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Congruence and quantum invariants of 3-manifolds. (English) Zbl 1161.57003

For an integer \(f\), the author defines the concepts of weak \(f\)-congruence, \(f\)-congruence, and strong \(f\)-congruence of \(3\)-manifolds. These equivalence relations are defined via Dehn surgery. An \(n/{\ell}\) surgery is a weak type-\(f\) surgery if \({\ell}\) is divisible by \(f\), a type-\(f\) surgery if \({\ell}\) is divisible by \(f\) and \(n\equiv \pm 1\pmod f\) and a strong type-\(f\) surgery if \({\ell}\) is divisible by \(f\) and \(n \equiv \pm 1\pmod\ell\). It is shown that a weak \(f\)-congruence between \(3\)-manifolds induces graded group isomorphisms between the homologies and respectively the cohomologies with \({\mathbb Z}_f\)-coefficients of the manifolds. If \(f\) is odd, this isomorphism preserves the ring structure, if \(f\) is even, it need not preserve it. The author shows that strong \(f\)-congruence coincides with the congruence mod \(f\) studied by Lackenby and extends Lackenby’s result that quantum \(SU(2)\) invariants behave well under this congruence to \(SO(3)\). The sphere \(S^3\), the Poincaré homology sphere, the Brieskorn homology sphere \(\Sigma(2,3,7)\) and their mirror images up to strong \(f\)-congruence, are compared. A finite list of the only possible \(f\) for which there might be strong \(f\)-congruence between \(S^3\) and the Brieskorn homology spheres \(\pm \Sigma(2,3,5)\) and \(\pm\Sigma(2,3,7)\) is given. The author also distinguishes the weak \(f\)-congruence classes of some manifolds with the same \({\mathbb Z}_f\)-cohomology ring structure.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
57R56 Topological quantum field theories (aspects of differential topology)

Software:

Mathematica
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