Van Est differentiation and integration

  • Eckhard Meinrenken
  • María Amelia SalazarEmail author


The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from Li-Bland and Meinrenken (Enseign Math 61(1–2):93–137, 2015), we revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.



E.M. was supported by an NSERC Discovery Grant. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance code 001. The authors would like to thank the hospitality of Fields Institute where some of this research was carried out.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Instituto de Matematica e EstatisticaUniversidade Federal FluminenseNiteróiBrazil

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