Finiteness theorems for matroid complexes with prescribed topology

TL;DR

This paper investigates finiteness properties of matroid-related simplicial complexes with prescribed topological invariants, establishing finiteness in certain cases and conjecturing it in others.

Contribution

It proves finiteness of complexes with fixed top-dimensional $h$-vector entries for independence and order complexes of geometric lattices, and conjectures the same for broken circuit complexes.

Findings

Finiteness holds for independence complexes with fixed $h_d$.

Finiteness holds for order complexes of geometric lattices with fixed $h_d$.

Conjecture: Finiteness also holds for broken circuit complexes with fixed $h_d$.

Abstract

It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$ for any natural number $k$. In this paper we study the question at the other end of the $h$-vector: Are there only finitely many $(d-1)$-dimensional simplicial complexes with $h_d=k$ for any given $k$? The answer is no if we consider general complexes, but when focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. We prove the answer is yes in cases (i) and (iii) and conjecture it is also true in case (ii).