The L-space conjecture for rational homology 3-spheres

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The L-space conjecture for rational homology 3-spheres

   Let $M$ be a rational homology 3-sphere. An L-space is a rational homology sphere whose Heegaard Floer homology satisfies

   $$\operatorname{rank}\widehat{HF}(M)=|H_{1}(M;\mathbb{Z})|.$$

   A group is left-orderable if it admits a strict total order invariant under left multiplication. The L-space conjecture. The fundamental group $\pi_{1}(M)$ is non-left-orderable if and only if $M$ is an L-space. This conjecture proposes an equivalence between a Floer-theoretic condition on a rational homology 3-sphere and the orderability of its fundamental group; the supplied source states it as a remarkable conjecture but gives no resolution.

   source: Tetsuya Ito, “Non-left-orderable double branched coverings”, arXiv:1106.1499 (2011).

2. The L-space conjecture for rational homology 3-spheres

   Let $Y$ be a rational homology 3-sphere. A group is left-orderable if it admits a total ordering $<$ such that $a<b$ implies $ca<cb$ for all $a,b,c$ in the group. The manifold $Y$ is an $L$-space if

   $$\operatorname{rank}\widehat{HF}(Y)=|H_1(Y;\mathbb{Z})|,$$

   where $\widehat{HF}(Y)$ denotes its hat-flavored Heegaard Floer homology. The $L$-space conjecture. The following conditions are equivalent: $Y$ is an $L$-space; $\pi_1(Y)$ is not left-orderable; and $Y$ does not admit a taut foliation. This conjecture seeks to relate Heegaard Floer homology, orderability of fundamental groups, and taut foliations in 3-manifold topology. Its general status is not specified in the supplied source context.

   source: Ollie Thakar, “Left-orderable surgeries on the knot 6_2 via hyperbolic PSL(2,R)-representations”, arXiv:2307.00107 (2023).