Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

TL;DR

This paper studies the classification of manifolds with 2-torus and torus group actions, providing new descriptions of their equivariant bordism groups and showing they are generated by specific types of manifolds.

Contribution

It offers new descriptions of equivariant bordism groups for 2-torus and unitary toric manifolds, and proves generation by small covers and quasitoric manifolds.

Findings

Determined the dimension of the 2-torus manifold bordism group as a z2-vector space.

Provided a new proof that bordism groups are generated by small covers.

Extended results to unitary toric manifolds, showing generation by quasitoric 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.