Congruence and quantum invariants of 3–manifolds

Abstract

Let $f$ be an integer greater than one. We study three progressively finer equivalence relations on closed 3–manifolds generated by Dehn surgery with denominator $f$: weak $f$–congruence, $f$–congruence, and strong $f$–congruence. If $f$ is odd, weak $f$–congruence preserves the ring structure on cohomology with ${\mathbb{Z}}_f$–coefficients. We show that strong $f$–congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum $SU\left(2\right)$ invariants are well-behaved under this congruence. We strengthen this result and extend it to the $SO\left(3\right)$ quantum invariants. We also obtain some corresponding results for the coarser equivalence relations, and for quantum invariants associated to more general modular categories. We compare $S^3$, the Poincaré homology sphere, the Brieskorn homology sphere $\Sigma \left(2,3,7\right)$ and their mirror images up to strong $f$–congruence. We distinguish the weak $f$–congruence classes of some manifolds with the same ${\mathbb{Z}}_f$–cohomology ring structure.