Let
be an integer greater than one. We study three progressively finer equivalence
relations on closed 3–manifolds generated by Dehn surgery with denominator
: weak
–congruence,
–congruence, and
strong –congruence.
If is odd, weak
–congruence
preserves the ring structure on cohomology with
–coefficients. We show
that strong –congruence
coincides with a relation previously studied by Lackenby. Lackenby showed that the
quantum
invariants are well-behaved under this congruence. We strengthen this result and extend
it to the
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 ,
the Poincaré homology sphere, the Brieskorn homology sphere
and their mirror images up
to strong –congruence. We
distinguish the weak –congruence
classes of some manifolds with the same
–cohomology
ring structure.
