Ring and module structures on $K$-theory of leaf spaces and their application to longitudinal index theory

TL;DR

This paper introduces a new ring-structured K-theory model for leaf spaces of foliations, contrasting with Connes' model, and interprets module multiplication via indices of twisted longitudinal elliptic operators.

Contribution

It constructs a ring-based K-theory model for leaf spaces using the stable Higson corona, providing a new algebraic framework and linking it to index theory.

Findings

New ring structure on leaf space K-theory

Connes' K-theory as a module over this ring

Interpretation of module multiplication via index theory

Abstract

Pursuing conjectures of John Roe, we use the stable Higson corona of foliated cones to construct a new $K$-theory model for the leaf space of a foliation. This new $K$-theory model is -- in contrast to Alain Connes' $K$-theory model -- a ring. We show that Connes' $K$-theory model is a module over this ring and develop an interpretation of the module multiplication in terms of indices of twisted longitudinally elliptic operators.