Bounded G-theory with fibred control

Abstract

We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The resulting theory can be thought of as a “pushout” of bounded K-theory with fibred control and bounded G-theory constructed and used by the authors. Bounded G-theory was geared toward constructing a G-theoretic version of assembly maps and proving the Novikov injectivity conjecture for them. The G-theory with fibred control is needed in the study of surjectivity of the assembly map. The relation between the K- and G-theories is the classical one: K-theory is meaningful, however G-theory is easier to compute, and the relationship is expressed via a Cartan map. This map turns out to be an equivalence under very mild constraints in terms of metric geometry such as finite decomposition complexity. The fibred theory is certainly more complicated than the absolute theory. This paper contains the non-equivariant theory including fibred controlled excision theorems known to be crucial for computations.