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

TL;DR

This paper introduces a bounded refinement of Quillen's K-theory for weighted rings, providing a functorial splitting that separates the classical K-theory from a relative part, enhancing understanding of bounded homotopy theory.

Contribution

It constructs a new bounded K-theory functor for weighted rings with a natural splitting, extending classical K-theory with control conditions.

Findings

Established a bounded refinement of Quillen's K-theory.

Proved the functorial splitting of the bounded K-theory.

Defined the homotopy fiber as a relative bounded K-theory component.

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 $BK(-) \simeq K(-) \times BK^{rel}(-)$ where $BK^{rel}(-)$ is the homotopy fiber of the comparison map.