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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The odd primary $ H$-structure of low rank Lie groups and its application to exponents


Author: Stephen D. Theriault
Journal: Trans. Amer. Math. Soc. 359 (2007), 4511-4535
MSC (2000): Primary 55P45, 55Q52, 57T20
Published electronically: April 17, 2007
MathSciNet review: 2309196
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Abstract: A compact, connected, simple Lie group $ G$ localized at an odd prime $ p$ is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of $ G$ is low. This holds for $ SU(n)$, for example, if $ n\leq (p-1)(p-3)$. The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to $ \Omega G$. This is applied to prove useful information about the torsion in the homotopy groups of $ G$, including an upper bound on its exponent.


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Additional Information

Stephen D. Theriault
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
Email: s.theriault@maths.abdn.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04304-8
PII: S 0002-9947(07)04304-8
Keywords: Lie group, exponent, Whitehead product, $H$-space
Received by editor(s): October 18, 2005
Published electronically: April 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society