  The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for
the face rings of the following classes of simplicial complexes: matroid
complexes, complexes of dimension one and two, and Gorenstein complexes of
dimension at most four. The lower bound part of this conjecture is also
established for the face rings of all doubly Cohen-Macaulay complexes whose
1-skeleton's connectivity does not exceed the codimension plus one as well as
for all (d-1)-dimensional d-Cohen-Macaulay complexes. The main ingredient of
the proofs is a new interpretation of the minimal shifts in the resolution of
the face ring via the Cohen-Macaulay connectivity of the skeletons of the
complex.
