The Boyer–Gordon–Watson–Juhász $L$-space conjecture

Let $Y$ be an irreducible rational homology sphere. A three-manifold is an $L$-space when its Heegaard Floer module is free, and a group is left-orderable when it admits an ordering $<$ such that

$$a<b\quad\Longleftrightarrow\quad ga<gb.$$

The $L$-space conjecture. The following statements are equivalent:

1. $Y$ is an $L$-space, that is, $\operatorname{HF}^-(Y)$ is free.
2. $Y$ does not support a taut foliation.
3. $\pi_1(Y)$ has no left-ordering.

This conjecture relates the Heegaard Floer module of a rational homology sphere to taut foliations and the orderability of its fundamental group. It is known for graph manifolds and several other families, but remains open in general.