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Maximal Homotopy Lie Subgroups of Maximal Rank

Published online by Cambridge University Press:  20 November 2018

John A. Frohliger
John A. Frohliger*
Affiliation:
St. Norbert College, De Pere, Wisconsin
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Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows:

(1)

Here π, ϕ, ϕ1 and ϕ2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes.

Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 4 , 01 August 1988 , pp. 769 - 787
Copyright
Copyright © Canadian Mathematical Society 1988

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