A Crossed Module giving the Godbillon-Vey Cocycle

TL;DR

This paper constructs crossed modules linked to the Godbillon-Vey class for various Lie algebras of vector fields, revealing new algebraic structures related to geometric and topological invariants.

Contribution

It introduces explicit crossed modules associated with the Godbillon-Vey class for several Lie algebras of vector fields, expanding the understanding of their algebraic and geometric properties.

Findings

Construction of crossed modules for Lie algebras of vector fields.

Connection between crossed modules and the Godbillon-Vey class.

Extension of known structures to new classes of vector fields.

Abstract

We find crossed modules, i.e. certain 4 term exact sequences, associated to the Godbillon-Vey class for W_1, Vect(S^1), Vect_{1,0}(\Sigma) and Hol(\Sigma_r), i.e. for the Lie algebras of formal vector fields in 1 variable, vector fields on the circle, differentiable vector fields of holomorphic type on a Riemann surface and holomorphic vector fields on an open Riemann surface.