Balanced vertex decomposable simplicial complexes and their h-vectors

TL;DR

This paper introduces a method to construct balanced, vertex decomposable simplicial complexes from any given complex, linking their h-vectors to the original complex's f-vector, and extends existing theories on f-vectors and h-vectors.

Contribution

It presents a new construction that generalizes whiskering, relates f-vectors to Betti numbers, and offers new insights into the classification of f-vectors and h-vectors of simplicial complexes.

Findings

Constructed balanced, vertex decomposable complexes from any simplicial complex.

Established that the h-vector of the new complex equals the original's f-vector.

Extended classification theorems and proved cases of conjectures on h-vectors.

Abstract

Given any finite simplicial complex \Delta, we show how to construct a new simplicial complex \Delta_{\chi} that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex \Delta_{\chi} is precisely the f-vector, denoted f(\Delta), of the original complex \Delta. We deduce this result by relating f(\Delta) with the graded Betti numbers of the Alexander dual of \Delta_{\chi}. Our construction generalizes the "whiskering" construction of Villarreal, and Cook and Nagel. As a corollary of our work, we add a new equivalent statement to a theorem of Bj\"orner, Frankl, and Stanley that classifies the f-vectors of simplicial complexes. We also prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the h-vectors of flag complexes.