2001/09/19 by Linus Kramer, Kramer, Linus
Mathematics · #51E12 #51H15 #53C42 #57T15 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.DG #math.GT #msc:51E12 #msc:51H15 #msc:53C42 #msc:57T15
paper · pdf · doi:10.48550/arxiv.math/0109133
To appear in Memoirs Amer. Math. Soc
arxiv created 2001/09/19 · openalex publication_date 2001/09/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.