Polyspherical Complexes

Gábor Hetyei

Department of Mathematics and Statistics, University of North Carolina, Charlotte NC 28223, USA.

ghetyei@uncc.edu

Annals of Combinatorics 9 (4) p.379-409 December, 2005

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.

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