K-theory of geometric modules with fibred control

TL;DR

This paper develops a framework for fibred control in controlled algebra, establishing localization and excision theorems, while addressing challenges posed by equivariant phenomena in geometric topology.

Contribution

It introduces a general framework for fibred control, proves key localization and excision theorems, and analyzes limitations due to equivariant phenomena.

Findings

Established fibred control framework for controlled algebra

Proved localization and fiberwise excision theorems

Identified breakdown of standard tools with equivariant phenomena

Abstract

Controlled algebra plays a central role in many recent advances in geometric topology. This paper studies the iteration construction that was present from the very origins of the theory but started being exploited only recently. We develop the general framework for fibred control, prove localization theorems required for fibrewise excision, and then prove several versions of fibrewise excision theorems. However, we also demonstrate how the standard tools break down in the presence of new equivariant phenomena which require advanced localization methods.