Universal $L^2$-torsion and sutured decomposition for 3-manifolds

TL;DR

This paper introduces a universal $L^2$-torsion invariant for 3-manifolds that detects fiberedness and characterizes product sutured manifolds, extending previous invariants and providing computational tools.

Contribution

It extends the universal $L^2$-torsion to sutured 3-manifolds and establishes its role in detecting fiberedness and product structures, with explicit computations.

Findings

Universal $L^2$-torsion detects fiberedness in most 3-manifolds.

Trivial universal $L^2$-torsion characterizes product sutured manifolds.

The invariant can be explicitly computed for sutured handlebodies.

Abstract

Given an admissible 3-manifold $M$ and a cohomology class $\phi\in H^1(M;\mathbb R)$, we prove that the universal $L^2$-torsion of $M$ detects the fiberedness of $\phi$, except when $M$ is a closed graph manifold that admits no non-positively curved metric. We further extend this invariant to sutured 3-manifolds and derive a decomposition formula for taut sutured decompositions. Moreover, we show that a taut sutured manifold is a product if and only if its universal $L^2$-torsion is trivial. Our methods are based on a detailed study of the leading term map over Linnell's skew field. As an application, we apply the theory to homomorphisms between finitely generated free groups, which enables explicit computations of the invariant for sutured handlebodies.