Algebraic & Geometric Topology

Topological classification of torus manifolds which have codimension one extended actions

Suyoung Choi and Shintarô Kuroki

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Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2655-2679.

Dates
Received: 5 November 2010
Revised: 6 August 2011
Accepted: 10 August 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715302

Digital Object Identifier
doi:10.2140/agt.2011.11.2655

Mathematical Reviews number (MathSciNet)
MR2846908

Zentralblatt MATH identifier
1231.57031

Subjects
Primary: 55R25: Sphere bundles and vector bundles
Secondary: 57S25: Groups acting on specific manifolds

Keywords
sphere bundle complex projective bundle torus manifold nonsingular toric variety quasitoric manifold cohomological rigidity problem toric topology

Citation

Choi, Suyoung; Kuroki, Shintarô. Topological classification of torus manifolds which have codimension one extended actions. Algebr. Geom. Topol. 11 (2011), no. 5, 2655--2679. doi:10.2140/agt.2011.11.2655. https://projecteuclid.org/euclid.agt/1513715302


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