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)

Keywords:

surgery; framed link; modular category; TQFT

Mathematica
Full Text:

References:

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