Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Abstract

The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group ${\mathbb{Z}}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group ${\mathcal{Z}}_n({\mathbb{Z}}_2^n)$ of 2-torus manifolds, and determine the dimention of ${\mathcal{Z}}_n({\mathbb{Z}}_2^n)$ as a ${\mathbb{Z}}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups ${\mathcal{Z}}_n({\mathbb{Z}}_2^n)$ are generated by small covers. We will give a new proof to this result. These results can be generalized to the equivariant bordism of unitary toric manifolds, that is, we will give a new description of the group ${\mathcal{Z}}_n^U(T^n)$ of unitary torus manifolds, and prove that ${\mathcal{Z}}_n^U(T^n)$ can be generated by quasitoric manifolds with omniorientations.