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 $ℤ_{f}$ –coefficients. We show that strong $f$ –congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum $S U (2)$ invariants are well-behaved under this congruence. We strengthen this result and extend it to the $S O (3)$ 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 Poincaré homology sphere, the Brieskorn homology sphere $Σ (2, 3, 7)$ 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.

Keywords

surgery, framed link, modular category, TQFT

Mathematical Subject Classification 2000

Primary: 57M99

Secondary: 57R56

References

Publication

Received: 27 June 2007

Accepted: 31 August 2007

Published: 18 December 2007

Authors

Patrick M Gilmer
Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA
http://www.math.lsu.edu/~gilmer/

Volume 7, issue 4 (2007)

Patrick M Gilmer

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors