A toric manifold is a compact non-singular toric variety. A torus
manifold is an oriented, closed, smooth manifold of dimension
with an effective action
of a compact torus
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
in the
family of torus manifolds with codimension one extended actions, and we give a topological
classification of .
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.