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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.
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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

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