Congruence and similarity 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 Z_f-coefficients. We show that strong f-congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) invariants 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 S^3, the Poincare homology sphere, the Brieskorn homology sphere Sigma(2,3,7) and their mirror images up to strong f-congruence. We distinguish the weak f-congruence classes of some manifolds with the same Z_f-cohomology ring structure. 
Patrick M. Gilmer