The L-space homology-sphere classification conjecture

A rational homology 3-sphere $Y$ is an L-space if the rank of its Heegaard Floer homology group $\widehat{\operatorname{HF}}(Y)$ equals the order of $H_1(Y;\mathbb{Z})$. An irreducible homology sphere is one that cannot be expressed as a nontrivial connected sum, and an incompressible torus is an embedded torus whose inclusion induces an injective map on fundamental groups. L-space homology-sphere classification conjecture. If $Y$ is an irreducible homology sphere that is an L-space, then $Y$ is homeomorphic to either $S^3$ or the Poincaré homology sphere. Thus, the conjecture predicts that L-spaces with the homology of the 3-sphere are connected sums of zero or more copies of the Poincaré sphere, with either orientation. It is known for manifolds obtained by Dehn surgery on knots in $S^3$ and for all Seifert fibered spaces, but remains open in general.