Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

2022/02/23 by Chen, Bo, Lü, Zhi, Tan, Qiangbo
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.11290

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 ℤ₂ⁿ and torus group Tⁿ, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group Z_n(ℤ₂ⁿ) of 2-torus manifolds, and determine the dimention of Z_n(ℤ₂ⁿ) as a ℤ₂-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups Z_n(ℤ₂ⁿ) 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 Z_n^U(Tⁿ) of unitary torus manifolds, and prove that Z_n^U(Tⁿ) can be generated by quasitoric manifolds with omniorientations.

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