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Topological classification of torus manifolds which have codimension one extended actions

Suyoung Choi and Shintarô Kuroki

Algebraic & Geometric Topology 11 (2011) 2655–2679

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.

sphere bundle, complex projective bundle, torus manifold, nonsingular toric variety, quasitoric manifold, cohomological rigidity problem, toric topology
Mathematical Subject Classification 2010
Primary: 55R25
Secondary: 57S25
Received: 5 November 2010
Revised: 6 August 2011
Accepted: 10 August 2011
Published: 22 September 2011
Suyoung Choi
Department of Mathematics
Ajou University
San 5
Suwon 443-749
Republic of Korea
Shintarô Kuroki
Department of Mathematical Sciences
335 Gwahangno
Daejeon 305-701
Republic of Korea