Congruence and Quantum Invariants of 3-manifolds

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 Zf -coefficients. We show that strong f -congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) are well-behaved under this congruence. We strengthen this result and extend it to the SO(3) 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 S3 , the Poincaré homology sphere, the Brieskorn homology sphere Σ(2, 3, 7) and their mirror images up to strong f -congruence. We distinguish 0-framed surgery on the Whitehead link and #2S1 × S2 up to weak f -congruence for f an odd prime greater than three. As a corollary, we recover a slightly strengthened version of a result of Dabkowski and Przytycki’s concerning rational moves on links.