The fundamental isomorphism conjecture via non-commutative motives

TL;DR

This paper introduces a fundamental additive functor for groups' orbit categories, demonstrating that isomorphism conjectures valid for it extend to various additive functors like K-theory and cyclic homology, and reduces K-theoretic conjectures to this fundamental case.

Contribution

It constructs a universal additive functor that unifies isomorphism conjectures across multiple invariants, simplifying their verification.

Findings

Validates the conjecture for the fundamental functor

Reduces K-theoretic conjectures to the fundamental case

Establishes a unifying framework for isomorphism conjectures

Abstract

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture for K-theoretic ones.