Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T12:49:22.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Delooping the total Stiefel–Whitney class

Published online by Cambridge University Press:  24 October 2008

A. Kozlowski
A. Kozlowski
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Let FH(X) denote the group of units of the classical cohomology ring H(X) = Πn≥0Hn(X; Z/2) of a CW-complex X. The total Stiefel–Whitney class can be viewed as a group homomorphism where is the reduced real K-theory of X. Both and FH( ) are representable functors, with representing spaces BO and FH, and thus w can be represented by a map w: BO → FH. By the Bott periodicity theorem, BO is an infinite loop space, and by a theorem of G. Segal[9] so is FH. However, it is well known that w is not an infinite loop map; this was first shown in [10]. The purpose of this paper is to prove the following:

Theorem 0·1. w: BO → FHis a loop map but not a double loop map.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 83 - 87
Copyright
Copyright © Cambridge Philosophical Society 1988

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. M. and Vogt, R. M.. Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Math. vol. 347 (Springer-Verlag, 1973).CrossRefGoogle Scholar
[2]Cohen, F. R., Lada, T. and May, J. P.. The Homology of Iterated Loop Spaces. Lecture Notes in Math. vol. 533 (Springer-Verlag, 1976).CrossRefGoogle Scholar
[3]Kochman, S. O.. Homology of the classical groups over the Dyer–Lashof algebra. Trans. Amer. Math. Soc. 185 (1973), 83–136.CrossRefGoogle Scholar
[4]Kozlowski, A.. Operations in Segal's cohomology. Math. Proc. Cambridge Philos. Soc. 95 (1984), 437–441.CrossRefGoogle Scholar
[5]Kozlowski, A.. The total Steenrod operation is induced by an A ∞ ring homomorphism. Math. Proc. Cambridge Philos. Soc. 101 (1987), 469–476.CrossRefGoogle Scholar
[6]May, J. P. (with contributions by N. Ray, F. Quinn and J. Tornehave). E∞ Ring Spaces and E∞ Ring Spectra. Lecture Notes in Math. vol. 577 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[7]May, J. P.. A ∞ ring spaces and algebraic K-theory. In Algebraic Topology, Aarhus 1982, Lecture Notes in Math. vol. 1051 (Springer-Verlag, 1984), pp. 240–315.Google Scholar
[8]Schwänzl, R. and Vogt, R. M.. Homotopy invariance of A ∞ and E ∞ ring spaces. In Algebraic Topology, Aarhus 1982, Lecture Notes in Math. vol. 1051 (Springer-Verlag, 1984), pp. 442–481.Google Scholar
[9]Segal, G. B.. The multiplicative group of classical cohomology. Quart. J. Math. 26 (1975), 289–293.CrossRefGoogle Scholar
[10]Snaith, V.. The total Chern and Stiefel–Whitney classes are not infinite loop maps. Illinois J. Math. 21 (1977), 300–303.CrossRefGoogle Scholar
[11]Steiner, R. J.. A canonical operad pair. Math. Proc. Cambridge Philos. Soc. 86 (1979), 443–449.CrossRefGoogle Scholar