Composition of simplicial complexes, polytopes and multigraded Betti numbers

Abstract

For a simplicial complex K on m vertices and simplicial complexes K1,...,Km a composed simplicial complex K(K1,...,Km) is introduced. This construction generalizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P(P1,...,Pm) defined by G. Agnarsson. The composition defines a structure of an operad on a set of finite simplicial complexes. We prove the following: (1) a composed complex K(K1,...,Km) is a simplicial sphere iff K is a simplicial sphere and Ki are the boundaries of simplices; (2) a class of spherical nerve-complexes is closed under the operation of composition (3) finally, we express multigraded Betti numbers of K(K1,...,Km) in terms of multigraded Betti numbers of K, K1,...,Km using a composition of generating functions.

22 pages, 3 figures

References in corpus (1)

• The flag polynomial of the Minkowski sum of simplices

Cited by in corpus (5)

• Families of minimally non-Golod complexes and their polyhedral products
• Simplicial complexes Alexander dual to boundaries of polytopes
• The Cohomology Algebra of Polyhedral Product Objects
• The Complement of Polyhedral Product Spaces and the Dual Simplicial Complexes
• Complement Spaces, Dual Complexes and Polyhedral Product Spaces