Euler class of taut foliations and Dehn filling

Ying Hu

Published 2019-12-03Version 1

We study the Euler class of co-orientable taut foliations on rational homology spheres. Given a rational homology solid torus $X$, we give necessary and sufficient conditions for the Euler class of taut foliations on Dehn fillings of $X$ that are transverse to the core of the filling solid torus to vanish, from which restrictions on the range of the filling slopes are derived. Precise calculations are done for taut foliations that are carried by certain nice branched surfaces. Implications of our results regarding left-orderability of $3$-manifold groups and the L-space conjecture are discussed. Our method also applies to non-integrable oriented plane fields over a rational homology sphere.