Polyspherical Complexes

Abstract.

We construct spherical CW-complexes whose face structure may be conveniently described using a system of polyspherical coordinates introduced by Vilenkin, Kuznetsov and Smorodinskii. We show that these complexes may be constructed by repeated use of CW-suspension, free join, and edge subdivision. We prove that all CW-spheres constructed in this way have non-negative cd-index and thus verify Stanley’s famous conjecture. Among the particular examples we find a new class of partially ordered sets whose order complexes encode the derivative polynomials for secant of even degree. The geometric constructions presented in this paper generalize CW-complexes introduced whose flag numbers are suitable to encode systems of orthogonal polynomials.