The L-space conjecture

Let $M$ be an irreducible, rational homology $3$-sphere other than $\mathbb{S}^3$. The L-space conjecture. The following are equivalent:

1. The fundamental group of $M$ is left-orderable.
2. $M$ supports a coorientable taut foliation.
3. $M$ is not an L-space.

The conjecture connects left-orderability, taut foliations, and Heegaard–Floer homology. It is a central open conjecture in the study of $3$-manifold groups; the exclusion of $\mathbb{S}^3$ avoids the convention that its trivial group is left-orderable.