Topological classification of torus manifolds which have codimension one extended actions

Choi, Suyoung¹ ; Kuroki, Shintarô²

¹Department of Mathematics, Ajou University, San 5, Woncheon-dong, Yeongtong-gu, Suwon 443-749, Republic of Korea
²Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2655-2679

Résumé

A toric manifold is a compact non-singular toric variety. A torus manifold is an oriented, closed, smooth manifold of dimension 2n with an effective action of a compact torus Tn 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 M in the family of torus manifolds with codimension one extended actions, and we give a topological classification of M. 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.

Détail

DOI : 10.2140/agt.2011.11.2655

Classification : 55R25, 57S25
Keywords: sphere bundle, complex projective bundle, torus manifold, nonsingular toric variety, quasitoric manifold, cohomological rigidity problem, toric topology

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