  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 PL $(d-1)$-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.
