A spectral sequence for polyhedral products

Bahri, Abbas; Bendersky, Martin; Cohen, Fred; Gitler, S.

doi:10.1016/j.aim.2016.12.028

Cited by 7 publications

(26 citation statements)

References 23 publications

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“…The latter is described in [40]. In [16] it is shown, using the results of [12], that this is indeed the ring structure in H * Z(K; (CA, A)) .…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 70%

“…Continuing in this way, we arrive at the (reduced) Poincaré series P H * ( Z(K; (X, A))) = t 9 + t 11 + 3t 12 + 5t 14 + 2t 16 .…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

“…We turn our attention now to the computation of the cohomology groups of a polyhedral product H * Z(K; (X, A)) . For a suitable collection of pairs of spaces (X, A), the cohomology of the polyhedral product Z K; (X, A) is computed using a spectral sequence by the authors in [16]. A computation using different methods by Q. Zheng can be found in [134].…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

“…Moreover, the theorem generalizes to any multiplicative cohomology theory h * for which (X, A) satisfies the natural formulation of the strong freeness condition for h * . More information about the cohomology ring may be found in [16] and [134]. The use of the methods developed in [16] is illustrated next with a few examples, each having a different flavor.…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

“…Remark. The ring structure for general (X, A) satisfying the strong freeness condition may be found in [12], [16] and [134].…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

See 4 more Smart Citations

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

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A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.The paper is organized as follows. A brief technical discussion of the origin of polyhedral products as moment-angle complexes in Sections 2 and 3, includes the original topological construction of toric manifolds by M. Davis and T. Januszkiewicz [52] and the reformulation of one of their main spaces as a moment-angle complex by V. Buchstaber and T. Panov, [34,35].A discussion follows in Section 4 about the independent discovery of moment-angle complexes as intersections of certain quadrics. We describe briefly the work of S. López de Medrano [100], S. López de Medrano and A. Verjovsky [101], F. Bosio and L. Meersseman [30], followed by that of S. López de Medrano and S. Gitler [65]. This approach has proved effective in identifying classes of moment-angle manifolds that are connected sums of products of spheres. This is followed in Section 5 by a sketch of the computation of the cohomology ring of moment-angle complexes.We begin our focus on more general polyhedral products in Section 6, by describing their behaviour with respect to the exponentiation of CW-pairs. This has led to an application to the study of toric manifolds [58,128,129,124,70,72,45], and orbifolds [18]. An application to topological joins is included as an example of the utility of this property with respect to exponentiation.The behaviour of polyhedral products with respect to fibrations, due to G. Denham and A. Suciu [55], is surveyed briefly in Section 7. We take advantage of the formalism to describe briefly the early work in this area by G. Porter [117,118], T. Ganea [64] and A. Kurosh [92]. and G. W. Whitehead. Instances of polyhedral products appear also in the work of D. Anick [5].In Section 8, we review the various fundamental unstable and stable splitting theorems for the polyhedral product. These theorems give access to the homotopy type of the polyhedral product in many of the most important cases and drive a number of applications. The identification of the various wedge summands which appear in the stable decompositions, occupies the second part of this section.A fine theorem of A. Al-Raisi [2] shows that, in certain cases, the stable splitting of the polyhedral product can be chosen to be equivariant with respect to the action of Aut(K). This and other related results, described in Sec...

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“…The latter is described in [40]. In [16] it is shown, using the results of [12], that this is indeed the ring structure in H * Z(K; (CA, A)) .…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 70%

“…Continuing in this way, we arrive at the (reduced) Poincaré series P H * ( Z(K; (X, A))) = t 9 + t 11 + 3t 12 + 5t 14 + 2t 16 .…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

“…Remark. The ring structure for general (X, A) satisfying the strong freeness condition may be found in [12], [16] and [134].…”

Section: The Cohomology Of Polyhedral Products and A Spectral Sequencementioning

confidence: 99%

See 3 more Smart Citations

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

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show abstract

On the Cohomology Ring of Real Moment-Angle Complexes

Vidaurre

2024

Fields Institute Communications

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Symmetric Products and a Cartan-Type Formula for Polyhedral Products

Bahri,

Bendersky,

Cohen

et al. 2024

Fields Institute Communications

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A spectral sequence for polyhedral products

Cited by 7 publications

References 23 publications

Polyhedral products and features of their homotopy theory

Polyhedral products and features of their homotopy theory

On the Cohomology Ring of Real Moment-Angle Complexes

Symmetric Products and a Cartan-Type Formula for Polyhedral Products

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