The Euler Class and Flux Homomorphisms under Non-Orientability

2025/08/18 by KyeongRo Kim, Kim, KyeongRo, Shuhei Maruyama +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Geometric Topology (math.GT) #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2508.12874

openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an orientable surface with an area form, there are two invariants of area-preserving dynamics, the flux homomorphism and the Calabi invariant. Tsuboi found a remarkable connection between the Calabi invariant on the closed disk and a topological invariant -- the Euler class. In this paper, we investigate a relationship between the Euler class and the flux homomorphism for non-orientable compact surfaces with one boundary component. Furthermore, we prove the simplicity of the kernel of the flux homomorphisms in this non-orientable setting, which implies the non-existence of invariants analogous to the Calabi invariant.

Citations

• Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band
• The sharp C⁰-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces
• Structure theorems for actions of homeomorphism groups

Related

• Topological extension of Calabi invariants and non-simpleness of the area preserving homeomorphism group of the disc
• Mapping class group orbit closures for non-orientable surfaces
• The canonical class and the C^∞ properties of Kähler surfaces
• The flux homomorphism and central extensions of diffeomorphism groups
• G-invariant quasimorphisms and symplectic geometry of surfaces