We discuss ways that the ring of coefficients for a topological quantum field theory (TQFT) can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers: ℤ[ζ2p] if p≡−1 (mod 4), and ℤ[ζ4p] if p≡1 (mod 4), where ζk is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs, but the modules are guaranteed only to be projective.
"Integrality for TQFTs." Duke Math. J. 125 (2) 389 - 413, 1 November 2004. https://doi.org/10.1215/S0012-7094-04-12527-8