Maximal Homotopy Lie Subgroups of Maximal Rank

Frohliger, John A.

Canadian journal of mathematics, Tome 40 (1988) no. 4, pp. 769-787

Résumé

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, φ 1 and φ 2 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.

Détail

DOI : 10.4153/CJM-1988-034-6

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