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2007 Congruence and quantum invariants of 3–manifolds
Patrick M Gilmer
Algebr. Geom. Topol. 7(4): 1767-1790 (2007). DOI: 10.2140/agt.2007.7.1767


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.


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Patrick M Gilmer. "Congruence and quantum invariants of 3–manifolds." Algebr. Geom. Topol. 7 (4) 1767 - 1790, 2007.


Received: 27 June 2007; Accepted: 31 August 2007; Published: 2007
First available in Project Euclid: 20 December 2017

zbMATH: 1161.57003
MathSciNet: MR2366177
Digital Object Identifier: 10.2140/agt.2007.7.1767

Primary: 57M99
Secondary: 57R56

Rights: Copyright © 2007 Mathematical Sciences Publishers


Vol.7 • No. 4 • 2007
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