Habiro-Ring Refinements of Three-Manifold Quantum Invariants

Definitions and mathematical statement

Let $Y$ be a rational homology $3$-sphere, meaning that $H_i(Y;{\mathbb{Q}})\cong H_i(S^3;{\mathbb{Q}})$ for every $i$. The quantum invariants below are understood in their usual setting of closed, connected, oriented $3$-manifolds; the supplied excerpt does not specify their orientation and normalization conventions. In this setting the source's hypothesis $b_1(Y)=0$, where $b_1(Y)=\dim_{{\mathbb{Q}}}H_1(Y;{\mathbb{Q}})$, is equivalent to the rational homology sphere condition. Write $\operatorname{Spin}^c(Y)$ for the set of isomorphism classes of $\operatorname{Spin}^c$ structures on $Y$.

Define the Habiro ring by \[ \widehat{{\mathbb{Z}}[q]}=\varprojlim_{n\geq1}{\mathbb{Z}}[q]\big/\left(\prod_{k=1}^{n}(1-q^k)\right). \] For a root of unity $\zeta$, evaluation induces a well-defined homomorphism $\operatorname{ev}_{\zeta}:\widehat{{\mathbb{Z}}[q]}\to{\mathbb{Z}}[\zeta]$: the defining product vanishes at $\zeta$ once $n$ is at least its order.

Put \[ A_Y=H^1(Y;{\mathbb{C}}/2{\mathbb{Z}}),\qquad B_Y=H^1(Y;{\mathbb{Z}}/2{\mathbb{Z}})\subseteq A_Y, \] using the inclusion of coefficient groups. The source uses the mod-$2$ cohomological refinement $\operatorname{WRT}_r(Y,\omega)$ for $\omega\in B_Y$, and the Costantino--Geer--Patureau-Mirand invariant $N_r(Y,\omega)$ for $\omega\in A_Y\setminus B_Y$, at admissible levels $r$. Let $\mathcal T(Y,\omega)$ denote the version of Reidemeister--Turaev torsion required by the source. Retain its explicit hypothesis \[ \mathcal T(Y,\omega)\ne0\qquad\text{for every }\omega\in A_Y\setminus B_Y. \] Subject to the admissibility and normalization conventions of the source, define \[ \widetilde N_r(Y,\omega)= \begin{cases} \operatorname{WRT}_r(Y,\omega),&\omega\in B_Y,\\[2pt] \displaystyle\frac{N_r(Y,\omega)}{\mathcal T(Y,\omega)},&\omega\in A_Y\setminus B_Y. \end{cases} \] The particular torsion, the invariant normalizations, and the admissible levels are not defined in the supplied excerpt.

Habiro-ring existence conjecture (source Conjecture 2). For every such $Y$, subject to the stated torsion hypothesis, there exists a family \[ \left(I_s(Y)\right)_{s\in\operatorname{Spin}^c(Y)}\in \prod_{s\in\operatorname{Spin}^c(Y)}\widehat{{\mathbb{Z}}[q]} \] such that, simultaneously for every source-admissible level $r$ and cohomology class $\omega\in A_Y$, \[ \widetilde N_r(Y,\omega)= \sum_{s\in\operatorname{Spin}^c(Y)} C_s(Y,\omega)\, \operatorname{ev}_{e^{2\pi i/r}}\left(I_s(Y)\right). \] The family depends on $Y$ and $s$, not on $r$ or $\omega$. The coefficients $C_s(Y,\omega)$ are prescribed universal coefficients from the source, with dependence on $r$ suppressed in its notation; they also involve certain homotopy-type invariants of $Y$. They are not additional unknowns to be chosen to satisfy the equation. Their formula and the full admissibility conditions are absent, so this is a qualified reconstruction rather than a self-contained specification of those data.

The source motivates this conjecture through its preceding Conjecture 1, attributed there to references [3, 2], asking for series $\widehat Z_s(Y)\in{\mathbb{Z}}[[q]]$, convergent for $|q|<1$, whose weighted limits at $q=e^{2\pi i/r}$ recover $\widetilde N_r$. It states that root-of-unity limits alone do not ensure uniqueness of such series, because nonzero series may have zero limits at all roots of unity. The displayed Habiro-ring conjecture explicitly asserts existence; a separate uniqueness assertion for the indexed family is not supplied. The excerpt reports verification of the preceding conjecture for an unspecified family in reference [2], a reduction to an unspecified analogous conjecture for links in $S^3$, and a possible analogous conjecture for $\mathfrak{sl}(2|1)$ versions of CGP invariants in reference [5]. It also says that the torsion hypothesis holds for a generic $3$-manifold, asks how it might be avoided, notes that its conventions differ slightly from [2], and mentions a possible version for $b_1(Y)>0$. None of these unspecified extensions or reported results supplies missing hypotheses for the statement above.

Short English statement

For each rational homology $3$-sphere satisfying the source's torsion nonvanishing hypothesis, seek one Habiro-ring element for each $\operatorname{Spin}^c$ structure. Their evaluations at roots of unity should recover the combined refined WRT and torsion-normalized CGP invariant through the source's prescribed coefficients, for every admissible level and cohomology class. The coefficient formulas, admissibility rules, and normalization conventions remain unspecified. This is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, supplied record with permanent ID $30005073$, source problem code catalogue entry, title "Habiro-Ring Refinements of Three-Manifold Quantum Invariants." No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: Mini-Workshop: Recent Developments in Representation Theory and Mathematical Physics (2023), Oberwolfach Reports, pp. 817--848, DOI https://doi.org/10.4171/owr/2022/16. The supplied literature list also labels this link "Oberwolfach Report 2022/16." The title year and report identifier differ and have not been reconciled here.
• Supplied literature reference: Costantino--Gukov--Putrov, SIGMA, https://doi.org/10.3842/SIGMA.2023.010. The supplied assessment attributes proofs of the $q$-series/root-of-unity relation under technical hypotheses and for several families to these authors.
• Literature assessment checked at 2026-08-22: this exact date records supplied dated curation, not independent verification. The registry labels the problem "partially solved" and states that the general existence and uniqueness conjecture for all rational homology spheres remains unproved. The supplied background explicitly requires independent verification of time-sensitive status and cited claims. The assessment's reference to uniqueness is broader than the explicit existence assertion in the excerpt.
• The supplied excerpt attributes the preceding conjecture to [3, 2], CGP invariants to Costantino, Geer, and Patureau-Mirand [1], the mod-$2$ WRT refinement to [6], and the $\mathfrak{sl}(2|1)$ analogues to [5]. Full bibliographic entries and a reliable mapping of these numbered references to the supplied links are absent.
• The registry records statement_status: corrected_verified and says the reconstruction was checked against the official report. These are supplied provenance claims; no source was visited or independently checked in preparing this draft.