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

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 \(\tau^G_r(M)\), 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 \(G=\mathrm{SU}(2)\). We show that the conjecture is false in general. With \(\tau(S^3)=1\): for \(\mathrm{SU}(4)\) and \(r=12\) the lens spaces \(L(8,1)\) and \(L(8,3)\) have \(|\tau|=16+8\sqrt3\) and \(56+32\sqrt3\); for \(\mathrm{SU}(5)\) and \(r=10\) one has \(|\tau(L(5,1))|^2\)\({}=3475+1550\sqrt5\) and \(|\tau(L(5,2))|\)\({}=45+20\sqrt5\); for \(\mathrm{Spin}(7)\) and \(r=15\) the moduli for \(L(10,1)\) and \(L(10,3)\) are nonzero with ratio \(2-\sqrt3\). 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 \(\mathrm{SU}(N)\) for every \(N\ge4\), for \(\mathrm{Spin}(m)\) with \(m=7\) and every \(m\ge9\), and for \(E_6\), \(E_7\), \(E_8\), and that it holds for \(\mathrm{Sp}(2n)\) for every \(n\ge1\) and for \(\mathrm{SU}(3)\), \(G_2\), \(\mathrm{Spin}(8)\) 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 \(\mathrm{SU}(2)\), \(\mathrm{SU}(3)\), \(\mathrm{Sp}(2n)\), \(G_2\), \(\mathrm{Spin}(8)\) and \(F_4\). We also record that two closed formulas of Guadagnini and Pilo for \(\mathrm{SU}(2)\) are inaccurate; their conclusion is not affected. This is an unrefereed note.