L-spaces, taut foliations, and graph manifolds

TL;DR

This paper establishes a deep connection between the existence of coorientable taut foliations, L-spaces, and the left-orderability of the fundamental group in closed orientable graph manifolds, unifying several conjectures.

Contribution

It proves that a closed orientable graph manifold admits a coorientable taut foliation if and only if it is not an L-space, linking foliation properties with L-space classification.

Findings

Y admits a coorientable taut foliation iff Y is not an L-space

Y is an L-space iff its fundamental group is not left-orderable

Unified foliation and L-space criteria for graph manifolds

Abstract

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\pi_1(Y)$ is not left-orderable.