Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-29T18:18:13.659Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable proofs of stable splittings

Published online by Cambridge University Press:  24 October 2008

Ralph L. Cohen
Ralph L. Cohen
Affiliation:
University of Chicago
Get access Rights & Permissions [Opens in a new window]

Extract

V. P. Snaith's theorem giving splittings of the suspension spectrum of ΩnΣnX for a connected space X (8) has been exploited in several areas of homotopy theory. (See (2), (5) and (6), for instance.) Although this is a theorem about spectra, Snaith's proof (as well as a subsequent proof of Cohen, May, and Taylor (4)) proceeds unstably, on the space level. While the methods employed in these proofs are useful, they are somewhat complicated. The purpose of this note is to give a simple proof of this theorem, using only spectrum level arguments.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 149 - 151
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Boardman, J. R. and Vogt, R. M.Homotopy invariant algebraic structures on topological spaces. Springer Lecture Notes in Mathematics, no. 347 (1973).CrossRefGoogle Scholar
(2)Brown, E. H. Jr and Peterson, F. P.On the stable decomposition of Ω2S r + 2. Trans. Amer. Math. Soc. 243 (1978), 287–298.CrossRefGoogle Scholar
(3)Bruner, R., Lewis, G., May, J. P., McClure, J. and Steinberger, M.H∞ ring spectra and their applications. Springer Lecture Notes in Mathematics. (To appear.)Google Scholar
(4)Cohen, F. R., May, J. P. and Taylor, L. R.Splitting of certain spaces CX. Math. Proc. Cambridge Philos. Soc. 84 (1978), 465–496.CrossRefGoogle Scholar
(5)Cohen, R. L. New infinite families in the stable homotopy groups of spheres and of Moore spaces. (To appear.)Google Scholar
(6)Mahowald, M.A new infinite family in . Topology 16 (1977), 249–254.CrossRefGoogle Scholar
(7)May, J. P.The geometry of iterated loop spaces. Springer Lecture Notes in Mathematics, no. 271 (1972).CrossRefGoogle Scholar
(8)Snaith, V. P.A stable decomposition for ΩnSnX. J. London Math. Soc. 7 (1974), 577–583.CrossRefGoogle Scholar