# 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 $K_0(C_{\Bbb F}(X))$ = $K_0\circ C_{\Bbb F}(X)$ be the $K_0$-group of the ring $C_{\Bbb F}(X)$ of ${\Bbb F}$-valued continuous functions on a topological space $X$, where ${\Bbb F}$ is the field of real or complex numbers or the quaternion algebra. It is known that the functor $K_0\circ C_{\Bbb F}$ 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 $K_0\circ C_{\Bbb F}$ has a homotopical presentation by the product of the ring of integers ${\Bbb Z}$ and the infinite Grassmannian $G_{\infty}(\Bbb F )$. This presentation makes it possible to calculate the groups $K_0(C_{\Bbb F}(X))$ 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