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**Combinatorics of simplicial cell complexes and torus actions.**
*(English.
Russian original)*
Zbl 1098.52003

Geometric topology and set theory. Collected papers. Dedicated to the 100th birthday of Professor Lyudmila Vsevolodovna Keldysh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 247, 33-49 (2004); translation from Tr. Mat. Inst. Steklova 247, 41-58 (2004).

Summary: Simplicial cell complexes are special cellular decompositions also known as virtual or ideal triangulations; incombinatorics, 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.

For the entire collection see [Zbl 1087.55002].

For the entire collection see [Zbl 1087.55002].

### MSC:

52B70 | Polyhedral manifolds |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |

57S25 | Groups acting on specific manifolds |