Almost Transitive Actions on Spaces with the Rational Homotopy of Sphere Products

Oliver Bletz-Siebert

Abstract: We determine the structure of transitive actions of compact Lie groups on spaces which have the dimension and the (rational) homotopy groups of a product $\SS^1\times\SS^m$ of spheres. \endgraf These homogeneous spaces arise in several geometric contexts and may be considered as $\SS^1$-bundles over certain spaces, e.g. over lens spaces and over certain quotients of Stiefel manifolds. \endgraf Furthermore, we show that if a non-compact simply connected Lie group acts transitively on such a space, then the orbits of the maximal compact subgroups are all simply connected rational cohomology spheres of codimension one and hence classified. We obtain this by giving a short proof of the existence and structure of the natural bundle of Gorbatsevich under these much less general assumptions. In this special case the proof gets considerably shorter by the use of the homotopy properties of the spaces in question and a theorem of Mostert on the structure of orbit spaces of compact Lie groups on manifolds.

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