A Crossed Module Representing the Godbillon–Vey Cocycle

Abstract

We exhibit crossed modules of Lie algebras (i.e. certain four-term exact sequences) corresponding to the Godbillon–Vey generator for the Lie algebras of formal vector fields in one variable, of vector fields on the circle, of differentiable vector fields of holomorphic type and of holomorphic vector fields on a punctured Riemann surface.