QUASITORIC MANIFOLDS OVER A PRODUCT OF SIMPLICES

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A quasitoric manifold (resp. a small cover) is a 2n-dimensional (resp. an n-dimensional) smooth closed manifold with an effective locally standard action of (S(1))(n) (resp. (Z(2))(n)) whose orbit space is combinatorially an n-dimensional simple convex polytope P. In this paper we study them when P is a product of simplices. A generalized Bott tower over F, where F = C or R, is a sequence of projective bundles of the Whitney sum of F-line bundles starting with a point. Each stage of the tower over F, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when F = C) and small covers (when F = R) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold M over a product of simplices is homeomorphic to a generalized Bolt manifold if M has the same cohomology ring as a product of complex projective spaces with Q coefficients.
Publisher
OSAKA JOURNAL OF MATHEMATICS
Issue Date
2010-03
Language
English
Article Type
Article
Keywords

BOTT TOWERS; CLASSIFICATION; POLYTOPES

Citation

OSAKA JOURNAL OF MATHEMATICS, v.47, pp.109 - 129

ISSN
0030-6126
URI
http://hdl.handle.net/10203/98486
Appears in Collection
MA-Journal Papers(저널논문)
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