Graph manifolds Z-homology 3-spheres and taut foliations

TL;DR

This paper proves that certain graph manifolds that are Z-homology 3-spheres admit horizontal foliations, establishing equivalences among several important properties like not being an L-space, having a left-orderable fundamental group, and admitting taut foliations.

Contribution

It demonstrates that all non-trivial Z-homology 3-sphere graph manifolds admit horizontal foliations, linking key properties in 3-manifold topology.

Findings

Non-trivial Z-homology 3-sphere graph manifolds admit horizontal foliations.

Equivalence of being non-L-space, having a left-orderable fundamental group, and admitting taut foliations for these manifolds.

Abstract

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincar\'e homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-oriented taut foliation, are equivalent for graph manifold Z-homology 3-spheres.