On cobordism of generalized (real) Bott manifolds

TL;DR

This paper investigates the cobordism properties of generalized real Bott manifolds, showing they are boundaries of some manifolds but not necessarily in an equivariant sense, and provides examples of null-cobordant manifolds that are not orientedly null-cobordant.

Contribution

It establishes that certain generalized Bott manifolds are boundaries and constructs examples of manifolds with specific cobordism properties.

Findings

Generalized real Bott manifolds over product of simplices are boundaries.

These manifolds do not necessarily bound equivariantly.

Existence of null-cobordant but not orientedly null-cobordant quasitoric manifolds.

Abstract

We show that all generalized (real) Bott manifolds which are (small covers) quasitoric manifolds over a product of simplices $\Delta^{n_1}\times\cdots\times\Delta^{n_r}\times\Delta^{1}$ are always boundaries of some manifolds. But these manifolds with the natural $(\mathbb{Z}_2)^n$ action do not necessarily bound equvariantly. In addition, we can construct some examples of null-cobordant but not orientedly null-cobordant manifolds among quasitoric manifolds.