$\mathbb{T}\sp 2$–cobordism of quasitoric $4$–manifolds

Abstract

We show the ${\mathbb{T}}^2$–cobordism group of the category of 4–dimensional quasitoric manifolds is generated by the ${\mathbb{T}}^2$–cobordism classes of ${ℂ\mathbb{P}}^2$. We construct nice oriented ${\mathbb{T}}^2$ manifolds with boundary whose boundaries are the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.