Skip to main content
SpringerLink
Log in

Weak identities, phantom maps andH-spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

For a pointed spaceX, let Aut(X) be the group of pointed homotopy classes of pointed self-homotopy equivalences ofX and let WI(X) be the normal subgroup of Aut(X) consisting of weak identities, that is, elements represented by maps weakly homotopic to the identity map. IfX is a path-connected CW-space satisfying certain finiteness conditions, then the author has shown elsewhere that the quotient group Aut(X)/WI(X) is a residually finite group. If, in addition,X supports a homotopy-associativeH-space structure, has a finite fundamental group, and has finitely generated higher homotopy groups, it is shown here that any normal subgroupN of Aut(X) rendering the quotient group Aut(X)/N residually finite must contain WI(X). The proof relies on establishing an isomorphism between WI(X) and the group Ph(X) of pointed homotopy classes of phantom self-maps ofX and making a detailed analysis of the group-theoretic structure of the latter, following W. Meier and A. Zabrodsky.

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. J. F. Adams and G. Walker,An example in homotopy theory, Proc. Camb. Phil. Soc.60 (1964), 699–700.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. M. Alonso,Fibrations that are cofibrations, Proc. Am. Math. Soc.87 (1983), 749–753.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. H. Bruck,A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1958.

    MATH  Google Scholar 

  4. B. I. Gray,Spaces of the same n-type, for all n, Topology5 (1966), 241–243.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Hilton, G. Mislin and J. Roitberg,Localization of Nilpotent Groups and Spaces, Notas de Matematica, North-Holland Mathematics Studies 15, Amsterdam, 1975.

  6. W. Meier,Localisation, complétion, et applications fantômes, C.R. Acad. Sci. Paris281 (1975), 787–789.

    MATH  MathSciNet  Google Scholar 

  7. W. Meier,Détermination de certains groupes d’applications fantômes, C.R. Acad. Sci. Paris283 (1976), 971–974.

    MATH  MathSciNet  Google Scholar 

  8. W. Meier,Pullback theorems and phantom maps, Quart. J. Math.29 (1978), 469–481.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Roitberg,Residually finite, Hopfian and co-Hopfian spaces, Contemp. Math.37 (1985), 131–144.

    MathSciNet  Google Scholar 

  10. A. Zabrodsky,Hopf Spaces, Notas de Matematica, North-Holland Mathematics Studies 22, Amsterdam, 1976.

  11. A. Zabrodsky,On phantom maps and a theorem of H. Miller, Isr. J. Math.58 (1987), 129–143.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hunter College, City University of New York, 10021, New York, NY, USA

    Joseph Roitberg

  2. Graduate Center, City University of New York, 10036, New York, NY, USA

    Joseph Roitberg

Authors
  1. Joseph Roitberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported (in part) by a grant from the City University of New York PSC-CUNY Research Award Program.

I thank Colleen Feeney for her excellent and efficient typing of the manuscript.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roitberg, J. Weak identities, phantom maps andH-spaces. Israel J. Math. 66, 319–329 (1989). https://doi.org/10.1007/BF02765901

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02765901

Keywords

Search

Navigation