Lower Bounds for thecd-index of Odd-dimensional Simplicial Manifolds

We obtain lower bounds on the coefficients of the cd -index of any (2k − 1)-dimensional simplicial manifold (or, more generally, any Eulerian Buchsbaum complex)Δ. These bounds imply that many of the coefficients of the cd -index of such Δ are positive and that[formula]χ denotes the reduced Euler characteristic andβ0 , β1, . . . are reduced Betti numbers of Δ.