On Losik classes of diffeomorphism pseudogroups

2025/05/27 by Yaroslav Bazaikin, Bazaikin, Yaroslav V., Yury D. Efremenko +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.20778

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Problem list on Foliations and Diffeomorphism groups
• Foliations and diffeomorphism groups
• Relativity of reductive chain complexes of non-abelian simplexes
• On holonomy groupoid of vertex operator algebra bundles on foliations
• Orbit spaces and leaf spaces of foliations as generalized manifolds
• On the continuous cohomology of diffeomorphism groups
• Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes

Related

• McDuff's secondary class and the Euler class of foliated sphere bundles
• Characteristic classes associated to Q-bundles
• Comparison of approaches to characteristic classes of foliations
• Haefliger's differentiable cohomology
• On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives