Non-negative cd-coefficients of Gorenstein∗ posets

We give a convolution formula for cd-index coefficients. The convolution formula, together with the proof by Davis and Okun of the Charney–Davis Conjecture in dimension 3, imply that certain cd-coefficients are non-negative for all Gorenstein∗ posets. Additional coefficients are shown to be non-negative by interpreting them in terms of the top homology of certain Cohen–Macaulay complexes. In particular we verify, up to rank 6, Stanleyʹs conjecture that the coefficients in the cd-index of a Gorenstein∗ ranked poset are non-negative.