Three-manifolds that do not admit veering triangulations

Abstract

We show that if a three-manifold $M$ admits a veering triangulation, then the dual spine of every essential triangulation of $M$ can be smoothed to a branched surface. We use this criterion to identify a large collection of cusped hyperbolic three-manifolds that do not admit veering triangulations. We then use the same criterion to construct the first infinite family of cusped hyperbolic three-manifolds without veering triangulations.