The L-space conjecture for irreducible rational homology 3-spheres

Equivalent formulations 7

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 irreducible rational homology 3-spheres

   Let $Y$ be a closed, connected, orientable $3$-manifold. It is an L-space if it is a rational homology sphere and

   $$\operatorname{rk}\widehat{\operatorname{HF}}(Y)=\left|H_1(Y;\mathbb{Z})\right|.$$

   A group is left-orderable if it admits a strict total ordering $<$ such that $g<h$ implies $fg<fh$ for all elements $f,g,h$. The L-space conjecture. An irreducible rational homology $3$-sphere is an L-space if and only if its fundamental group is not left-orderable. This conjecture would provide a characterization of L-spaces using the topology of their fundamental groups rather than Heegaard Floer homology. The equivalence is known in several important classes of $3$-manifolds, but remains open in general.

   source: Steven Boyer, Cameron McA. Gordon and Liam Watson, “On L-spaces and left-orderable fundamental groups”, arXiv:1107.5016 (2011).

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

   Let $M$ be an irreducible $3$-manifold that is a rational homology sphere. An L-space is a closed rational homology $3$-sphere whose Heegaard–Floer homology has minimal possible rank, namely $\operatorname{rank}\widehat{HF}(M)=|H_1(M,\mathbb Z)|$. A group is left-orderable if it admits a left-invariant total ordering. The L-space conjecture. $M$ is an L-space if and only if $\pi_1(M)$ is not left-orderable. This conjecture proposes a relationship between Heegaard–Floer homology and the orderability of 3-manifold groups; it is stated in the source as unsolved, although it is known in various classes of manifolds.

   source: Michel Boileau, Steven Boyer, Dale Rolfsen and Shicheng Wang, “1-domination of knots”, arXiv:1511.07073 (2015).

3. The L-space Conjecture for irreducible rational homology 3-spheres

   Let $Y$ be an irreducible rational homology 3-sphere. A non-L-space is a rational homology 3-sphere whose Heegaard Floer homology is not simple. The group $\pi_1(Y)$ is left-orderable if it admits a strict total order invariant under left multiplication. A taut foliation on $Y$ is a foliation satisfying the usual tautness condition.

   The L-space Conjecture. The following are equivalent:

   1. $Y$ is a non-L-space.
   2. $\pi_1(Y)$ is left-orderable.
   3. $Y$ admits a taut foliation.

   This conjecture predicts an equivalence between Floer-homological, algebraic, and geometric properties of closed 3-manifolds. Its status is not resolved in the supplied source context.

   source: Siddhi Krishna, “Taut Foliations, Positive 3-Braids, and the L-Space Conjecture”, arXiv:1809.03959 (2020).

4. The L-space conjecture for irreducible rational homology 3-spheres

   Let $M$ be an irreducible $\mathbb{Q}$-homology 3-sphere. A group is left-orderable if it admits a strict total ordering invariant under left multiplication, and an L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. A coorientable taut foliation is a coorientable taut foliation of $M$.

   The L-space conjecture. The following are equivalent:

   1. $\pi_1(M)$ is left-orderable.
   2. $M$ is not an L-space.
   3. $M$ admits a coorientable taut foliation.

   The conjecture connects left-orderability, Heegaard Floer theory, and foliations in 3-manifold topology. Its general status is not specified in the source.

   source: Khanh Le, “Left orderability for surgeries on the [1,1,2,2,2j] two-bridge knots”, arXiv:2109.05957 (2021).

5. The L-space conjecture for irreducible rational homology 3-spheres

   Let $M$ be an irreducible oriented rational homology $3$-sphere. An L-space is a rational homology $3$-sphere whose Heegaard Floer homology is minimal, and a group is left orderable if its fundamental group admits a left-invariant total order. L-space conjecture. The following conditions are equivalent:

   1. $M$ supports a cooriented taut foliation.
   2. $M$ is not an L-space.
   3. $\pi_1(M)$ is left orderable.

   This conjecture relates taut foliations, Heegaard Floer homology, and orderability of fundamental groups. The paper proves the equivalence between the first two conditions for rational homology spheres obtained by Dehn surgery on the Whitehead link, and, together with other results, establishes the conjecture for all rational homology spheres arising as integer surgeries on the Whitehead link; the general statement remains open.

   source: Diego Santoro, “L-spaces, taut foliations and the Whitehead link”, arXiv:2201.01211 (2026).

6. The L-space conjecture for irreducible rational homology 3-spheres

   Let $Y$ be an irreducible rational homology 3-sphere. A non-L-space is a 3-manifold whose Heegaard Floer homology is not "small"; a group is left-orderable when it admits a total ordering invariant under left multiplication. A taut foliation on $Y$ is a foliation admitting a simple closed curve meeting every leaf transversely. The L-space conjecture. The following conditions are equivalent:

   1. $Y$ is a non-L-space.
   2. $\pi_1(Y)$ is left-orderable.
   3. $Y$ admits a taut foliation.

   The conjecture seeks to organize Floer homological, algebraic, and geometric properties of 3-manifolds, since L-spaces cannot admit taut foliations. Its resolution status is not specified in the supplied text.

   source: Siddhi Krishna, “Taut foliations, braid positivity, and unknot detection”, arXiv:2312.00196 (2025).

7. The L-space conjecture for irreducible rational homology 3-spheres

   Let $M$ be an irreducible rational homology $3$-sphere. The $L$-space conjecture. The following are equivalent:

   1. $M$ carries a cooriented taut foliation.
   2. $\pi_1(M)$ is left-orderable.
   3. $M$ is not an $L$-space.

   This conjecture proposes a correspondence between taut foliations, left-orderability, and Heegaard Floer-theoretic complexity of rational homology spheres. The source describes the problem as open.

   source: Thomas Massoni, “Taut foliations and contact pairs in dimension three”, arXiv:2405.15635 (2024).