A formula for the Euler class of foliations

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 ${\mathrm{\Gamma }}_m(\mathcal{B})$ representing the Poincaré 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.