Parity-Refined Heegaard–Floer and Lattice Cohomology Correspondence

Definitions and mathematical statement

Let $M$ be an oriented negative-definite plumbed rational homology $3$-sphere: thus $H_*(M;{\mathbb{Q}})\cong H_*(S^3;{\mathbb{Q}})$, and $M$ is presented as the oriented boundary of a plumbed $4$-manifold whose intersection form is negative definite. Write $-M$ for the oppositely oriented manifold and $\operatorname{Spin}^c(M)$ for its set of $\operatorname{Spin}^c$ structures. The notation on $-M$ uses the corresponding structure under orientation reversal.

The supplied clean statement uses $HF^+_{\mathrm{red}}(-M,\sigma)$ for reduced plus Heegaard--Floer homology and $\mathbb H^q_{\mathrm{red}}(M,\sigma)$ for reduced lattice cohomology in cohomological degree $q$, associated with the plumbing and $\sigma$. The subscripts $\mathrm{even}$ and $\mathrm{odd}$ on the former refer to its parity grading. The coefficient system, the precise lattice-cohomology construction, and the grading normalizations are not specified in the supplied record. The symbol $d(M,\sigma)$ denotes the grading-shift quantity appearing there; its definition and normalization are also not supplied. Brackets denote a grading shift, with the sign convention requiring clarification.

Recorded conjectural correspondence. For every such $M$ and every $\sigma\in\operatorname{Spin}^c(M)$, the supplied clean statement asks for both isomorphisms \[ \begin{aligned} HF^+_{\mathrm{red,even}}(-M,\sigma) &\cong \bigoplus_{\substack{q\in{\mathbb{Z}}\\q\equiv0\pmod 2}} \mathbb H^q_{\mathrm{red}}(M,\sigma)[-d(M,\sigma)],\\ HF^+_{\mathrm{red,odd}}(-M,\sigma) &\cong \bigoplus_{\substack{q\in{\mathbb{Z}}\\q\equiv1\pmod 2}} \mathbb H^q_{\mathrm{red}}(M,\sigma)[-d(M,\sigma)]. \end{aligned} \] Here $q$ is the cohomological degree; the internal grading on lattice cohomology is a separate grading. The intended algebraic category of the isomorphisms is unspecified.

Unresolved source qualifications. These formulas reproduce the clean statement rather than resolve its conventions. The original statement writes $H^q_{\mathrm{red}}$ instead of $\mathbb H^q_{\mathrm{red}}$, and says "up to the shift $d(M,\sigma)$" while also displaying $[-d(M,\sigma)]$. Whether that wording describes the displayed shift or permits an additional shift is unclear. Consequently, the exact grading-preserving assertion requires source review.

Short English statement

The recorded conjecture compares reduced Heegaard--Floer homology of $-M$ with reduced lattice cohomology of a negative-definite plumbing for $M$, matching even and odd parity separately for every $\operatorname{Spin}^c$ structure. Coefficients, grading conventions, the shift, and the algebraic structure preserved by the proposed isomorphisms require clarification. The supplied open label is a curation claim, not an independently verified status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003628$, exact source code catalogue entry, titled Parity-Refined Heegaard--Floer and Lattice Cohomology Correspondence; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Interactions between Low-dimensional Topology and Complex Algebraic Geometry (2018), Oberwolfach Reports, pp. 2987--3033, DOI 10.4171/owr/2017/50, https://doi.org/10.4171/owr/2017/50. The citation's 2018 date and the report identifier 50/2017 are retained as supplied.
• Supplied literature source: Oberwolfach Report 50/2017, https://doi.org/10.4171/owr/2017/50. Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open but says the extracted record was an incomplete continuation ending at the word "conjecture," without the assertion needed for a literature determination. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement corrected_verified and reports that the official report was checked directly and definitions restored. These are supplied verification claims; this draft does not independently confirm them. The record nevertheless leaves the conventions and conflicting extraction assessment described above unresolved.