Configuration spaces and polyhedral products

Beben, Piotr; Grbić, Jelena

doi:10.1016/j.aim.2017.05.001

Cited by 12 publications

(26 citation statements)

References 22 publications

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“…The faces of K are the vertices {1}, {2}, {3}, {4} and the edges (1, 2), (1,3), (1,4), (2,3) and (2,4). For a vertex {i} we have X {i} = X i and for an edge (i, j) we have X (i,j) = X i ∧ X j .…”

mentioning

confidence: 99%

“…The missing faces of K are (3, 4), (1, 2, 3), (1, 2, 4), (1,3,4), (2,3,4) and (1,2,3,4). The full subcomplexes K I of K on these vertex sets are: 2,4) contributes an S 6 to ΣZ K .…”

mentioning

confidence: 99%

“…The full subcomplexes K I of K on these vertex sets are: 2,4) contributes an S 6 to ΣZ K . Third, observe that |K (1,3,4) | is contractible, so Σ |I|+2 |K I | in this case is * . Similarly, K (2,3,4) contributes a point to ΣZ K .…”

mentioning

confidence: 99%

“…Third, observe that |K (1,3,4) | is contractible, so Σ |I|+2 |K I | in this case is * . Similarly, K (2,3,4) contributes a point to ΣZ K . Finally, observe that…”

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confidence: 99%

“…In [23], a desuspension was proved for a strictly larger family of simplicial complexes called sequential Cohen-Macauley. In [3,23] it was shown that a desuspension exists if and only if (CX, X) K is a co-H-space, and near characterizations of when this occurs were given. The strategy for desuspending the decomposition in Theorem 4.6 in the shifted case was exactly the strategy that we have been working with: the shifted property implies -after some work -that there is a reordering of the vertices with the property that the map (CX,…”

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confidence: 99%

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Toric Homotopy Theory

Theriault¹

2017

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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These notes describe some of the homotopy theory surrounding DavisJanuszkiewicz spaces, moment-angle complexes and their generalizations to polyhedral products. These spaces are defined by gluing together products formed from pairs of spaces (X, A), where the gluing is determined by the faces of a simplicial complex K. The emphasis is on determining homotopy types -of the spaces themselves, their suspensions and their based loop spaces -and showing how these homotopy types depend on a beautiful interplay between the topology of the pairs (X, A) and the combinatorics of the simplicial complex K.

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confidence: 99%

“…The missing faces of K are (3, 4), (1, 2, 3), (1, 2, 4), (1,3,4), (2,3,4) and (1,2,3,4). The full subcomplexes K I of K on these vertex sets are: 2,4) contributes an S 6 to ΣZ K .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Third, observe that |K (1,3,4) | is contractible, so Σ |I|+2 |K I | in this case is * . Similarly, K (2,3,4) contributes a point to ΣZ K . Finally, observe that…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Toric Homotopy Theory

Theriault¹

2017

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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Orbit configuration spaces of small covers and quasi-toric manifolds

Chen

Lü

2019

Sci. China Math.

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In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of Euler characteristic for the classical configuration space, which generalizes the Félix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retract of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which turns out that we are able to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

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