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Topological
classification of torus manifolds which have codimension one
extended actions
Suyoung Choi and Shintarô Kuroki |
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Algebraic & Geometric Topology 11 (2011) 2655–2679
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Abstract |
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A toric manifold is a compact non-singular toric variety. A torus manifold is an oriented, closed, smooth manifold of dimension with an effective action of a compact torus having a non-empty fixed point set. Hence, a torus manifold can be thought of as a generalization of a toric manifold. In the present paper, we focus on a certain class in the family of torus manifolds with codimension one extended actions, and we give a topological classification of . As a result, their topological types are completely determined by their cohomology rings and real characteristic classes. The problem whether the cohomology ring determines the topological type of a toric manifold or not is one of the most interesting open problems in toric topology. One can also ask this problem for the class of torus manifolds. Our results provide a negative answer to this problem for torus manifolds. However, we find a sub-class of torus manifolds with codimension one extended actions which is not in the class of toric manifolds but which is classified by their cohomology rings. |
Keywords
sphere bundle, complex projective bundle, torus manifold,
nonsingular toric variety, quasitoric manifold,
cohomological rigidity problem, toric topology
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Mathematical Subject Classification 2010
Primary: 55R25
Secondary: 57S25
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References |
Publication
Received: 5 November 2010
Revised: 6 August 2011
Accepted: 10 August 2011
Published: 22 September 2011
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Authors |
| Suyoung Choi | |
| Department of Mathematics Ajou University San 5 Woncheon-dong Yeongtong-gu Suwon 443-749 Republic of Korea |
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| Shintarô Kuroki | |
| Department of Mathematical
Sciences KAIST 335 Gwahangno Yuseong-gu Daejeon 305-701 Republic of Korea |
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