A formula for the Euler class of foliations

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 $Γ(\mathcal B)$ representing the Poincaré dual of the Euler class of $\mathcal F$. This construction provides an effective way to track how the Euler class of a foliation carried by a hierarchy changes when the orientation of a decomposing disk is reversed. We also show how our formula generalises previous results of Lackenby and Dunfield.