Integrality for Tqfts

We discuss two ways that the ring of coeffients for a TQFT can be reduced if one may restrict somewhat the allowed cobordisms. When we apply both methods to a TQFT associated to SO(3) and 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[ζp], if p ≡ −1 (mod 4), and Z[ζ4p], if p ≡ 1 (mod 4), where ζk is a primitive kth root of unity. As a corollary, we obtain representations of central extensions of the mapping class groups of surfaces defined over rings of integers. We obtain some results on the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We discuss new invariants coming from strong shift equivalence and integrality.