The L-space conjecture for 3-manifolds

Let $Y$ be a closed, connected $3$-manifold. A group is left-orderable if it is nontrivial and admits a total order invariant under left multiplication, and $Y$ is an L-space if its Heegaard Floer homology satisfies $\widehat{\operatorname{HF}}(Y)\cong\mathbb{Z}^{\lvert H_1(Y;\mathbb{Z})\rvert}$. The L-space conjecture.

$$\pi_{1}(Y)\text{ is not left-orderable if and only if }Y\text{ is an L-space}.$$

This conjecture proposes an equivalence between an algebraic property of the fundamental group and the simplest possible Heegaard Floer homology for a closed, connected $3$-manifold. It is closely related to the existence of taut foliations and is motivated by established results connecting left-orderability, foliations, and Heegaard Floer theory; its general validity remains open.