Algebraic & Geometric Topology

Congruence and quantum invariants of 3–manifolds

Patrick M Gilmer

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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.

Article information

Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1767-1790.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M99: None of the above, but in this section
Secondary: 57R56: Topological quantum field theories

surgery framed link modular category TQFT


Gilmer, Patrick M. Congruence and quantum invariants of 3–manifolds. Algebr. Geom. Topol. 7 (2007), no. 4, 1767--1790. doi:10.2140/agt.2007.7.1767.

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