Topological classification of torus manifolds which have codimension one extended actions

TL;DR

This paper classifies a specific class of torus manifolds with codimension one extended actions, showing their topological types are determined by cohomology rings and characteristic classes, and addresses open problems in toric topology.

Contribution

It provides a topological classification of a certain class of torus manifolds and explores the extent to which cohomology rings determine their topological types.

Findings

Topological types are determined by cohomology rings and characteristic classes.

Cohomology rings do not always determine the topological type of torus manifolds.

A subclass of torus manifolds is classified by their cohomology rings, despite not being toric manifolds.

Abstract

A toric manifold is a compact non-singular toric variety equipped with a natural half-dimensional compact torus action. A torus manifold is an oriented, closed, smooth manifold of dimension $2n$ with an effective action of a compact torus $T^{n}$ 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 $\mM$ in the family of torus manifolds with codimension one extended actions, and we give a topological classification of $\mM$. 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 even if its orbit spaces are highly structured. 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.