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arXiv · 1602.06887

Van Est isomorphism for homogeneous cochains

Alejandro Cabrera·Thiago Drummond
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Abstract

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, 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. 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.

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Alejandro Cabrera, Thiago Drummond. 2016-02-22. Van Est isomorphism for homogeneous cochains. https://doi.org/10.2140/pjm.2017.287.297

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