Volume 7, issue 4 (2007)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Congruence and quantum invariants of 3–manifolds

Patrick M Gilmer

Algebraic & Geometric Topology 7 (2007) 1767–1790

arXiv: math.GT/0612282


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 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 S3, 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 f–cohomology ring structure.

surgery, framed link, modular category, TQFT
Mathematical Subject Classification 2000
Primary: 57M99
Secondary: 57R56
Received: 27 June 2007
Accepted: 31 August 2007
Published: 18 December 2007
Patrick M Gilmer
Department of Mathematics
Louisiana State University
Baton Rouge LA 70803