Abstract:

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) $ \Delta$. These bounds imply that many of the coefficients of the $ cd$-index of such $ \Delta$ are positive and that

$\displaystyle (-1)^l\tilde{\chi}($Skel$\displaystyle \,_l(\Delta)) > 1 +\left\{ \begin{array}{ll} {2k-1 \choose k}\be... ...{\min\{k-1,l\}}{2k-2 \choose j}\beta_j & 0\leq l\leq 2k-4, \end{array}\right. $

where $ \tilde{\chi}$ denotes the reduced Euler characteristic and $ \beta_0, \beta_1, \ldots $ are reduced Betti numbers of $ \Delta$.