The Euler Class and Flux Homomorphisms under Non-Orientability

TL;DR

This paper explores the relationship between the Euler class and flux homomorphisms on non-orientable surfaces, establishing the simplicity of flux kernels and the absence of Calabi-like invariants in this setting.

Contribution

It extends the understanding of flux homomorphisms and Euler class interactions to non-orientable surfaces, proving the simplicity of flux kernels and showing no Calabi invariant analog exists.

Findings

Proves the relationship between Euler class and flux homomorphism on non-orientable surfaces.

Establishes the simplicity of the kernel of flux homomorphisms in this context.

Shows the non-existence of Calabi invariant analogs for non-orientable surfaces.

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.