Almost Integral Tqfts from Simple Lie Algebras

Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective Witten-Reshetikhin-Turaev invariant τ for closed 3-manifolds. As a corollary, one can show that τ is an algebraic integer for certain prime integers. The result in this paper can be used to prove that τ M satisfies some Murasugi type equivalence relation if M is a homology sphere and admits a cyclic group action with fixed point set a circle.