On the gauge invariance of the Kuperberg invariant of certain high genus framed 3-manifolds

Technical Summary: On the Gauge Invariance of the Kuperberg Invariant of Certain High Genus Framed 3-Manifolds

Problem Statement
The paper addresses a fundamental question in the theory of finite-dimensional Hopf algebras: whether the Kuperberg invariants of framed 3-manifolds constitute gauge invariants. A gauge invariant is a quantity assigned to a Hopf algebra that remains unchanged under tensor equivalences of its representation categories (or equivalently, under Drinfeld twists). While previous work by Chang, Ng, and Wang [CNW25] established that Kuperberg invariants for framed lens spaces and certain genus-2 3-manifolds are gauge invariants, those examples were non-hyperbolic. This left open the question of whether gauge invariance holds for hyperbolic 3-manifolds, which constitute the generic class of 3-manifolds. Specifically, the authors investigate whether the Kuperberg invariant of the Weeks manifold (the unique hyperbolic 3-manifold of smallest volume) and the 3-torus are gauge invariants for general finite-dimensional Hopf algebras over arbitrary base fields.

Methodology
The authors employ a combinatorial approach based on Heegaard diagrams and the algebraic structure of Hopf algebras. The methodology proceeds in the following steps:

1. Heegaard Diagram Construction: The authors construct specific framed Heegaard diagrams for the Weeks manifold (genus 2) and the 3-torus (genus 3). These diagrams include two admissible vector fields ($b_1$ and $b_2$) that define the framing. The vector fields determine rotation angles ($\theta$) and twist front crossings ($\phi$) for the upper and lower curves of the diagram.
2. Kuperberg Invariant Formulation: Using the Heegaard diagram, the Kuperberg invariant $Z(M, f, H)$ is expressed as a trace of a linear map on a tensor power of the Hopf algebra $H$. This involves:
   • Assigning coproduct factors of normalized integrals ($\Lambda$) to intersection points.
   • Applying powers of the antipode $S$ and the map $T(x) = g^{-1}S^2(x)g$ based on the rotation data at intersections.
   • Contracting the resulting tensor using the dual integral $\lambda$.
3. Gauge Transformation Analysis: To prove gauge invariance, the authors analyze the invariant under a Drinfeld twist $F$. They utilize the properties of normalized 2-cocycles and the resulting twisted Hopf algebra $H_F$. The proof relies heavily on:
   • Trace Formulas: Utilizing Theorem 1 (from [Rad94] and [CNW25]) which expresses the trace of an endomorphism in terms of integrals.
   • Cocycle Identities: Applying a series of algebraic identities regarding 2-cocycles (Proposition 1) and the distinguished group-like elements $g$ and $u$ associated with the twist.
   • Technical Lemmas: The authors derive specific lemmas (Lemmas 1, 2, and 3) to manipulate the complex tensor expressions involving the twisted antipode $S_F$ and the integrals, showing that the twisted invariant $Z_F$ simplifies back to the original invariant $Z$ (up to known gauge invariant factors).

Key Contributions and Results

• The Weeks Manifold (Main Theorem): The authors prove that for any finite-dimensional Hopf algebra $H$ and any framing $f$ on the Weeks manifold $W$, the Kuperberg invariant $Z(W, f, H)$ is a gauge invariant.
  • They first construct a specific framed Heegaard diagram ($f_0$) and explicitly compute the invariant as a trace of a linear map on $H \otimes H$.
  • Through a rigorous algebraic reduction involving the twist $F$, they demonstrate that $Z(W, f_0, H)$ is invariant under gauge transformations.
  • They extend this result to all framings on the Weeks manifold by noting that framings differ by degrees in $H^3(W, \mathbb{Z})$. A change in degree by $+1$ multiplies the invariant by $\alpha(g)$, which is already known to be a gauge invariant. Thus, the invariant for any framing is a product of gauge invariants.
• The 3-Torus: The authors also prove that the Kuperberg invariant of the 3-torus $T^3$ (genus 3) with a specific framing is a gauge invariant. This is achieved by expressing the invariant as a trace on $H^{\otimes 3}$ and performing a similar, albeit more complex, algebraic simplification under the twist.

Significance and Claims
The paper claims to provide the first example of gauge invariants for general (not necessarily semisimple) finite-dimensional Hopf algebras derived from hyperbolic 3-manifolds.

• Support for Conjecture: The results provide decisive supporting evidence for the affirmative answer to Question 1.1 in [CNW25], which asks if all framed Kuperberg invariants are gauge invariants.
• Geometric Independence: By proving invariance for a hyperbolic manifold, the authors suggest that the gauge invariance of Kuperberg invariants is likely independent of the geometric type (hyperbolic vs. non-hyperbolic) of the 3-manifold.
• Systematic Construction: The work supports the idea proposed in [CNW25] that topological methods can systematically produce infinite families of gauge invariants for Hopf algebras, which is particularly valuable for the classification of finite-dimensional Hopf algebras and the study of non-semisimple quantum invariants.

The authors remain modest, framing their work as providing "very important evidence" rather than a complete proof for all 3-manifolds, acknowledging that the Weeks manifold and 3-torus are specific cases within a broader, open problem.