Joint torsion of several commuting operators

TL;DR

This paper introduces a new invariant called joint torsion for multiple commuting operators, generalizing existing concepts and linking to classical invariants like Lefschetz numbers and Cauchy integrals.

Contribution

It defines joint torsion for several commuting operators satisfying a Fredholm condition, extending previous pairwise torsion concepts and exploring its algebraic properties and applications.

Findings

Joint torsion generalizes Carey-Pincus torsion for pairs of operators.

Under certain conditions, joint torsion recovers multiplicative Lefschetz numbers.

Links are established between joint torsion and the Cauchy integral formula for Toeplitz operators.

Abstract

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalizes the Carey-Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a symmetry property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.