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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 5(335), Pages 3–106 (Mi umn320)  

This article is cited in 45 scientific papers (total in 46 papers)

Torus actions, combinatorial topology, and homological algebra

V. M. Buchstaber, T. E. Panov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This 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. Attention is concentrated on simplicial and cubical subdivisions of manifolds, and especially on spheres. Important constructions are described that enable one to study these combinatorial objects by using commutative and homological algebra. The proposed approach to combinatorial problems is based on the theory of moment-angle complexes recently developed by the authors. The crucial construction assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\mathscr Z_K$ with special bigraded cellular decomposition. In the framework of this theory, well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. It is shown that diverse invariants of simplicial complexes and related combinatorial-geometric objects can be expressed in terms of bigraded cohomology rings of the corresponding moment-angle complexes. Finally, it is shown that the new relationships between combinatorics, geometry, and topology lead to solutions of some well-known topological problems.


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English version:
Russian Mathematical Surveys, 2000, 55:5, 825–921

Bibliographic databases:

UDC: 515.164+515.142+515.145+514.172.45
MSC: 52B70, 57Q15, 57R19, 14M25, 52B05, 13F55, 05B35
Received: 10.08.2000

Citation: V. M. Buchstaber, T. E. Panov, “Torus actions, combinatorial topology, and homological algebra”, Uspekhi Mat. Nauk, 55:5(335) (2000), 3–106; Russian Math. Surveys, 55:5 (2000), 825–921

Citation in format AMSBIB
\by V.~M.~Buchstaber, T.~E.~Panov
\paper Torus actions, combinatorial topology, and homological algebra
\jour Uspekhi Mat. Nauk
\yr 2000
\vol 55
\issue 5(335)
\pages 3--106
\jour Russian Math. Surveys
\yr 2000
\vol 55
\issue 5
\pages 825--921

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