Congruence and Similarity of 3-manifolds

Let f be an integer greater than one. Type-f surgery is a kind of surgery along a knot in a 3-manifold which generalizes the notion of n/f surgery in a homology sphere. Such surgeries preserves the cohomology groups with Zf coefficients. Type-f surgery generates an equivalence relation on 3manifolds which we call similarity modulo f . If f is odd, we show that typef surgery also preserves the cohomology ring structure with Zf coefficients. Lackenby considered a finer equivalence relation on 3-manifolds called congruence modulo f which is generated a move which increments the framing on a component of framed link description by f . Congruence and similarity between three manifolds are reflected in their quantum invariants. We refine a relation between quantum invariants of congruent manifolds found by Lackenby, and find a relation reflecting similarity. Let p be an odd prime. We show that the quantum SO(3) invariant at a pth root of unity has a very simple surgery formula for type-p surgeries. As a corollary, we distinguish, up to similarity modulo p for p ≥ 5, two 3-manifolds with the same cohomology rings: 0-framed surgery to the Whitehead link and #S × S. We also show that the quantum SO(3) invariant at a pth root of unity is preserved, up to phase, by congruence modulo p. In this way, we show that the 3-sphere, the Poincare homology sphere and the Breiskorn homology sphere Σ(2, 3, 7) lie in distinct congruence modulo p classes, but in the same congruence class modulo f for f = 2, 3, or 4. We generalize a theorem of Cochran and Melvin about the divisibility of quantum invariants of 3-manifolds with non-zero first cohomology with Zp coefficients to the case where the 3-manifolds may contain colored framed links. We strengthen a result of Masbaum and the author on the divisibility of certain quantum invariants.