A formula for the Euler class of foliations

TL;DR

This paper introduces a new formula for the Euler class of foliations using the dual graph of a branched surface, enabling classification of realizable classes and generalizing prior results.

Contribution

It provides a novel simplicial 1-cycle formula for the Euler class of foliations and extends existing classifications and theorems in the field.

Findings

Classified which homology classes in the Whitehead link exterior are realizable as Euler classes.

Generalized previous results of Lackenby and Dunfield.

Established a Combinatorial Transverse Surface Theorem for certain branched surfaces.

Abstract

Given a cooriented branched surface $\mathcal B$ fully carrying a foliation $\mathcal F$, we use the dual graph of $\mathcal B$ to define a simplicial 1-cycle $\Gamma_m(\mathcal B)$ representing the Poincar\'e dual of the Euler class of $\mathcal F$ relative to the boundary. As an example, we complete the classification of which homology classes in the Whitehead link exterior are realisable as relative Euler classes of taut foliations. We also show how our formula generalises previous results of Lackenby and Dunfield. Finally, we observe that cooriented branched surfaces whose complement is a union of balls satisfy a Combinatorial Transverse Surface Theorem, in the sense of Landry--Minsky--Taylor.