Advertisement

Life after the Telescope Conjecture

Chapter
Part of the NATO ASI Series book series (ASIC, volume 407)

Abstract

We discuss the chromatic filtration in stable homotopy theory and its connections with algebraic K-theory, specifically with some results of Thomason, Mitchell, Waldhausen and McClure-Staffeldt. We offer a new definition (suggested by the failure of the telescope conjecture) of the chromatic filtration, in which all of the localization functors used are finite.

Keywords

Direct Limit Homotopy Group Algebraic Topology Inverse Limit Localization Functor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BD92]
    M. Bendersky and D. M. Davis. 2-primary v1-periodic homotopy groups of SU(n). American Journal of Mathematics, 114: 465 - 494, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [BDM]
    M. Bendersky, D. M. Davis, and M. Mimura. vi-periodic homotopy groups of exceptional Lie groups—torsion-free cases. To appear in Transactions of the American Mathematical Society.Google Scholar
  3. [Ben92]
    M. Bendersky. The v1-periodic unstable Novikov spectral sequence. Topology, 31: 47 - 64, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [13òu75]
    A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14: 133 - 150, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [Bou79]
    A. K. Bousfield. The localization of spectra with respect to homology. Topology, 18: 257 - 281, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Dav91]
    D. M. Davis. The v1-periodic homotopy groups of SU(n) at odd primes. Proceedings of the London Mathematical Society (3), 43: 529 - 544, 1991.zbMATHCrossRefGoogle Scholar
  7. [DHS88]
    E. Devinatz, M. J. Hopkins, and J. H. Smith. Nilpotence and stable homotopy theory. Annals of Mathematics, 128: 207 - 242, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [DM92]
    DM92] D. M. Davis and M. E. Mahowald. Some remarks on v1-periodic homotopy groups. In N. Ray and G. Walker, editors, Adams Memorial Symposium on Algebraic Topology Volume 2, pages 55 - 72, Cambridge University Press, Cambridge, 1992.Google Scholar
  9. [GZ67]
    P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory. Springer-Verlag, New York, 1967.zbMATHCrossRefGoogle Scholar
  10. [Hop87]
    M. J. Hopkins. Global methods in homotopy theory. In J. D. S. Jones and E. Rees, editors, Proceedings of the 1985 LMS Symposium on Homotopy Theory, pages 73 - 96, 1987.Google Scholar
  11. [HRa]
    M. J. Hopkins and D. C. Ravenel. A proof of the smash product conjecture. To appear.Google Scholar
  12. [HRb]
    M. J. Hopkins and D. C. Ravenel. Suspension spectra are harmonic. To appear in Bol. Soc. Math. Mexicana.Google Scholar
  13. [HS]
    M. J. Hopkins and J. H. Smith. Nilpotence and stable homotopy theory II. Submitted to Annals of Mathematics. Google Scholar
  14. [Mah82]
    M. E. Mahowald. The image of J in the EHP sequence. Annals of Mathematics, 116: 65 - 112, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Mil]
    H. R. Miller. Finite localizations. To appear.Google Scholar
  16. [Mita]
    Mita] S. A. Mitchell. Harmonic localization of algebraic K-theory spectra. To appear in Transactions of the American Mathematical Society.Google Scholar
  17. [Mitb]
    S. A. Mitchell. On the Lichtenbaum-Quillen conjectures from a stable homotopytheoretic viewpoint. To appear in Topics in Algebraic Topology and its Applications, MSRI publication series, Springer Verlag (1992).Google Scholar
  18. [Mit85]
    S. A. Mitchell. Finite complexes with A(n)-free cohomology. Topology, 24: 227248, 1985.Google Scholar
  19. [Mit90]
    S. A. Mitchell. The Morava K-theory of algebraic K-theory spectra. K-Theory, 3: 607 - 626, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [MRW77]
    H. R. Miller, D. C. Ravenel, and W. S. Wilson. Periodic phenomena in the Adams-Novikov spectral sequence. Annals of Mathematics, 106: 469 - 516, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [MSa]
    MSa] M. E. Mahowald and H. Sadofsky. vn-telescopes and the Adams spectral sequence. To appear.Google Scholar
  22. [MSb]
    J. E. McClure and R. E. Staffeldt. The chromatic convergence theorem and a tower in algebraic K-theory. To appear in Proceedings of the American Mathematical Society.Google Scholar
  23. [Ray]
    D. C. Ravenel. A counterexample to the telescope conjecture. To appear.Google Scholar
  24. [Rav84]
    D. C. Ravenel. Localization with respect to certain periodic homology theories. American Journal of Mathematics, 106: 351 - 414, 1984.MathSciNetCrossRefGoogle Scholar
  25. [Rav86]
    D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. Academic Press, New York, 1986.zbMATHGoogle Scholar
  26. [Rav87]
    D. C. Ravenel. The geometric realization of the chromatic resolution. In W. Browder, editor, Algebraic topology and algebraic K-theory, pages 168 - 179, 1987.Google Scholar
  27. [Rav90]
    D. C. Ravenel. The nilpotence and periodicity theorems in stable homotopy theory. Astérisque, 189-190: 399 - 428, 1990.MathSciNetGoogle Scholar
  28. [Rav92a]
    D. C. Ravenel. Nilpotence and periodicity in stable homotopy theory. Volume 128 of Annals of Mathematics Studies, Princeton University Press, Princeton, 1992.Google Scholar
  29. [Rav92b]
    D. C. Ravenel. Progress report on the telescope conjecture. In N. Ray and G. Walker, editors, Adams Memorial Symposium on Algebraic Topology Volume 2, pages 1 - 21, Cambridge University Press, Cambridge, 1992.Google Scholar
  30. [Smi]
    J. Smith. Finite complexes with vanishing lines of small slope. To appear.Google Scholar
  31. [Tho85]
    R. W. Thomason. Algebraic K-theory and étale cohomology. Ann. Scient. Écoles Norm. Sup., 13: 437 - 552, 1985.MathSciNetGoogle Scholar
  32. [Tho90]
    R. J. Thompson. Unstable vi-periodic homotopy at odd primes. Transactions of the American Mathematical Society, 319: 535 - 559, 1990.MathSciNetzbMATHGoogle Scholar
  33. [Wa184]
    F. Waldhausen. Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy theory. In I. Madsen and R. Oliver, editors, Algebraic Topology Aarhus 1982, pages 173-195, Lecture Notes in Mathematics, 1051, Springer-Verlag, 1984.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA

Personalised recommendations