A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres

Abstract

We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, τ_ζ(M) is an algebraic integer. Moreover, it follows that τ_ζ(M) as a function on ζbehaves like an ``analytic function'' defined on the set of roots of unity. That is, the τ_ζ(M) for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τ_ζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q=1.