Ozsváth–Szabó Conjecture for Integer Homology Spheres

Definitions and mathematical statement

An integer homology $3$-sphere is a closed, connected, orientable $3$-manifold $M$ satisfying \[ H_i(M;{\mathbb{Z}})\cong H_i(S^3;{\mathbb{Z}})\qquad\text{for every }i\geq 0. \] For the Heegaard Floer terminology, take a smooth structure and an orientation on $M$. With the conventional coefficient choice ${\mathbb{F}}_2={\mathbb{Z}}/2{\mathbb{Z}}$, a rational homology $3$-sphere $Y$ is an L-space if \[ \dim_{{\mathbb{F}}_2}\widehat{HF}(Y;{\mathbb{F}}_2)=|H_1(Y;{\mathbb{Z}})|, \] where $\widehat{HF}$ denotes hat Heegaard Floer homology. A non-L-space is a rational homology sphere that does not satisfy this equality. In particular, for an integer homology sphere the L-space condition is $\dim_{{\mathbb{F}}_2}\widehat{HF}(M;{\mathbb{F}}_2)=1$. The supplied source does not specify coefficients; the choice ${\mathbb{F}}_2$ is an explicit draft convention requiring confirmation.

Let $\Sigma(2,3,5)$ denote the Poincaré homology sphere. Write $\cong_{\mathrm{unor}}$ for homeomorphism after forgetting orientation. This interpretation includes both orientations of the exceptional homology sphere; the source does not explicitly specify its orientation convention.

Question attributed to Ozsváth--Szabó, as recorded. Is it true that every integer homology $3$-sphere $M$ satisfies \[ M\text{ is a non-L-space} \quad\Longleftrightarrow\quad \left(M\not\cong_{\mathrm{unor}}S^3\right) \ \text{and}\ \left(M\not\cong_{\mathrm{unor}}\Sigma(2,3,5)\right)? \] Unresolved scope qualification. Both the extracted statement and the quoted conjecture omit irreducibility, whereas the supplied literature assessment describes the conjecture for irreducible integer homology spheres. Here irreducible means that every smoothly embedded $2$-sphere bounds a $3$-ball. Thus the assessment concerns the displayed equivalence with the additional hypothesis that $M$ is irreducible. The record does not resolve whether that hypothesis was omitted from the extraction or assumed in the surrounding report. The unrestricted wording and the irreducible formulation must therefore remain distinguished pending mathematical review.

Short English statement

The recorded question asks whether the only integer homology $3$-spheres that are L-spaces are $S^3$ and the Poincaré sphere, including its reversed orientation under the stated draft convention. The supplied literature assessment restricts the conjecture to irreducible manifolds, but the extracted question omits that hypothesis. Coefficient and orientation conventions also require confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $30005367$, source problem code catalogue entry, title "Ozsváth--Szabó Conjecture for Integer Homology Spheres." The supplied record attributes the conjecture to Ozsváth--Szabó; its separate proposed_by field is empty.
• Low-dimensional Topology (2023), Oberwolfach Reports, pp. 151--213, DOI 10.4171/owr/2023/3. Supplied source report and literature-source label "original Oberwolfach report": https://doi.org/10.4171/owr/2023/3.
• Supplied literature-source label "Lidman--Pinzon-Caicedo--Zentner, toroidal homology spheres": https://doi.org/10.1112/topo.12275. No fuller bibliographic information is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes the Ozsváth--Szabó claim as unresolved for general irreducible integer homology spheres, with substantial families, including graph and toroidal cases, proved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The contextual passage credits Eftekhary, under reference label [Eft], with the toroidal non-L-space result. It also uses the unresolved bibliographic labels [BGH1], [BC2], [BC1], [RR], and [HRW]. The final sentence attributes an argument to Hanselman--Rasmussen but is truncated. These are supplied contextual attributions, not independently checked references.