Bounded homotopy theory and the $K$-theory of weighted complexes

Abstract

Given a bounding class $B$, we construct a bounded refinement $BK(-)$ of Quillen's $K$-theory functor from rings to spaces. $BK(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $BK\to K$ induced by "forgetting control". In contrast to the situation with $B$-bounded cohomology, there is a functorial splitting $BK(-)\simeq K(-)\times B{K}^{rel}(-)$ where $B{K}^{rel}(-)$ is the homotopy fiber of the comparison map.

25 pages, 13 diagrams; comments welcome