The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f -vector; and moreover, among all such 2-Cohen-Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also verified that the l-skeleton of a flag (d−1)-dimensional 2-CM complex is 2(d−l)-CM, while the l-skeleton of a flag piecewise linear (d−1)-sphere is 2(d−l)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.

Upper bounds on the Betti numbers over an arbitrary field of Vietoris-Rips complexes are established, and examples of such complexes with large Betti numbers are given.

a b s t r a c tWe prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs.