# Homotopical Presentations and Calculations of Algebraic $K_{0}$-Groups for Rings of Continuous Functions

### Hiroshi Kihara

The University of Aizu, Aizu-Wakamatsu City, Fukushima, Japan### Nobuyuki Oda

Fukuoka University, Japan

## Abstract

Let $K_{0}(C_{F}(X))$ = $K_{0}∘C_{F}(X)$ be the $K_{0}$-group of the ring $C_{F}(X)$ of $F$-valued continuous functions on a topological space $X$, where $F$ is the field of real or complex numbers or the quaternion algebra. It is known that the functor $K_{0}∘C_{F}$ is representable on the category of compact Hausdorff spaces. It is a homotopy functor which is not representable on the category of topological spaces. Making use of the compactly-bounded homotopy set, which is a variant of the homotopy set, the functor $K_{0}∘C_{F}$ has a homotopical presentation by the product of the ring of integers $Z$ and the infinite Grassmannian $G_{∞}(F)$. This presentation makes it possible to calculate the groups $K_{0}(C_{F}(X))$ explicitly for some infinite dimensional complexes $X$ by use of the results of H. Miller on Sullivan conjecture.

## Cite this article

Hiroshi Kihara, Nobuyuki Oda, Homotopical Presentations and Calculations of Algebraic $K_{0}$-Groups for Rings of Continuous Functions. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 65–82

DOI 10.2977/PRIMS/61