L-space conjecture (equivalence of taut foliation, non–L-space, and left-orderability)

Establish the equivalence, for any irreducible oriented rational homology 3-sphere M, of the following three properties: (i) M supports a cooriented taut foliation; (ii) M is not an L-space, meaning rank \widehat{HF}(M) equals |H1(M, Z)|; and (iii) the fundamental group \pi1(M) is left-orderable. The goal is to fully characterize rational homology 3-spheres by the coincidence of these geometric, Floer-homological, and algebraic conditions.

Background

The L-space conjecture posits deep connections among geometry/dynamics (taut foliations), Heegaard Floer homology (L-space condition), and algebra (left-orderability of the fundamental group) for irreducible rational homology 3-spheres. It has been proven in various special cases, notably that the existence of a cooriented taut foliation implies a manifold is not an L-space, and for all graph manifolds the three properties coincide, but the full equivalence remains unresolved in general.

This paper contributes evidence toward the conjecture by constructing cooriented taut foliations for broad families of surgeries on knots and links under diagrammatic conditions, and by verifying the equivalence of (1) and (2) for all surgeries on the Borromean link. The conjecture serves as the overarching motivation for the results and applications developed here.