Let = be the -group of the ring of -valued continuous functions on a topological space , where is the field of real or complex numbers or the quaternion algebra. It is known that the functor 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 has a homotopical presentation by the product of the ring of integers and the infinite Grassmannian . This presentation makes it possible to calculate the groups explicitly for some infinite dimensional complexes by use of the results of H. Miller on Sullivan conjecture.
Cite this article
Hiroshi Kihara, Nobuyuki Oda, Homotopical Presentations and Calculations of Algebraic -Groups for Rings of Continuous Functions. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 65–82DOI 10.2977/PRIMS/61