Google Scholar