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Quasitoric manifolds homeomorphic to homogeneous spaces

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Wiemeler, M. Osaka J. Math. 50 (2013), 153–160 QUASITORIC MANIFOLDS HOMEOMORPHIC TO HOMOGENEOUS SPACES MICHAEL WIEMELER (Received February 4, 2011, revised May 30, 2011) Abstract We present some classification results for quasitoric manifolds M with p1(M) D � P a2i for some ai 2 H 2(M) which admit an action of a compact connected Lie- group G such that dim M=G � 1. In contrast to Kuroki’s work [7, 6] we do not require that the action of G extends the torus action on M . 1. Introduction Quasitoric manifolds are certain 2n-dimensional manifolds on which an n- dimensional torus acts such that the orbit space of this action may be identified with a simple convex polytope. They were first introduced by Davis and Januszkiewicz [2] in 1991. In [7, 6] Kuroki studied quasitoric manifolds M which admit an extension of the torus action to an action of some compact connected Lie-group G such that dim M=G � 1. Here we drop the condition that the G-action extends the torus action in the case where the first Pontrjagin-class of M is equal to the negative of a sum of squares of elements of H 2(M). In this note all cohomology groups are taken with coefficients in Q. We have the following two results. Theorem 1.1. Let M be a quasitoric manifold with p1(M) D � P a2i for some ai 2 H 2(M) which is homeomorphic (or diffeomorphic) to a homogeneous space G=H with G a compact connected Lie-group. Then M is homeomorphic (diffeomorphic) to Q S2. In particular, all Pontrjagin-classes of M vanish. Theorem 1.2. Let M be a quasitoric manifold with p1(M) D � P a2i for some ai 2 H 2(M). Assume that the compact connected Lie-group G acts smoothly and al- most effectively on M such that dim M=G D 1. Then G has a finite covering group of the form Q SU(2) or Q SU(2) � S1. Furthermore M is diffeomorphic to a S2-bundle over a product of two-spheres. 2000 Mathematics Subject Classification. 57S15, 57S25. Part of the research was supported by SNF Grants Nos. 200021-117701 and 200020-126795. 154 M. W

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