L-Space Conjecture

Definitions and mathematical statement

A closed manifold is compact and has empty boundary. A connected closed orientable $3$-manifold $W$ is prime if every connected-sum decomposition $W\cong W_1\#W_2$ has a summand homeomorphic to $S^3$.

A group $G$ is left-orderable if it admits a strict total order $<$ such that \[ \forall g,h,k\in G,\qquad g<h\ \Longrightarrow\ kg<kh. \] Under this definition the trivial group is left-orderable. Some formulations of the conjecture additionally require the group to be nontrivial; the supplied record does not specify this convention.

A rational homology $3$-sphere is a closed connected $3$-manifold $W$ satisfying $H_i(W;{\mathbb{Q}})\cong H_i(S^3;{\mathbb{Q}})$ for every $i$. To make the coefficient dependence explicit, for a fixed coefficient field $\mathbf{k}$, define an $L$-space with coefficients in $\mathbf{k}$ to be a rational homology $3$-sphere satisfying \[ \dim_{\mathbf{k}}\widehat{HF}(W;\mathbf{k}) =\lvert H_1(W;{\mathbb{Z}})\rvert, \] where $\widehat{HF}$ denotes hat Heegaard--Floer homology and the right-hand side is the finite order of the first integral homology group. Manifolds that are not rational homology spheres are not $L$-spaces under this definition. The source does not specify a coefficient convention; the parameter $\mathbf{k}$ here explains that unresolved choice and is not an additional universal quantifier in the source problem.

The supplied question is whether, for every closed, connected, orientable, prime $3$-manifold $W$, \[ \pi_1(W)\text{ is left-orderable} \quad\Longleftrightarrow\quad W\text{ is not a Heegaard--Floer }L\text{-space}. \] The quantifier is retained exactly, including $S^3$; no nontrivial-fundamental-group hypothesis is supplied. A definitive formulation requires clarification of the trivial-group convention and the intended $L$-space coefficient convention. No exclusion or additional hypothesis is silently imposed.

Short English statement

The supplied conjecture asks whether every closed, connected, orientable, prime $3$-manifold has a left-orderable fundamental group exactly when it is not a Heegaard--Floer $L$-space. The treatment of the trivial group and the homology coefficients remain unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30002172$, source problem code catalogue entry, title "L-Space Conjecture," in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year $2012$ and supplies no named proposer.
• Supplied report citation: Low-Dimensional Topology and Number Theory (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/42, pp. 2541--2596. Supplied source URL and literature reference "OWR report 42": https://doi.org/10.4171/owr/2012/42.
• Supplied literature reference: Boyer--Gordon--Watson, https://arxiv.org/abs/1107.5018.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that the conjecture is proved for broad classes, including graph manifolds and many surgery families, while the equivalence remains unknown for all prime orientable $3$-manifolds. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement-status label is corrected_verified, accompanied by a claim of manual semantic, typographic, and LaTeX review. This records the supplied editorial assessment and does not establish mathematical correctness or resolve the conventions identified above.