Universal $L^2$-torsion, polytopes and applications to $3$-manifolds

TL;DR

This paper introduces a universal $L^2$-torsion invariant for $L^2$-acyclic complexes, exploring its properties and connections to various invariants, including the Thurston norm polytope for 3-manifolds.

Contribution

It defines the universal $L^2$-torsion in terms of the chain complex of the universal cover and studies its key properties and applications to 3-manifold invariants.

Findings

Universal $L^2$-torsion is homotopy invariant.

It relates to invariants like $L^2$-torsion and Thurston norm.

The image of the torsion can be identified with several geometric invariants.

Abstract

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincar\'e duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group $\operatorname{Wh}^w(G)$ to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal $L^2$-torsion can be identified with many invariants such as the $L^2$-torsion, the $L^2$-torsion function, twisted $L^2$-Euler characteristics and, in the case of a $3$-manifold, the dual Thurston norm polytope.