L-spaces, taut foliations and the Whitehead link

TL;DR

This paper characterizes when rational homology spheres obtained from Whitehead link surgeries are L-spaces or support taut foliations, establishing the L-space conjecture for these manifolds and exploring related structures.

Contribution

It proves the equivalence between being an L-space and supporting a taut foliation for Whitehead link surgeries, and analyzes the structure of L-space slopes and taut foliations in related contexts.

Findings

Rational homology spheres from Whitehead link surgeries are L-spaces iff they lack taut foliations.

Constructed taut foliations with vanishing Euler class relate to left orderability.

All integer surgeries on the Whitehead link satisfy the L-space conjecture.

Abstract

We prove that if $M$ is a rational homology sphere that is a Dehn surgery on the Whitehead link, then $M$ is not an $L$-space if and only if $M$ supports a coorientable taut foliation. The left orderability of some of these manifolds is also proved, by determining which of the constructed taut foliations have vanishing Euler class. We also present some more general results about the structure of the $L$-space surgery slopes for links whose components are unknotted and with pairwise linking number zero, and about the existence of taut foliations on the fillings of a $k$-holed torus bundle over the circle with some prescribed monodromy. Our results, combined with some results from Roberts--Shareshian--Stein, also imply that all the rational homology spheres that arise as integer surgeries on the Whitehead link satisfy the L-space conjecture.