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A polynomial invariant of rational homology \(3\)-spheres. (English) Zbl 0855.57016
Let \(\tau_r^{\text{SO} (3)}\) denote the quantum SO(3) invariant associated with an odd integer \(r\). It was introduced by R. Kirby and P. Melvin [ibid. 105, 473-545 (1991; Zbl 0745.57006)] as a refinement of the quantum SU(2) invariant introduced by N. Reshetikhin and V. G. Turaev [ibid. 103, 547-597 (1991; Zbl 0725.57007)], which is inspired by a work of E. Witten [Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)]. The reviewer proved that \(\tau_r^{\text{SO} (3)} (M)\) belongs to \(\mathbb{Z} [q]\) if \(M\) is a rational homology 3-sphere and \(r\) is an odd prime [Math. Proc. Camb. Philos. Soc. 117, 237-249 (1995; Zbl 0854.57016)].
In the paper under review, the author proves that for a rational homology 3-sphere \(M\), there exist series of topological invariants \(\lambda_n (M)\) \((n=0, 1, 2, \dots)\) independent of \(r\) such that (roughly speaking) \(\tau_r^{\text{SO} (3)} (M)\) is congruent to \(\sum_n \lambda_n (M) (q- 1)^n\) modulo \(r\). In particular, \(\lambda_0 (M)= 1/|H_1 (M; \mathbb{Z} )|\) and \(\lambda_1 (M)\) is (a constant times) the Casson-Walker invariant. This is a generalization of his previous work [ibid. 117, 83-112 (1995; Zbl 0843.57019)], where the existence of \(\lambda_n\) is proved for integral homology 3-spheres.
Reviewer: H.Murakami (Tokyo)

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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