Mathematics – Geometric Topology
Scientific paper
2006-05-11
Mathematics
Geometric Topology
66 pages, 8 figures
Scientific paper
10.1007/s00222-007-0071-0
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 \zeta gives the the SU(2) Witten-Reshetikhin-Turaev invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on \zeta behaves like an ``analytic function'' defined on the set of roots of unity. That is, the \tau_\zeta(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, \tau_\zeta(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q=1.
Habiro Kazuo
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