Bounds for the Betti numbers of successive stellar subdivisions of a simplex

TL;DR

This paper establishes bounds on the Betti numbers of Stanley-Reisner rings after stellar subdivisions of simplicial complexes, using unprojection theory, and demonstrates the bounds are sharp through examples.

Contribution

It introduces a new bound for Betti numbers of stellar subdivisions, applicable to iterated subdivisions of a simplex boundary, based solely on the number of subdivisions.

Findings

Derived a bound for Betti numbers of stellar subdivisions

Applied unprojection theory to simplicial complexes

Constructed examples confirming the bound's sharpness

Abstract

We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.