Torus actions, combinatorial topology, and homological algebra
By V. Buchstaber, T. Panov. 2000
107 citations recorded by the source corpus.
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 with vertices a -space 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/02148629e79c2ef4c10a9dd8273f14e31f511497
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In Algebra and Number Theory
In Computational Theory and Mathematics
In Discrete Mathematics and Combinatorics
- S. Gitler
- A. Bahri
- Michael Joswig
- Martin Bendersky
- Frederick R. Cohen
- T. Panov
- V. Buchstaber
- S. Theriault
Citations in the corpus
- Operations on polyhedral products and a new topological construction of infinite families of toric manifolds (2010, background): “Polyhedral products generalize the spaces called moment-angle complexes which were developed first by Buchstaber and Panov in [5] and correspond to the case (Xi, Ai) = (D (2), S(1)), i = 1, 2, .”
- Projectivities in simplicial complexes and colorings of simple polytopes (2001, background): “... of the facets F =( F1 ,...,F n) and let T be the n-dimensional complex algebraic torus (C \{ 0}) F . On the product P × T define an equivalence relation ∼, where (p, s) ∼ (q, t) if and only if p = q and the i-th component of the quotient st −1 in the group T is trivial for all facets Fi not cont”
- The homotopy types of moment-angle complexes for flag complexes (2012, unknown): “Other important cases of polyhedral products include DJ(K) = (CP, ∗) , which is referred to as the Stanley–Reisner space [6] or the Davis–Januszkiewicz space [22], and the complement of the complex coordinate subspace arrangement corresponding to K U(K) = (C,C) = C \ ⋃”