A Conjecture of Guadagnini and Pilo on Quantum Invariants and the Fundamental Group: Counterexamples and a Classification by Gauge Group

Abstract

Guadagnini and Pilo conjectured that for a closed oriented 3-manifold M and a compact gauge group G the absolute value of the quantum invariant τ^G_r(M), when it does not vanish, depends only on the fundamental group of M; this is Conjecture 7.5 of Ohtsuki's problem list. They proved it for lens spaces and G = SU(2). We show that the conjecture is false in general. With τ(S^3) = 1: for SU(4) and r = 12 the lens spaces L(8,1) and L(8,3) have |τ| = 16 + 8√3 and 56 + 32√3; for SU(5) and r = 10 one has |τ(L(5,1))|^2 = 3475 + 1550√5 and |τ(L(5,2))| = 45 + 20√5; for Spin(7) and r = 15 the moduli for L(10,1) and L(10,3) are nonzero with ratio 2 − √3. These values are obtained by exact computation from the Reshetikhin–Turaev surgery formula. On the basis of the lens space formula of Hansen and Takata we then determine the simply connected compact simple groups for which the conjecture holds. We prove a structure result for the lattice Gauss sums in that formula: only a coset of a stabilizer in the Weyl group contributes. From it we deduce that the lens-space statement fails for SU(N) for every N ≥ 4, for Spin(m) with m = 7 and every m ≥ 9, and for E_6, E_7, E_8, and that it holds for Sp(2n) for every n ≥ 1 and for SU(3), G_2, Spin(8) and F_4 (for the last two with a computer-assisted lemma). By geometrization the conjecture for G is equivalent to its lens-space case, so, given that formula, it holds exactly for SU(2), SU(3), Sp(2n), G_2, Spin(8) and F_4. We also record that two closed formulas of Guadagnini and Pilo for SU(2) are inaccurate; their conclusion is not affected. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-103-0120 (ulamai/UnsolvedMath; Conjecture 7.5 of Ohtsuki's list "Problems on invariants of knots and 3-manifolds", due to Guadagnini and Pilo).