Skip to main content
SpringerLink
Log in

Comultiplications on a wedge of two spheres

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study the set of comultiplications on a wedge of two spheres. We are primarily interested in the size of this set and properties of the comultiplications such as associativity and commutativity. Our methods involve Whitehead products in wedges of spheres and the Hopf-Hilton invariants. We apply our results to specific examples and determine the number of comultiplications, associative comultiplications and commutative comultiplications in these cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arkowitz M. Co-H-spaces, Handbook of Algebraic Topology. New York: North-Holland, 1995, 1143–1173

    Book  Google Scholar 

  2. Arkowitz M, Gutierrez M. Comultiplications on free groups and wedge of circles. Trans Amer Math Soc, 1998, 350: 1663–1680

    Article  MATH  MathSciNet  Google Scholar 

  3. Arkowitz M, Lupton G. Rational co-H-spaces. Comment Math Helv, 1991, 66: 79–108

    Article  MATH  MathSciNet  Google Scholar 

  4. Arkowitz M, Lupton G. Equivalence classes of homotopy-associative comultiplications of finite complexes. J Pure Appl Algebra, 1995, 102: 109–136

    Article  MATH  MathSciNet  Google Scholar 

  5. Berstein I. A note on spaces with non-associative comultiplication. Math Proc Cambridge Philos Soc, 1964, 60: 353–354

    Article  MATH  MathSciNet  Google Scholar 

  6. Berstein I, Hilton P J. Categories and generalized Hopf invariants. Illinois J Math, 1960, 4: 437–451

    MATH  MathSciNet  Google Scholar 

  7. Ganea T. Cogroups and suspensions. Invent Math, 1970, 9: 185–197

    Article  MATH  MathSciNet  Google Scholar 

  8. Golasinski M, Goncalves D L. Comultiplications of the wedge of two Moore spaces. Colloq Math, 1998, 76: 229–242

    MATH  MathSciNet  Google Scholar 

  9. Hilton P J. On the homotopy groups of the union of spheres. J London Math Soc, 1955, 30: 154–172

    Article  MATH  MathSciNet  Google Scholar 

  10. Hilton P J. Homotopy Theory and Duality. New York: Gordon and Breach, 1965

    Google Scholar 

  11. Naylor C M. On the number of co-multiplications of a suspension. Illinois J Math, 1968, 12: 620–622

    MATH  MathSciNet  Google Scholar 

  12. Spanier E. Algebraic Topology. New York: McGraw-Hill, 1966

    MATH  Google Scholar 

  13. Toda H. Composition Methods in Homotopy Groups of Spheres. Princeton: Princeton University Press, 1962

    MATH  Google Scholar 

  14. Whitehead G W. Elements of Homotopy Theory. Graduate Texts in Math 61. New York-Berlin: Springer-Verlag, 1978

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, USA

    Martin Arkowitz

  2. Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju, Jeonbuk, 561-756, Korea

    Dae-Woong Lee

Authors
  1. Martin Arkowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dae-Woong Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dae-Woong Lee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arkowitz, M., Lee, DW. Comultiplications on a wedge of two spheres. Sci. China Math. 54, 9–22 (2011). https://doi.org/10.1007/s11425-010-4061-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4061-0

Keywords

MSC(2000)

Search

Navigation