Skip to main content

On conic spaces

  • Survey Articles
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1474))

Abstract

In this paper, the notions of conic spaces, Thom spaces, Hopf Invariants and construction X k are surveyed.

This is a survey paper of material related to the author's Ph.D. thesis. The thesis was carried out under supervision of Prof. M.G.Barratt. I wish to thank Prof. Barratt for his help during many years. I wish to thank Prof. Mahowald for many discussions concerning this study, and for presenting me with [CDGM]. I wish to thank Prof. B. Gray for presenting me with his unpublished work [Gr1] and discussing it with me, and Prof. H. Marcum for discussing with me his work [Mar1] and [Mar3]. I wish to thank Bob Oliver for suggestions concerning the presentation of the paper. I wish to thank Greg Henderson for proof reading the paper. I wish to thank Alex Nofech for mentioning [May2].

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.F. Adams, On the Non-Existence of Elements with Hopf Invariant One, Ann. of Math. 72 (1960), 20–104.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Adem, The Iteration of Steenrod Squares in Algebraic Topology, Proc. Natl. Acad. Sci. 38 (1952), 720–726.

    Article  MathSciNet  MATH  Google Scholar 

  3. M.F. Atiyah, Thom Complexes, Proc. Lond. Math. Soc. 11 (1961), 291–310.

    Article  MathSciNet  MATH  Google Scholar 

  4. W.D. Barcus and M.G. Barratt, On the Homotopy Classification of the Extensions of a Fixed Map, Trans. Amet Math. Soc. 88 (1957), 57–74.

    Article  MathSciNet  MATH  Google Scholar 

  5. M.G. Barratt, Track Groups I,II. Proc. Lond. Math. Soc. 5 (1955), 71–106, 285–329.

    Article  MathSciNet  MATH  Google Scholar 

  6. ____, Higher Hopf Invariants, Mimeographed notes, University of Chicago.

    Google Scholar 

  7. ____, Remark on James invariants, “Conference on Algebraic Topology,” Aarhus University 1962, pp. 102–103.

    Google Scholar 

  8. ____, Spaces of Finite Characteristic, Quat. Jour. of Math. 11 (1960), 124–136.

    MathSciNet  MATH  Google Scholar 

  9. ____, On the Construction X k , Unpublished.

    Google Scholar 

  10. ____, Twisted Lie Algebras, “Geometric applications of homotopy theory II (Northwestern 1977) M.G.Barratt and M.E.Mahowald edtrs,” Lect. Notes in Math. 658, pp. 9–15.

    Google Scholar 

  11. ____, Taming Hopf Invariants, “Combinatorial methods in topology and algebraic geometry (Rochester 1978) J.R.Harper and R.Mandelbaum edtrs,” Contemporary mathematics, Amer. Math. Soc. vol 44, pp. 201–205.

    Google Scholar 

  12. M.G.Barratt, J.D.S.Jones and M.E.Mahowald, The Kervaire Invariant and the Hopf Invariant, “Algebraic Topology Proceedings(Seattle 1985) H.Miller and D.C.Ravenel edtrs,” Lect. Notes in Math. 1286, pp. 135–173.

    Google Scholar 

  13. I. Berstein and P.J. Hilton, On Suspensions and Comultiplications, Topology 2 (1963), 73–82.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.M. Boardman and B. Steer, On Hopf Invariants, Comm. Math. Helv. 42 (1967), 180–221.

    Article  MathSciNet  MATH  Google Scholar 

  15. F.R. Cohen D.M. Davis P.G. Goerss and M.E. Mahowald, Integral Brown-Gitler spectra, Proc. Amer. Math. Soc. 103 (1988), 1299–1304.

    Article  MathSciNet  MATH  Google Scholar 

  16. E.S. Devinatz M.J. Hopkins and J.H. Smith, Nilpotence and Stable Homotopy theory I, Ann. of Math. 128 (1988), 207–241.

    Article  MathSciNet  MATH  Google Scholar 

  17. G.Dula, The Conic Structure of Thom Spaces, Thesis, Northwestern University, Aug. 1988.

    Google Scholar 

  18. Y. Félix, S. Halperin, J.M. Lemaire and J.C. Thomas, Mod p Loop Spaces Homology, Inv. Math. 95 (1989), 247–262.

    Article  MATH  Google Scholar 

  19. T. Ganea, On the Homotopy Suspension, Comm. Math. Helv. 43 (1968), 225–234.

    Article  MathSciNet  MATH  Google Scholar 

  20. B.Gray, Ring Spectra and the James Construction, Mimeographed Notes, 1968.

    Google Scholar 

  21. ____, Operations and Two Cell Complexes, “Conference on Algebraic Topology, V.Gugenheim edtr,” University of Illinois at Chicago Circle,1968, pp. 61–68.

    Google Scholar 

  22. ____, Ring Spectra, “Conference on Algebraic Topology,” Aarhus University 1970, pp. 143–153.

    Google Scholar 

  23. C.H.Hanks, On the Hopf Construction Associated to a Composition, Thesis, Northwestern University. Aug. 1975.

    Google Scholar 

  24. ____, On the Hopf Construction Associated with a Composition, “Geometric Applications of Homotopy Theory II, M.G. Barratt and M.E.Mahowald edtrs,” Lect. Notes in Math. 658, pp. 191–205.

    Google Scholar 

  25. R.P. Held and D.K. Sjerve, On the Homotopy Properties of Thom Complexes, Math. Zeit. 135 (1974), 315–323.

    Article  MathSciNet  MATH  Google Scholar 

  26. P.J. Hilton, On the Homotopy Groups of the Union of Spheres, Jour. Lond. Math. Soc. 30 (1955), 154–172.

    Article  MathSciNet  MATH  Google Scholar 

  27. ____, “Homotopy Theory and Duality,” Notes in Mathematics and its Applications, Gordon and Breach, 1965.

    Google Scholar 

  28. H. Hopf, Über die Abbildungen der Dreidimensionalen Sphare auf die Kugelflache, Math. Ann. 104 (1931), 639–665.

    Article  MathSciNet  MATH  Google Scholar 

  29. , Über die Abbildungen von Spharen auf Spharen Niedriger dimension, Fund. Math. 25 (1935), 427–440.

    MATH  Google Scholar 

  30. M.J.Hopkins, Thesis, Northwestern University, Aug. 1985.

    Google Scholar 

  31. I.M. James, Reduced Product Spaces, Ann. of Math. 62 (1955), 170–197.

    Article  MathSciNet  MATH  Google Scholar 

  32. ____, On the Suspension Triad, Ann. of Math. 63 (1956), 191–247.

    Article  MathSciNet  MATH  Google Scholar 

  33. W.Komornicki, Multiplication in two-cell spectra, “Geometric Applications of Homotopy Theory II, M.G.Barratt and M.E.Mahowald edtrs,” Lect. Notes in Math. 658, pp. 215–223.

    Google Scholar 

  34. U. Koschorke and B. Sanderson, Self intersections and higer Hopf Invariants, Topology (1978), 283–290.

    Google Scholar 

  35. M.E. Mahowald, Ring Spectra which are Thom Complexes, Duke Math. Jour. 46 (1979), 549–559.

    Article  MathSciNet  MATH  Google Scholar 

  36. H.J. Marcum, Fibrations over double mapping cylinders, Ill. jour. of Math. 24 (1980), 344–358.

    MathSciNet  MATH  Google Scholar 

  37. ____, Homotopy decompositions for product spaces, “Atas do Décimo Primero Colóquío Brasileiro de Matemática,” IMPA, Rio de Janeiro, 1978, pp. 665–680.

    Google Scholar 

  38. ____, Functional Properties of Hopf Invariant, to appear in Quaestiones Mathematicae.

    Google Scholar 

  39. J.P. May, “The Geometry of Iterated Loop Spaces,” Springer Verlag, Lecture Notes in Mathematics, 1972.

    Google Scholar 

  40. ____, The dual Whitehead theorems, “Topological topics, article presented to P.J.Hilton on his sixtieth birthday,” London Mathematical Society, Lecture Notes in Mathematics, I.M.James edtr., 1983, pp. 46–64.

    Google Scholar 

  41. J.W.Milnor, On the Construction FK, Mimeographed Lecture Notes, Princeton University, 1956.

    Google Scholar 

  42. ____. Construction of Universal Bundles II, Ann. of Math. 63 (1956), 430–436.

    Article  MathSciNet  MATH  Google Scholar 

  43. G.J. Porter, Higher Order Whitehead Products, Topology 3 (1965), 123–165.

    Article  MathSciNet  MATH  Google Scholar 

  44. D.C. Ravenel, Localization with Respect to Certain Periodic Homology Theories, Amer. Jour. of Math. 106 (1984) 351–414.

    Article  MathSciNet  MATH  Google Scholar 

  45. N.E. Steenrod, Cohomology Invariants of Mappings, Ann. of Math. 50 (1949), 954–988.

    Article  MathSciNet  MATH  Google Scholar 

  46. H.Toda, “Composition Methods in the Homotopy Groups of Spheres,” Annalls of Math. Studies, Princeton University, 1962.

    Google Scholar 

  47. M. Walker, Homotopy Pullbacks and the Hopf Invariant, Jour. Lond. Math. Soc. 19 (1979), 153–158.

    Article  MathSciNet  MATH  Google Scholar 

  48. C.T.C. Wall, Poincaré Complexes I, Ann. of Math. 86 (1967), 213–245.

    Article  MathSciNet  MATH  Google Scholar 

  49. G.W. Whitehead, On the Homotopy Groups of Spheres and Rotation Groups, Ann. of Math. 43 (1942), 634–640.

    Article  MathSciNet  MATH  Google Scholar 

  50. ____, A Generalization of the Hopf Invariant, Ann. of Math. 51 (1950), 192–237.

    Article  MathSciNet  MATH  Google Scholar 

  51. ____, On the Freüdenthal Theorems, Ann. of Math. 57 (1953), 209–228.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Stefan Jackowski Bob Oliver Krzystof Pawałowski

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Dula, G. (1991). On conic spaces. In: Jackowski, S., Oliver, B., Pawałowski, K. (eds) Algebraic Topology Poznań 1989. Lecture Notes in Mathematics, vol 1474. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084736

Download citation

  • DOI: https://doi.org/10.1007/BFb0084736

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54098-4

  • Online ISBN: 978-3-540-47403-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics