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Séminaire Lotharingien de Combinatoire, 82B.99 (2019), 12 pp.

# Federico Castillo and José Alejandro Samper

# Finiteness theorems for matroid complexes with prescribed topology

**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: given *d* and *k* there are only finitely many *-1-dimensional independence complexes, broken circuit complexes, and order complexes of geometric lattices (without coloops) with **h*_{d}=*k*. This suggests new upper/lower bound programs for these types of simplicial complexes.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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