## Algebraic & Geometric Topology

### Congruence and quantum invariants of 3–manifolds

Patrick M Gilmer

#### Abstract

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 $ℤf$–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 $ℤf$–cohomology ring structure.

#### Article information

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

Dates
Accepted: 31 August 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796772

Digital Object Identifier
doi:10.2140/agt.2007.7.1767

Mathematical Reviews number (MathSciNet)
MR2366177

Zentralblatt MATH identifier
1161.57003

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

#### Citation

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

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