Math Conjectures

LL-space conjecture

OPENMajorConjectureProposed 2013–2015 · Standard version

Canonical statement

Let YY be a closed, connected, orientable, irreducible rational-homology 33-sphere, and let HF^(Y;F2)\widehat{HF}(Y;\mathbb F_2) denote its hat Heegaard Floer homology. The following are equivalent: (i) rank⁡F2HF^(Y;F2)>∣H1(Y;Z)∣\operatorname{rank}_{\mathbb F_2}\widehat{HF}(Y;\mathbb F_2)> |H_1(Y;\mathbb Z)|, so YY is not an LL-space; (ii) π1(Y)\pi_1(Y) admits a total order << satisfying g<h⇒fg<fhg<h\Rightarrow fg<fh for all f,g,hf,g,h; (iii) YY admits a coorientable taut codimension-one foliation.
View source LaTeX
Let \(Y\) be a closed, connected, orientable, irreducible rational-homology \(3\)-sphere, and let \(\widehat{HF}(Y;\mathbb F_2)\) denote its hat Heegaard Floer homology. The following are equivalent: (i) \(\operatorname{rank}_{\mathbb F_2}\widehat{HF}(Y;\mathbb F_2)> |H_1(Y;\mathbb Z)|\), so \(Y\) is not an \(L\)-space; (ii) \(\pi_1(Y)\) admits a total order \(<\) satisfying \(g<h\Rightarrow fg<fh\) for all \(f,g,h\); (iii) \(Y\) admits a coorientable taut codimension-one foliation.
GEO-TOP-019:[BoyerGordonWatson2013Orderable][Juhasz2015FloerSurvey][BoyerClay2024OrderDetection][BIRS2025LSpace]
Geometry and topologyAlgebra and representation theory

Notes

An LL-space is a rational homology 33-sphere YY whose Heegaard Floer homology is as small as its ordinary homology allows, rank⁡HF^(Y)=∣H1(Y;Z)∣\operatorname{rank}\widehat{HF}(Y)=|H_1(Y;\mathbb Z)|. The LL-space conjecture predicts that for a closed, connected, orientable, irreducible rational homology sphere three properties of very different origins coincide: not being an LL-space, left-orderability of π1(Y)\pi_1(Y), and the existence of a coorientable taut foliation. The Floer–orderability link was formulated by Boyer, Gordon and Watson in 2013 [BoyerGordonWatson2013Orderable], and the three-way equivalence consolidated over the following years [Juhasz2015FloerSurvey]; the date range 2013–2015 reflects this gradual formulation.

The conjecture is established for graph manifolds and for a number of surgery and branched-cover families, with gluing and order-detection techniques on knot manifolds driving much recent progress [BoyerClay2024OrderDetection]; the area remains highly active [BIRS2025LSpace]. In full generality only one implication is known: a coorientable taut foliation forces YY to fail to be an LL-space.

The remaining implications — recovering a foliation or an order from Floer-homological largeness, and relating orderability to foliations directly — are open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.