Skip to main content
SpringerLink
Log in

Fibrations with Hopfian properties

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

Abstract

This study explores the homotopy-theoretic meeting-point of topics in differential topology, combinatorial group theory and algebraicK-theory. The first two are due to H. Hopf and date from around 1930. The third arose in the author’s characterisation of plus-constructive fibrations. LetF ( ί→ )E →B be a fibration such thati induces an isomorphism of homology with trivial integer coefficients; what is the effect ofi on fundamental groups? In particular, when one passes to hypoabelianisations by factoring out perfect radicals, doesi induce an epimorphism? Numerous conditions are determined which force an affirmative answer. On the other hand, negative examples of a non-finitary nature are also provided. This leaves the question open in the finitely generated case, where it forms a homological version of the dual to Hopf’s original, famous question in group theory.

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. A. J. Berrick,An Approach to Algebraic K-Theory, Research Notes in Math. 56, Pitman, London, 1982.

    MATH  Google Scholar 

  2. A. J. Berrick,The plus-construction and fibrations, Quart. J. Math. (2)33 (1982), 149–157.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. J. Berrick,Characterisation of plus-constructive fibrations, Adv. in Math.48 (1983), 172–176.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. J. Berrick,Group epimorphisms preserving perfect radicals, and the plus-construction, inAlgebraic K-Theory, Number Theory, Geometry and Analysis — Proc. Bielefeld 1982, Lecture Notes in Math.1046 Springer-Verlag, Berlin, 1984, pp. 1–12.

    Chapter  Google Scholar 

  5. A. J. Berrick,Remarks on the structure of acyclic groups, in preparation.

  6. A. J. Berrick and B. Hartley,Perfect radicals and homology of group extensions, Topology and its Applications25 (1987), 65–73.

    Article  MathSciNet  Google Scholar 

  7. E. Dror,A generalization of the Whitehead theorem, Lecture Notes in Math.249, Springer-Verlag, Berlin, 1971, pp. 13–22.

    Google Scholar 

  8. J. L. Dyer and E. Formanek,Characteristic subgroups and complete automorphism groups, Am. J. Math.99 (1977), 713–753.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Evens,The cohomology ring of a finite group, Trans. Am. Math. Soc.101 (1961), 224–239.

    Article  MATH  MathSciNet  Google Scholar 

  10. J.-C. Hausmann,Every finite complex has the homology of a duality group, to appear.

  11. A. Heller,On the homotopy theory of topogenic groups and groupoids, Illinois J. Math.24 (1980), 576–605.

    MATH  MathSciNet  Google Scholar 

  12. P. Hilton and J. Roitberg,On the Zeeman comparison theorem for the homology of quasinilpotent fibrations, Quart. J. Math. (2)27 (1976), 433–444.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Hilton and J. Roitberg,Note on epimorphisms and monomorphisms in homotopy theory, Proc. Am. Math. Soc.90 (1984), 316–320.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Hopf,Zur Topologie der Abbildungen von Mannigfaltigkeiten, II, Math. Ann.102 (1930), 562–623.

    Article  MathSciNet  Google Scholar 

  15. H. Hopf,Beiträge zur Klassifizierung der Flächenabbildungen, J. Reine Angew. Math.165 (1931), 225–236.

    MATH  Google Scholar 

  16. D. M. Kan and W. P. Thurston,Every connected space has the homology of a K (π, 1), Topology15 (1976), 253–258.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. C. Lyndon and P. E. Schupp,Combinatiorial Group Theory, Ergeb. der Math. 89, Springer-Verlag, Berlin, 1977.

    Google Scholar 

  18. J. R. Stallings,Homology and central series of groups, J. Algebra2 (1965), 170–181.

    Article  MATH  MathSciNet  Google Scholar 

  19. U. Stammbach,Anwendungen der Homologietheorie der Gruppen auf Zentralreihen und auf Invarianten von Präsentierungen, Math. Z.94 (1966), 157–177.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 0511, Kent Ridge, Singapore

    A. J. Berrick

Authors
  1. A. J. Berrick
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berrick, A.J. Fibrations with Hopfian properties. Israel J. Math. 66, 41–53 (1989). https://doi.org/10.1007/BF02765885

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02765885

Keywords

Search

Navigation