On face numbers of manifolds with symmetry

Abstract

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