From differences to differential forms

Alan Weinstein, University of California, Berkeley, Mathematics

The van Est homomorphism takes functions on a Lie group and its cartesian powers and produces skew-symmetric forms on the Lie algebra. By generalizing this construction to groupoids, we obtain in particular a way to go from functions of (n+1)-tuples of points in a manifold M to differential n-forms on M. The construction can be further extended (this was done by Bott, Shulman, and Stasheff in the 1970's) to allow the domain to consist of forms rather than functions. The objects in the range now consist of more complicated geometric objects which are perhaps best understood in the language of supermanifolds.