Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T07:03:23.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational Cup Product and Algebraic K0-Groups of Rings of Continuous Functions

Part of: Fiber spaces and bundles Homological methods Grothendieck groups and $K_0$

Published online by Cambridge University Press:  10 April 2018

Hiroshi Kihara  and
Nobuyuki Oda
Hiroshi Kihara*
Affiliation:
Center for Mathematical Sciences, University of Aizu, Tsuruga, Ikki-machi, Aizu-Wakamatsu City, 965-8580 Fukushima, Japan ([email protected])
Nobuyuki Oda
Affiliation:
Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan ([email protected])
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

A connected space is called a C0-space if its rational cup product is trivial. A characterizing property of C0-spaces is obtained. This property is used to calculate the algebraic K0-group K0(C𝔽(X)) of the ring of continuous functions for infinite-dimensional complexes X.

Keywords

K-groupSerre–Swan theoremSullivan conjecture

MSC classification

Primary: 55R50: Stable classes of vector space bundles, $K$-theory
Secondary: 16E20: Grothendieck groups, $K$-theory, etc. 19A49: $K_0$ of other rings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 607 - 622
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Arkowitz, M., Co-H-spaces, In Handbook of algebraic topology, pp. 1143–1173 (North-Holland, Amsterdam, 1995).CrossRefGoogle Scholar
2.Gabriel, P. and Zisman, M., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 (Springer-Verlag, New York, 1967).Google Scholar
3.Goerss, P. G. and Jardine, J. F., Simplicial homotopy theory, Progress in Mathematics, Volume 174 (Birkhäuser Verlag, Basel, 1999).Google Scholar
4.Husemoller, D., Fibre bundles, Graduate Texts in Mathematics, Volume 20 (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
5.Kihara, H., Groups of homotopy classes of phantom maps, Algebr. Geom. Topol. 18(1) (2018), 583–612.CrossRefGoogle Scholar
6.Kihara, H., Commutativity and cocommutativity of cogroups in the category of connected graded algebras, Topology Appl. 189 (2015), 107–121.Google Scholar
7.Kihara, H. and Oda, N., Homotopical presentations and calculations of algebraic K 0-groups for rings of continuous functions, Publ. Res. Inst. Math. Sci. 48(1) (2012), 65–82.Google Scholar
8.MacLane, S., Categories for the working mathematician, 2nd edition (Springer-Verlag, New York, 1998).Google Scholar
9.May, J. P., Simplicial objects in algebraic topology (University of Chicago Press, Chicago, 1967).Google Scholar
10.Roitberg, J. and Touhey, P., The homotopy fiber of profinite completion, Topology Appl. 103 (2000), 295–307.CrossRefGoogle Scholar
11.Scheerer, H., On rationalized H- and co-H-spaces, with an appendix on decomposable H- and co-H-spaces, Manuscripta Math. 51 (1985), 63–87.Google Scholar
12.Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38 (Cambridge University Press, Cambridge, 1994).Google Scholar
13.Whitehead, G. W., Elements of homotopy theory, Graduate Texts in Mathematics, Volume 61 (Springer-Verlag, New York, 1978).CrossRefGoogle Scholar