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Adjoint action of a finite loop space. II

Published online by Cambridge University Press:  14 November 2011

Norio Iwase  and
Akira Kono
Norio Iwase
Affiliation:
Graduate School of Mathematics, Kyushu University Ropponmatsu, Fukuoka 810, Japan
Akira Kono
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan
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Extract

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 773 - 785
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Araki, S. and Kudo, T.. Topology of Hn-spaces and H-squaring operations. Mem. Fac. Sci. Kyushu Univ. A 10 (1956), 85–120.Google Scholar
2Browder, W.. Torsion in Hopf space. Ann. Math. 74 (1961), 24–51.CrossRefGoogle Scholar
3Browder, W.. Homotopy commutative H-space. Ann. Math. 75 (1962), 283–311.CrossRefGoogle Scholar
4Browder, W.. On differential Hopf algebras. Trans. Am. Math. Soc. 107 (1963), 153–176.CrossRefGoogle Scholar
5Browder, W.. Higher torsion in H-spaces. Trans. Am. Math. Soc. 108 (1963), 353–375.Google Scholar
6Clark, A.. Homotopy commutativity and the Moore spectral sequence. Pacific. J. Math. 15 (1965), 65–74.CrossRefGoogle Scholar
7Cook, A. L. and Crabb, M. C.. Fibrewise Hopf structures on sphere-bundles. J. Lond. Math. Soc. 48 (1993), 365–384.CrossRefGoogle Scholar
8Hubbuck, J.. On homotopy commutative H-spaces. Topology 8 (1969), 119–126.CrossRefGoogle Scholar
9Iwase, N.. Adjoint action of a finite loop space. Proc. Am. Math. Soc. 125 (1997), 2753–2757.CrossRefGoogle Scholar
10Kane, R.. Primitivity and finite H-spaces. Q. J. Math. Oxford 3 (1975), 185–197.Google Scholar
11Kono, K. and Kozima, K.. The adjoint action of a Lie group on the space of loops. J. Math. Soc. Japan 45 (1993), 495–509.CrossRefGoogle Scholar
12Kono, K. and Kozima, K.. The adjoint action of the Dwyer-Wilkerson H-space on its loop space. J. Math. Kyoto Univ. 35 (1995), 53–62.Google Scholar
13Lin, J. P.. Torsion in H-spaces. II. Ann. Math. 107 (1978), 41–88.CrossRefGoogle Scholar
14Milnor, J. and Moore, J. C.. On the structure of Hopf algebras. Ann. Math. 81 (1965), 211–264.CrossRefGoogle Scholar
15Moore, J. C.. suspension, La. Seminar H. Cartan 12 (1959/1960), expose 6.Google Scholar
16Quillen, D.. The mod 2 cohomology rings of extra-special 2-groups and the spinor groups. Math. Ann. 194 (1971), 197–212.CrossRefGoogle Scholar
17Thomas, E.. On functional cup-products and the transgression operator. Arch. Math. 12 (1961), 435–444.CrossRefGoogle Scholar
18Zabrodsky, A.. Implications in the cohomology of H-space. Illinois J. Math. 16 (1971), 363–375.Google Scholar