## Algebraic & Geometric Topology

### Topological classification of torus manifolds which have codimension one extended actions

#### 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
Revised: 6 August 2011
Accepted: 10 August 2011
First available in Project Euclid: 19 December 2017

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

#### 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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