European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-09-26 , DOI: 10.1016/j.ejc.2020.103239 Federico Castillo , José Alejandro Samper
There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of -vectors, there are finitely many simplicial complexes of bounded dimension with for any natural number . In this paper we study the question at the other end of the -vector: Are there only finitely many -dimensional simplicial complexes with for any given ? 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.
中文翻译:
具有拟定拓扑的拟阵复合体的有限性定理
具有给定数量的顶点的有限多个单纯形复形(直至同构)。将此事实翻译为-向量,有有限个维的有限单纯形复 对于任何自然数 。在本文中,我们将研究另一端的问题-vector:仅有限地存在 维简单复形与 对于任何给定 ?如果不考虑一般的复数,答案是否定的,但我们将重点放在来自类阵的三种情况:(i)独立复数,(ii)断路复数和(iii)几何格的阶数复数。令人惊讶的是,在所有三种情况下答案都是肯定的。