The classification of compact Lie groups acting transitively on spheres due to
Montgomery, Samelson and Borel was one of the main achievements in the early
theory of compact transitive Lie transformation groups. Later, Hsiang and Su
classified compact transitive groups on (sufficiently highly connected) Stiefel mani-
folds. Their results were extended by Scheerer, Schneider and other authors. Given
a compact 1-connected manifold X, it is in general difficult to classify all compact
Lie groups which act transitively on X. The problem becomes more complicated if
only certain homotopy invariants of X such as the cohomology ring are known. If
the Euler characteristic of X is positive, then results of Borel, De Siebenthal and
Wang can be used. However, if the Euler characteristic is 0, there is no general
classification method.
* •¥ H
In this book we classify all 1-connected homogeneous spaces G/H of compact Lie
groups which have the same rational cohomology as a product of spheres
Sni x § U 2 ,
with 3 n\ ri2 and n2 odd. Note that this implies that the Euler characteristic
of G/H is 0. Examples of such spaces are besides products of spheres Stiefel
manifolds of orthonormal 2-frames in real, complex, or quaternionic vector spaces;
another class of examples are certain homogeneous sphere bundles. The following
theorem is a direct consequence of this rational classification.
Theorem Let X = G/H be a 1-connected compact homogeneous space of a compact
connected Lie group G. Assume that G acts effectively and contains no normal
transitive subgroup, and that X has the same integral cohomology as a product of
with 3 n\ n2 and n2 odd. There are the following possibilities for G/H and
the numbers (ni,n2).
(1) Stiefel manifolds
SO(2n)/SO(2n-2) = V2{R2n) ( 2 n - 2 , 2 n - l )
SU(n)/SU(n - 2) = V2 (Cn) (2n - 3, 2n - 1)
Sp(n)/Sp(n - 2) =
(4rc - 5, An - 1).
(2) Certain homogeneous sphere bundles
Sp(ra) x Sp(2)/Sp(n - 1) Sp(l) (7,4n - 1)
Sp(n) x SU(3)/Sp(/i - 1) Sp(l) (5,4n - 1)
Sp(n) x Sp(2)/Sp(™ - 1) Sp(l) Sp(l) (4,4n - 1).
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