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 $T^n$ 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 $\mathfrak{M}$ in the family of torus manifolds with codimension one extended actions, and we give a topological classification of $\mathfrak{M}$. 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.