Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

Details Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

2001-09-19

arxiv.org/abs/math/0109133v1

Mathematics

Geometric Topology

To appear in Memoirs Amer. Math. Soc

Scientific paper

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparametric hypersurfaces which admit a transitive isometry group on at least one focal manifold. This generalizes the classification of homogeneous isoparametric hypersurfaces by Hsiang and Lawson and gives a new, independent proof of their result. Secondly, we classify certain compact highly connected Tits buildings which admit a vertex transitive automorphism group. Such buildings arise as compactifications of symmetric spaces as well as from isoparametric submanifolds. This extends the recent classification of all compact connected Tits buildings which admit a chamber transitive automorphism group by Grundhofer, Knarr, and the author.

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