Decompositions of polyhedral products for shifted complexes

Iriye, Kouyemon; Kishimoto, Daisuke

doi:10.1016/j.aim.2013.05.003

Cited by 51 publications

(54 citation statements)

References 9 publications

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“…So it is particularly important to describe the homotopy types of polyhedral products. In studying the homotopy types of polyhedral products, the decomposition of suspensions of polyhedral products due to Bahri, Bendersky, Cohen, and Gitler [BBCG] is fundamental as in [GT,IK1,IK2], and we here recover this decomposition from Theorem 2.3. For I ⊂ [m], put K I := {σ ⊂ I | σ ∈ K} and (X I , A I ) := {(X i , A i )} i∈I .…”

Section: Applications Of Theorem 23mentioning

confidence: 98%

Decompositions of suspensions of spaces involving polyhedral products

Iriye

Kishimoto

2016

Algebr. Geom. Topol.

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Abstract. Two homotopy decompositions of supensions of spaces involving polyhedral products are given. The first decomposition is motivated by the decomposition of suspensions of polyhedral products in [BBCG], and is a generalization of the retractile argument of James [J]. The second decomposition is on the union of an arrangement of subspaces called diagonal subspaces, and generalizes the result in [La].

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Section: Applications Of Theorem 23mentioning

confidence: 98%

Decompositions of suspensions of spaces involving polyhedral products

Iriye

Kishimoto

2016

Algebr. Geom. Topol.

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“…When K is shifted, each |K I | is homotopy equivalent to a wedge of spheres, so |K I | * X I ∈ W m . The explicit version of Theorem 5.3 proved in [GT3,IK1] says that the decomposition for Σ(CX, X) K desuspends.…”

Section: K\{m}mentioning

confidence: 99%

“…The definition of a shifted complex implies that if K is shifted then both link K (m) and K\{m} are shifted. In [GT3,IK1] the main work is in showing that the map (CX, X) link K (m) −→ (CX, X) K\{m} is null homotopic, implying the following.…”

Section: K\{m}mentioning

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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“…A particular case of the following construction, which is the most important one for us here, appeared firstly in the work of Buchstaber and Panov [6] and then was studied intensively and generalized in the works of Bahri, Bendersky, Cohen, Gitler [1], Grbić and Theriault [12], Iriye and Kishimoto [16], and others.…”

Section: Introductionmentioning

confidence: 99%

Topology of moment-angle manifolds arising from flag nestohedra

Limonchenko

2017

Chin. Ann. Math. Ser. B

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Abstract. We construct a family of manifolds, one for each n ≥ 2, having a nontrivial Massey n-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus T m -action called moment-angle manifolds Z P , whose orbit spaces are simple n-dimensional polytopes P obtained from a n-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes P are flag nestohedra but not graph-associahedra. We compute some bigraded Betti numbers β −i,2(i+1) (Q) for an associahedron Q in terms of its graph structure and relate it to the structure of the loop homology (Pontryagin algebra) H * (ΩZ Q ). We also study triple Massey products in H * (Z Q ) for a graph-associahedron Q.

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Decompositions of polyhedral products for shifted complexes

Cited by 51 publications

References 9 publications

Decompositions of suspensions of spaces involving polyhedral products

Decompositions of suspensions of spaces involving polyhedral products

Homotopy theory in toric topology

Topology of moment-angle manifolds arising from flag nestohedra

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