The L-Space and Left-Orderability Conjecture

Definitions and mathematical statement

Use the usual category of closed, connected, oriented smooth $3$-manifolds. A rational homology $3$-sphere is such a manifold $W$ satisfying \[ H_i(W;{\mathbb{Q}})\cong H_i(S^3;{\mathbb{Q}})\qquad\text{for every }i\geq 0. \] It is irreducible if every smoothly embedded $2$-sphere in $W$ bounds an embedded $3$-ball. Its group $H_1(W;{\mathbb{Z}})$ is finite; write $\lvert H_1(W;{\mathbb{Z}})\rvert$ for its order.

For an explicit standard coefficient convention in this draft, put ${\mathbb{F}}_2={\mathbb{Z}}/2{\mathbb{Z}}$. An L-space is a rational homology $3$-sphere satisfying \[ \dim_{{\mathbb{F}}_2}\widehat{HF}(W;{\mathbb{F}}_2)=\lvert H_1(W;{\mathbb{Z}})\rvert, \] where $\widehat{HF}$ denotes hat Heegaard Floer homology. The supplied record does not specify coefficients; this convention requires confirmation against the cited source.

A group $G$ is left-orderable if it admits a strict total order $<$ such that \[ \forall g,a,b\in G,\qquad a<b\Longrightarrow ga<gb. \] Under this definition the trivial group is left-orderable.

The supplied conjecture asserts that, for every irreducible rational homology $3$-sphere $W$, \[ W\text{ is not an L-space} \quad\Longleftrightarrow\quad \pi_1(W)\text{ is left-orderable}. \] There is an unresolved boundary qualification: the stated hypotheses include $S^3$, which is an L-space and has trivial fundamental group. Thus the displayed assertion is incompatible with the definition of left-orderability above at $S^3$. A consistent intended formulation needs an explicit qualification, such as excluding $S^3$, or using a convention that requires a left-orderable group to be nontrivial. The supplied record does not specify either qualification, so neither is silently imposed here.

Short English statement

The supplied conjecture says that an irreducible rational homology $3$-sphere fails to be an L-space exactly when its fundamental group admits a left-invariant total order. The treatment of $S^3$ and the coefficient convention for Heegaard Floer homology require clarification before this becomes a fully resolved formulation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002641$, source problem code catalogue entry, titled The L-Space and Left-Orderability Conjecture, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2014.
• Supplied source citation: Low-dimensional Topology and Number Theory (2015), Oberwolfach Reports, pp. 2115--2176, DOI 10.4171/owr/2014/38. Original OWR report: https://doi.org/10.4171/owr/2014/38. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source labelled "2024 primary treatment": https://ems.press/content/serial-article-files/48584?nt=1. No author, title, or more precise bibliographic identification is supplied for this link.
• Literature assessment checked on 2026-08-21: the supplied curation labels the problem open and reports that the L-space equivalence with left-orderability is proved for graph manifolds and other special classes, while the irreducible rational-homology-sphere conjecture remains widely open in general. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. These are supplied registry assertions, not mathematical verification of this draft.