On the cd-index and γ-vector of S*-shellable CW-spheres

We show that the γ-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable CW-spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing c = 1 and considering the resulted polynomial in d, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.