Vector fields and genus in dimension 3

Pierre Dehornoy (1) , Ana Rechtman (2)

1 IF - Institut Fourier
2 IRMA - Institut de Recherche Mathématique Avancée

Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to the helicity.

Domaines: Systèmes dynamiques [math.DS]; Topologie géométrique [math.GT]

Pierre Dehornoy, Ana Rechtman. Vector fields and genus in dimension 3. International Mathematics Research Notices, 2022, 2022 (5), pp.3262-3277. ⟨10.1093/imrn/rnaa255⟩. ⟨hal-02307726v2⟩