  For positive integers k,n, we investigate the simplicial complex NM_k(n) of
all graphs G on vertex set [n] such that every matching in G has size less than
k. This complex (along with other associated cell complexes) is found to be
homotopy equivalent to a wedge of spheres. The number and dimension of the
spheres in the wedge are determined, and (partially conjectural) links to other
combinatorially defined complexes are described. In addition we study for
positive integers r,s and k the simplicial complex BNM_k(n) of all bipartite
graphs G on bipartition [r] \cup [\bar{s}] such that there is no matching of
size k in G, and obtain results similar to those obtained for NM_k(n).
