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

Let K0(CF(X)) = K0 ◦CF(X) be the K0-group of the ring CF(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 K0 ◦CF is representable on the category of compact Hausdorff spaces. It is a homotopy functor which is not representable on the category of topological spaces. With the use of the notion of a compactly-bounded homotopy set, which is a variant of a homotopy set, the functor K0 ◦CF has a homotopical presentation by means of the product of the ring of integers Z and the infinite Grassmannian G∞(F). This presentation makes it possible to calculate the groups K0(CF(X)) explicitly for some infinite-dimensional complexes X by using the results of H. Miller on Sullivan’s conjecture. 2010 Mathematics Subject Classification: Primary 55R50; Secondary 16E20, 19A49.