Necessary conditions on the face numbers of Cohen-Macaulay
simplicial complexes admitting a proper action of the
cyclic group

of a prime order are given. This result is extended
further to necessary conditions on the face numbers and
the Betti numbers of Buchsbaum simplicial complexes with a
proper

-action.
Adin's upper bounds on the face numbers of Cohen-Macaulay
complexes with symmetry are shown to hold for
all

-dimensional Buchsbaum complexes with symmetry
on

vertices.
A generalization of Kühnel's conjecture on the Euler
characteristic of

-dimensional manifolds and Sparla's
analog of this conjecture for centrally-symmetric

-manifolds are verified for all

-manifolds on

vertices. Connections
to the Upper Bound Theorem are discussed and its new version
for centrally symmetric manifolds is established.