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, L\"u 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.