Moment-angle complexes and polyhedral products for convex polytopes

Abstract

Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K_P which carries complete information about the combinatorial type of P. In the case when P is simple, K_P is the same as dP*, where P* is a polar dual polytope. Using the canonical embedding of a polytope P into nonnegative orthant, we introduce a moment-angle space Z_P for a polytope P. It is known, that in the case when P is simple the space Z_P is homeomorphic to the polyhedral product (D^2,S^1)^{K_P}. When P is not simple, we prove that the space Z_P is homotopically equivalent to the space (D^2,S^1)^{K_P}. This allows to introduce bigraded Betti numbers for any convex polytope. A Stanley-Reisner ring of a polytope P can be defined as a Stanley-Reisner ring of a simplicial complex K_P. All these considerations lead to a natural question: which simplicial complexes arise as K_P for some polytope P? We have proceeded in this direction by introducing a notion of a polytopic simplicial complex. It has the following property: link of each simplex in a polytopic complex is either contractible, or retractible to a subcomplex, homeomorphic to a sphere. The complex K_P is a polytopic simplicial complex for any polytope P. Links of so called face simplices in a polytopic complex are polytopic complexes as well. This fact is sufficient enough to connect face polynomial of a simplicial complex K_P to the face polynomial of a polytope P, giving a series of inequalities on certain combinatorial characteristics of P. Two of these inequalities are equalities for each P and represent Euler-Poincare formula and one of Bayer-Billera relations for flag f-numbers. In the case when P is simple all inequalities turn out to be classical Dehn-Sommerville relations.