  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).
