JournalsprimsVol. 48, No. 1pp. 65–82

Homotopical Presentations and Calculations of Algebraic K0K_0-Groups for Rings of Continuous Functions

  • Hiroshi Kihara

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

    Fukuoka University, Japan
Homotopical Presentations and Calculations of Algebraic $K_0$-Groups for Rings of Continuous Functions cover
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Abstract

Let K0(CF(X))K_0(C_{\Bbb F}(X)) = K0CF(X)K_0\circ C_{\Bbb F}(X) be the K0K_0-group of the ring CF(X)C_{\Bbb F}(X) of F{\Bbb F}-valued continuous functions on a topological space XX, where F{\Bbb F} is the field of real or complex numbers or the quaternion algebra. It is known that the functor K0CFK_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 K0CFK_0\circ C_{\Bbb F} has a homotopical presentation by the product of the ring of integers Z{\Bbb Z} and the infinite Grassmannian G(F)G_{\infty}(\Bbb F ). This presentation makes it possible to calculate the groups K0(CF(X))K_0(C_{\Bbb F}(X)) explicitly for some infinite dimensional complexes XX by use of the results of H. Miller on Sullivan conjecture.

Cite this article

Hiroshi Kihara, Nobuyuki Oda, Homotopical Presentations and Calculations of Algebraic K0K_0-Groups for Rings of Continuous Functions. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 65–82

DOI 10.2977/PRIMS/61