The Euler Class and Flux Homomorphisms under Non-Orientability

KyeongRo Kim; Shuhei Maruyama

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arXiv · 2508.12874

The Euler Class and Flux Homomorphisms under Non-Orientability

KyeongRo Kim·Shuhei Maruyama

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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.

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BibTeX RIS

KyeongRo Kim, Shuhei Maruyama. 2025-08-18. The Euler Class and Flux Homomorphisms under Non-Orientability. https://arxiv.org/abs/2508.12874

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