Three-manifold invariants and their relation with the fundamental group

TL;DR

This paper explores the relationship between 3-manifold invariants derived from Chern-Simons theory and the fundamental group, providing proofs and numerical evidence for the conjecture that the invariant's absolute value depends solely on the fundamental group.

Contribution

It proves the conjecture for lens spaces with SU(2) gauge group and provides numerical support for SU(3), linking the invariant to the fundamental group of the manifold.

Findings

|I(M)| depends only on (M) for nonzero invariants

Proof of the conjecture for lens spaces with SU(2)

Numerical evidence supporting the conjecture for SU(3)

Abstract

We consider the 3-manifold invariant I(M) which is defined by means of the Chern-Simons quantum field theory and which coincides with the Reshetikhin-Turaev invariant. We present some arguments and numerical results supporting the conjecture that, for nonvanishing I(M), the absolute value | I(M) | only depends on the fundamental group \pi_1 (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture.