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Quantum \(SO(3)\)-invariants dominate the \(SU(2)\)-invariant of Casson and Walker. (English) Zbl 0854.57016

In a recent paper [Math. Proc. Camb. Philos. Soc. 115, No. 2, 253-281 (1994; Zbl 0832.57005)] the author gave a relation between the quantum SU(2) invariants and the Casson invariant for an integral homology sphere. That work is generalized in the present paper to the relation between the quantum SO(3)-invariants and Walker’s invariant for a rational homology sphere. The main result follows from the following interesting Lemma:
Let \(r\) be a fixed odd prime. Let \(S\) be a \(\mu \times \mu\) symmetric matrix in \(\mathbb{Z}\). Then there exist integers \(n_1, n_2,\dots\) and \(n_\nu\) with \(n_\xi\) coprime to \(r\) \((1 \leq \xi \leq \nu)\) and a \((\mu+\nu) \times (\mu+\nu)\) unimodular matrix \(P\) in \(\mathbb{Z}\) such that \(P(S \oplus (n_1) \oplus (n_2) \oplus \dots \oplus (n_\nu)) P^T\) is a diagonal matrix. Here \(\oplus\) denotes the block sum and \(P^T\) is the transpose of \(P\).
The author says that the Lemma was suggested to him by T. Ohtsuki.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0832.57005
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References:

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