TY - JOUR
T1 - Finiteness theorems for matroid complexes with prescribed topology
AU - Castillo, Federico
AU - Samper, José Alejandro
N1 - Funding Information:
We would like to thank Richard Stanley for interesting conversations and for pointing out the reference in his book to Theorem 5.3. Thanks to Ed Swartz for reminding us of Example 2.13. An anonymous referee pointed out the connections between geometric lattices and cosimple matroids that inspired Corollary 5.5. We are specially indebted to Isabella Novik for various interesting conversations and helpful suggestions on preliminary versions. We are grateful to the University of Washington and University of Kansas where parts of this project were carried out. The second named author also thanks the University of Miami where he was employed when most of the project was carried out. This project was completed while both authors were members of the Max-Planck Institute for Mathematics in the Sciences.
PY - 2021/2
Y1 - 2021/2
N2 - There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=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 hd=k for any given k? The answer is no if we consider general complexes, but we focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. Surprisingly, the answer is yes in all three cases.
AB - There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=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 hd=k for any given k? The answer is no if we consider general complexes, but we focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. Surprisingly, the answer is yes in all three cases.
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U2 - 10.1016/j.ejc.2020.103239
DO - 10.1016/j.ejc.2020.103239
M3 - Article
AN - SCOPUS:85091557508
VL - 92
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
SN - 0195-6698
M1 - 103239
ER -