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

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

2011-02-02

arxiv.org/abs/1102.0497v2

Proc. Steklov Inst. Math. 275 (2011), 199 - 215

Mathematics

K-Theory and Homology

25 pages, 13 diagrams; comments welcome

Scientific paper

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.

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