Homotopical Presentations and Calculations of Algebraic $K_0$-Groups for Rings of Continuous Functions

Abstract

Let $K_0(C_{\mathbb{F}}(X))$ = $K_0\circ C_{\mathbb{F}}(X)$ be the $K_0$-group of the ring $C_{\mathbb{F}}(X)$ of $\mathbb{F}$-valued continuous functions on a topological space $X$, where $\mathbb{F}$ is the field of real or complex numbers or the quaternion algebra. It is known that the functor $K_0\circ C_{\mathbb{F}}$ 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 $K_0\circ C_{\mathbb{F}}$ has a homotopical presentation by the product of the ring of integers $\mathbb{Z}$ and the infinite Grassmannian $G_{\infty }(\mathbb{F})$. This presentation makes it possible to calculate the groups $K_0(C_{\mathbb{F}}(X))$ explicitly for some infinite dimensional complexes $X$ by use of the results of H. Miller on Sullivan conjecture.