Joint torsion equals the determinant invariant

TL;DR

This paper proves that the algebraic K-theory determinant invariant for almost commuting Fredholm operators equals the homologically defined joint torsion, confirming conjectures and elucidating properties of these invariants.

Contribution

It establishes the equality of the determinant invariant and joint torsion for Fredholm operators, confirming a conjecture and clarifying their properties.

Findings

Determinant invariant equals joint torsion for almost commuting Fredholm operators.

Joint torsion is norm continuous and depends only on operators modulo trace class.

Determinant invariant for commuting operators can be computed via finite-dimensional determinants.

Abstract

A determinant in algebraic $K$-theory is associated to any two almost commuting Fredholm operators. On the other hand, one can calculate a homologically defined invariant known as joint torsion. We answer in the affirmative a conjecture of Richard Carey and Joel Pincus, namely that these two invariants agree. In particular, this implies that joint torsion is norm continuous, depends only on the images of the operators modulo trace class, and satisfies the expected Steinberg relations. Moreover, we show that the determinant invariant of two commuting operators can be computed simply as a determinant on a finite dimensional vector space.