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A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres. (English) Zbl 1144.57006
For integral homology $$3$$-spheres $$M$$, the author introduces an invariant $$J_M$$ which unifies the $$SU(2)$$ Witten-Reshetikhin-Turaev (WRT) invariants at all roots of unity. It should be compared with the fact that the Ohtsuki series by T. Ohtsuki [Math. Proc. Camb. Philos. Soc. 117, No. 1, 83–112 (1995; Zbl 0843.57019)] unifies the WRT invariants at roots of unity of odd prime orders. This invariant was announced in a previous paper by the author [Geom. Topol. Monogr. 4, 55–68 (2002; Zbl 1040.57010)].
It takes values in a completion $$\widehat{\mathbb{Z}[q]}$$ of the polynomial ring $$\mathbb{Z}[q]$$, and its evaluation at any root of unity $$\zeta$$ gives the WRT invariant $$\tau_\zeta(M)$$. As an immediate consequence, $$\tau_\zeta(M)\in \mathbb{Z}[\zeta]$$, which generalizes the integrality result by H. Murakami [Math. Proc. Camb. Philos. Soc. 115, No. 2, 253–281 (1994; Zbl 0832.57005)].
Let $$\mathcal{Z}\subset \mathbb{C}$$ be the set of all roots of unity. For each $$M$$, the WRT function $$\tau(M):\mathcal{Z}\to\mathbb{C}$$ is defined by $$\tau(M)(\zeta)=\tau_\zeta(M)$$. Then it is shown that $$J_M$$ and $$\tau(M)$$, also $$J_M$$ and the Ohtsuki series $$\tau^O(M)$$, have the same strength in distinguishing integral homology spheres.

MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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