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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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