On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen-Macaulay simplicial complexes admitting a proper action of the cyclic group Z/pZ 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 Z/pZ-action. Adin's upper bounds on the face numbers of Cohen-Macaulay complexes with symmetry are shown to hold for all (d−1)-dimensional Buchsbaum complexes with symmetry on n⩾3d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2k-manifolds are verified for all 2k-manifolds on n⩾6k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.