$\TT^2$-cobordism of Quasitoric 4-Manifolds

TL;DR

This paper investigates the $ ext{T}^2$-cobordism group of 4-dimensional quasitoric manifolds, demonstrating it is generated by classes of $ ext{CP}^2$, using constructions involving boundary manifolds and Hirzebruch surfaces.

Contribution

It establishes the generators of the $ ext{T}^2$-cobordism group for 4D quasitoric manifolds and introduces constructions with boundary manifolds related to Hirzebruch surfaces.

Findings

The $ ext{T}^2$-cobordism group is generated by $ ext{CP}^2$ classes.

Constructed oriented $ ext{T}^2$ manifolds with boundary as Hirzebruch surfaces.

Applied quasitoric manifold theory to cobordism classification.

Abstract

We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.