On Losik classes of diffeomorphism pseudogroups

TL;DR

This work extends Losik's approach to characteristic classes of diffeomorphism pseudogroups by formulating explicit differential-form representatives for the Godbillon-Vey-Losik (GVL) class and the first Chern-Losik (CL) class on second-order frame spaces associated with a quotient $M/P$. It shows how these classes can be realized as a volume form on a $D_{2n+1}$-space $A(M/P)$ and a symplectic form on a $D_{2n}$-space $B(M/P)$, and introduces simplified variants via $\mathrm{SL}(n,\mathbb{R})$ and $\mathrm{SL}(n,\mathbb{R})\times\mathbb{Z}_2$ reductions to obtain efficient local models. The paper also connects these invariants to vector-field dynamics by providing criteria for triviality and illustrating nontrivial examples on $D^2$, highlighting how singularity data influence the Losik characteristic classes. These results deepen the link between diffeomorphism pseudogroups, Gelfand-Fuchs cohomology, and geometric structures on reduced frame bundles, with potential applications to foliations and dynamical systems.

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.

Theorems & Definitions (20)

• Definition 1
• Proposition 1
• Lemma 1
• Definition 2
• Definition 3
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6