Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 7, Number 4 (2007), 1767-1790.
Congruence and quantum invariants of 3–manifolds
Let be an integer greater than one. We study three progressively finer equivalence relations on closed 3–manifolds generated by Dehn surgery with denominator : weak –congruence, –congruence, and strong –congruence. If is odd, weak –congruence preserves the ring structure on cohomology with –coefficients. We show that strong –congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum invariants are well-behaved under this congruence. We strengthen this result and extend it to the 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 , the Poincaré homology sphere, the Brieskorn homology sphere and their mirror images up to strong –congruence. We distinguish the weak –congruence classes of some manifolds with the same –cohomology ring structure.
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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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. https://projecteuclid.org/euclid.agt/1513796772