Torus actions, combinatorial topology, and homological algebra

Бухштабер, Виктор Матвеевич; Panov, Taras

doi:10.1070/rm2000v055n05abeh000320

Cited by 83 publications

(133 citation statements)

References 68 publications

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“…. , x n ] be the polynomial ring with variables in degree 2, and let I be the ideal generated by all monomials corresponding to non-faces of K. As shown by Buchstaber and Panov [9], the cohomology ring of the moment-angle complex, H * (Z K , k), is isomorphic to Tor S (S/I, k), the Toralgebra of the Stanley-Reisner ring S/I.…”

Section: 2mentioning

confidence: 99%

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Moment-angle Complexes, Monomial Ideals and Massey Products

Denham¹,

Suciu²

2007

Pure and Applied Mathematics Quarterly

123

127

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Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z K ; if K is a triangulation of a sphere, Z K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.

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Section: 2mentioning

confidence: 99%

“…A construction due to Davis and Januszkiewicz [12] and studied in detail by Buchstaber and Panov [9] associates to every simplicial complex K on n vertices a finite cellular complex Z K , endowed with a natural action by the n-torus, and whose orbit space is the cone over K.…”

Section: Moment-angle Complexesmentioning

confidence: 99%

Moment-angle Complexes, Monomial Ideals and Massey Products

Denham¹,

Suciu²

2007

Pure and Applied Mathematics Quarterly

123

127

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“…Buchstaber and Panov [BP1] showed that the moment-angle complex Z K is a T m -equivariant retract of U (K). Consequently, there is a homotopy equivalence…”

Section: Vm] (Z[k] Z)mentioning

confidence: 99%

“…The story starts with Davis and Januszkiewicz's work [DJ] which uses simple polytopes to construct new families of manifolds with torus actions. However, it is more convenient for us to begin with Buchstaber and Panov's generalisation [BP1] of their construction to simplicial complexes. Let K be a simplicial complex on the vertex set [m].…”

Section: Part 1 a Survey Of Homotopy Theory In Toric Topologymentioning

confidence: 99%

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Homotopy theory in toric topology

Grbić¹,

Theriault²

2016

Russ. Math. Surv.

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Abstract. In toric topology, to each simplicial complex K on m vertices one associates two key spaces, the Davis-Januskiewicz space DJ K and the moment-angle complex Z K , which are related by a homotopy fibrationA great deal of work has been done to study properties of DJ K and Z K , their generalisations to polyhedral products, and applications to algebra, combinatorics and geometry.In the first part of this paper we survey some of the main results in the study of the homotopy theory of these spaces. In the second part we break new ground by initiating a study of the map w.We show that, for a certain family of simplicial complexes K, the map w is a sum of higher and iterated Whitehead products.

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Graded rings and equivariant sheaves on toric varieties

Perling

2003

Mathematische Nachrichten

172

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In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of graded ring theory and this way we considerably clarify earlier constructions of Kaneyama and Klyachko. We also connect the formalism to the theory of fine-graded modules over Cox' homogeneous coordinate ring of a toric variety. As an application we construct minimal resolutions of equivariant vector bundles of rank two on toric surfaces.

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Torus actions, combinatorial topology, and homological algebra

Cited by 83 publications

References 68 publications

Moment-angle Complexes, Monomial Ideals and Massey Products

Moment-angle Complexes, Monomial Ideals and Massey Products

Homotopy theory in toric topology

Graded rings and equivariant sheaves on toric varieties

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