## Geometry & Topology

### Unified quantum invariants for integral homology spheres associated with simple Lie algebras

#### Abstract

For each finite-dimensional, simple, complex Lie algebra $g$ and each root of unity $ξ$ (with some mild restriction on the order) one can define the Witten–Reshetikhin–Turaev (WRT) quantum invariant $τMg(ξ) ∈ ℂ$ of oriented $3$–manifolds $M$. We construct an invariant $JM$ of integral homology spheres $M$, with values in $ℤ[q]̂$, the cyclotomic completion of the polynomial ring $ℤ[q]$, such that the evaluation of $JM$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case $g = sl2$ proved by Habiro. It follows that $JM$ unifies all the quantum invariants of $M$ associated with $g$ and represents the quantum invariants as a kind of “analytic function” defined on the set of roots of unity. For example, $τM(ξ)$ for all roots of unity are determined by a “Taylor expansion” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants $τM(ξ)$ for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at $q = 1$, and hence by the Lê–Murakami–Ohtsuki invariant. Another consequence is that the WRT quantum invariants $τMg(ξ)$ are algebraic integers. The construction of the invariant $JM$ is done on the level of quantum group, and does not involve any finite-dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, “representation-free” definition of the quantum invariants of integral homology spheres.

#### Article information

Source
Geom. Topol., Volume 20, Number 5 (2016), 2687-2835.

Dates
Accepted: 25 October 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859044

Digital Object Identifier
doi:10.2140/gt.2016.20.2687

Mathematical Reviews number (MathSciNet)
MR3556349

Zentralblatt MATH identifier
1362.57019

#### Citation

Habiro, Kazuo; Lê, Thang T Q. Unified quantum invariants for integral homology spheres associated with simple Lie algebras. Geom. Topol. 20 (2016), no. 5, 2687--2835. doi:10.2140/gt.2016.20.2687. https://projecteuclid.org/euclid.gt/1510859044

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