Abstract: It is well known that the independence complex of any matroid without coloops is homotopy equivalent to a wedge of $k > 0$ equidimensional spheres. Similar results are also known to hold for order complexes of geometric lattices and (reduced) broken circuit complexes. We prove that if the dimension and the number of spheres is fixed, then only finitely many such complexes exist. This counterintuitive property leads to new structural questions such as upper and lower bound theorems/conjectures for such complexes based on the two parameters mentioned and many enumeration properties of the finite classes of complexes. This is joint work with $F$. Castillo.