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On the β-Construction in K-Theory

Published online by Cambridge University Press:  20 November 2018

C. M. Naylor
C. M. Naylor*
Affiliation:
University of California, Irvine, California
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The β-construction assigns to each complex representation φ of the compact Lie group G a unique element β(φ) in (G). For the details of this construction the reader is referred to [1] or [5]. The purpose of the present paper is to determine some of the properties of the element β(φ) in terms of the invariants of the representation φ. More precisely, we consider the following question. Let G be a simple, simply-connected compact Lie group and let f : S3 →G be a Lie group homomorphism. Then (S3) ⋍ Z with generator x = β(φ1), φ1 the fundamental representation of S3 , so that if φ is a representation of G,f*(φ) = n(φ)x, where n(φ) is an integer depending on φ and f . The problem is to determine n(φ).

Since G is simple and simply-connected we may assume that ch2, the component of the Chern character in dimension 4 takes its values in H4(SG,Z)≅Z. Let u be a generator of H4(SG,Z) so that ch2(β (φ)) = m(φ)u, m(φ) an integer depending on φ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 819 - 824
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Atiyah, M. F., On the K-theory of compact Lie groups, Topology 4 (1965), 95–99.Google Scholar
2. Bott, R., Lectures on K(X) (Benjamin, New York, 1969).Google Scholar
3. Bott, R. and Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.Google Scholar
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