Stellar subdivisions, wedges and Buchstaber numbers

TL;DR

This paper investigates operations on PL spheres, such as stellar subdivision and wedge, demonstrating their preservation of Buchstaber numbers and polytopality, and constructs new polytopal seeds, confirming the tightness of a known inequality.

Contribution

It introduces methods to generate new polytopal seeds via stellar subdivision and wedge operations, maintaining key properties and confirming an existing inequality's tightness.

Findings

Stellar subdivision and wedge preserve Buchstaber numbers.

Constructed new polytopal toric colorable seeds.

Proved the tightness of the toric colorable seed inequality.

Abstract

A seed is a PL sphere that is not obtainable by a wedge operation from any other PL sphere. In this paper, we study two operations on PL spheres, known as the stellar subdivision and the wedge, that preserve the maximality of Buchstaber numbers and polytopality. We construct a new polytopal toric colorable seed from these two operations. As a corollary, we prove that the toric colorable seed inequality established by Choi and Park is tight.