Controlled K-theory I: Basic theory

Abstract

This paper provides a full controlled version of algebraic $K$-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There is a careful treatment of spectral cosheaf homology and related tools, including an ``iterated homology identity'' giving a spectrum-level version of the Leray-Serre spectral sequence.