On Losik classes of diffeomorphism pseudogroups

TL;DR

This paper explores characteristic classes of diffeomorphism pseudogroups, providing explicit formulas and geometric representations for these classes, with applications to specific low-dimensional examples.

Contribution

It introduces explicit expressions for Godbillon-Vey-Losik and Chern-Losik classes and constructs geometric models representing these classes on reduced frame bundles.

Findings

Explicit formulas for characteristic classes are derived.

Geometric models represent classes as volume and symplectic forms.

Examples in dimension 2 illustrate the theory.

Abstract

Let $P$ be a pseudogroup of local diffeomorphisms of an $n$-dimensional smooth manifold $M$. Following Losik we consider characteristic classes of the quotient $M/P$ as elements of the de~Rham cohomology of the second order frame bundles over $M/P$ coming from the generators of the Gelfand-Fuchs cohomology. We provide explicit expressions for the classes that we call Godbillon-Vey-Losik class and the first Chern-Losik class. Reducing the frame bundles we construct bundles over $M/P$ such that the Godbillon-Vey-Losik class is represented by a volume form on a space of dimension $2n+1$, and the first Chern-Losik class is represented by a symplectic form on a space of dimension $2n$. Examples in dimension 2 are considered.