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**Torus actions, combinatorial topology, and homological algebra.**
*(English.
Russian original)*
Zbl 1010.52011

Russ. Math. Surv. 55, No. 5, 825-921 (2000); translation from Usp. Mat. Nauk 55, No. 5, 3-106 (2000).

The paper is a survey of new results and open problems connected with fundamental combinatorial concepts, including polytopes, simplicial complexes, cubical complexes and arrangements of subspaces. Many important constructions are described that enable to study combinatorial objects by using commutative and homological algebra. The results of the paper establish solutions of some well-known topological problems.

The authors’ remarkable techniques which they have carefully developed is a strong unifying theme of the paper. The amount of material covered in the paper is very large. The authors’ presentation is attractive and lucid, quite suitable for a valuable reference for further research of the topic.

The authors’ remarkable techniques which they have carefully developed is a strong unifying theme of the paper. The amount of material covered in the paper is very large. The authors’ presentation is attractive and lucid, quite suitable for a valuable reference for further research of the topic.

Reviewer: Akrur Behera (Rourkela)

### MSC:

52B70 | Polyhedral manifolds |

57Q15 | Triangulating manifolds |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

57R19 | Algebraic topology on manifolds and differential topology |