Combinatorics of Simplicial Cell Complexes and Torus Actions

Simplicial cell complexes are special cellular decompositions also known as virtual or ideal triangulations; in combinatorics, appropriate analogues are given by simplicial partially ordered sets. In this paper, combinatorial and topological properties of simplicial cell complexes are studied. Namely, the properties of $f$-vectors and face rings of simplicial cell complexes are analyzed and described, and a number of well-known results on the combinatorics of simplicial partitions are generalized. In particular, we give an explicit expression for the operator on $f$- and $h$-vectors that is defined by a barycentric subdivision, derive analogues of the Dehn–Sommerville relations for simplicial cellular decompositions of spheres and manifolds, and obtain a generalization of the well-known Stanley criterion for the existence of regular sequences in the face rings of simplicial cell complexes. As an application, a class of manifolds with a torus action is constructed, and generalizations of some of our previous results on the moment–angle complexes corresponding to triangulations are proved.