I've just started looking through Quantum Invariants of Knots and 3-Manifolds by V.G Turaev and want to understand what exactly is breaking in the construction of a 3d-TQFT when one considers the quantum group $U_{q}\mathfrak{sl}_{2}$ when $q$ is not a root of unity.
In particular if convergence of expressions for the invariants of all manifolds isn't guaranteed this wouldn't be so much of an issue for my interests. For example something like $S^{1}\times\Sigma$ having infinite invariant isn't so bad for me (i.e having infinite dimensional vector spaces on boundaries).
Thanks in advance!