A non-commutative Reidemeister-Turaev torsion of homology cylinders

TL;DR

This paper introduces a new non-commutative Reidemeister-Turaev torsion invariant for homology cylinders, linking it to finite-type invariants and known quantum invariants, enriching the understanding of 3-manifold topology.

Contribution

It computes a non-commutative torsion invariant for homology cylinders and relates it to finite-type invariants and quantum invariants like the LMO homomorphism.

Findings

The torsion takes values in the $K_1$-group of the $I$-adic completion.

Its reduction is a finite-type invariant of degree $d$.

The leading term recovers the 1-loop part of the LMO homomorphism and the Enomoto-Satoh trace.

Abstract

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the $K_1$-group of the $I$-adic completion of the group ring $\mathbb{Q}\pi_1\Sigma_{g,1}$, and prove that its reduction to $\widehat{\mathbb{Q}\pi_1\Sigma_{g,1}}/\hat{I}^{d+1}$ is a finite-type invariant of degree $d$. We also show that the $1$-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.