Van Est isomorphism for homogeneous cochains

Alejandro Cabrera and Thiago Drummond

Vol. 287 (2017), No. 2, 297–336

DOI: 10.2140/pjm.2017.287.297

Abstract

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. We show that their differentiable cohomology admits a refinement by considering the complex of cochains which are $k$-homogeneous on the linear fiber. Our main result is a van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case $k=1$ yields a van Est map for representations up to homotopy on $2$-term graded vector bundles and, moreover, to a new proof of a rigidity conjecture posed by Crainic and Moerdijk. Arbitrary $k$-homogeneous cochains on suitable VB-groupoids lead to a novel van Est theorem for differential forms on Lie groupoids with values in a representation.

Keywords

Lie groupoids, van Est theorem

Mathematical Subject Classification 2010

Primary: 22A22, 53D17

Milestones

Received: 14 May 2016

Revised: 28 June 2016

Accepted: 21 August 2016

Published: 9 March 2017