Noncommutative stable homotopy theory

TL;DR

This paper develops a stable homotopy theoretic framework for Kasparov's KK and E-theory, enabling broader applications in categories like bornological algebras and extension groups.

Contribution

It introduces a novel stable homotopy construction of KK and E-theory, expanding their applicability to more general algebraic categories.

Findings

Constructed KK and E-theory via stable homotopy methods

Applied the framework to groups of extensions

Extended the scope of bivariant theories to bornological algebras

Abstract

We construct Kasparov's bifunctor $KK$ and $E$-theory by stable homotopy theoretic methods. This is motivated by results concerning constructions of bivariant theories on more general categories such as, for example, bornological algebras. The details of the construction have interesting applications to groups of extensions.