The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

By A. Bahri, Martin Bendersky, Frederick R. Cohen, S. Gitler. 2007

239 citations recorded by the source corpus.

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\it partial product spaces} which arise as {\it polyhedral product functors} described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \cite{goresky.macpherson}, Hochster\cite{hochster}, Baskakov \cite{baskakov}, Panov \cite{panov}, and Buchstaber-Panov \cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter \cite{porter} and Ganea \cite{ganea}. The spirit of the work here follows that of Denham-Suciu in \cite{denham.suciu}.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25e92e371aa0326db70d282de94a5d05f4537225

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Citations in the corpus

Complex-analytic structures on moment-angle manifolds (2010, background): “K) = (C,C×)K, where C× = C\{0}. We obviously have ZK ⊂ U(K). Moreover, ZK is a Tm-equivariant deformation retractof U(K) foreveryK [BP2, Th. 8.9].Formoregeneralisationsandapplications of K-powers see [BBCG]. It is shown in [BP2, Lemma 6.13] that ZK is a (closed) topological manifold whenever |K| is ”
Toric Topology (2012, methodology): “One of the most important contributions was the 2010 work of Bahri, Bendersky, Cohen and Gitler [14] establishing a decomposition of the suspension of a polyhedral product (X,A) into a wedge of suspensions corresponding to subsets I ⊂ [m].”
The homotopy types of moment-angle complexes for flag complexes (2012, background): “But stably these attaching maps are all null homotopic since, by [1], the homotopy equivalence in (3.”
THE HOMOTOPY TYPE OF THE POLYHEDRAL PRODUCT FOR SHIFTED COMPLEXES (2011, methodology): “owers in [BP1], appeared without THE HOMOTOPY TYPE OF THE POLYHEDRAL PRODUCT FOR SHIFTED COMPLEXES 3 fanfare in [GT1, GT2], and came to prominence in recent work of Bahri, Bendersky, Cohen and Gitler [BBCG1]. Following Ganea’s and Porter’s results, it is natural to ask when the homotopy type of the ”
Intersections of Quadrics, Moment-angle Manifolds and Connected Sums (2009, methodology): “ld Massociated to P. 7In the notation of section 2.1, Z=D ( + );Z0 0) and Z0 TOB +). 11 In fact, all our constructions can be expressed within the context of the generalized moment-angle complexes of [B-B-C-G] (but not inside any of the many previous generalizations of moment-angle manifolds) and th”
Operations on polyhedral products and a new topological construction of infinite families of toric manifolds (2010, background): “i=1 Wi, where Wi = ( Xi if i ∈σ Ai if i ∈[m]−σ. Here, the colimit is a union given by Z(K;(X,A)) = [ σ∈K D(σ). Detailed background information about polyhedral products may be found in [7], [9], [1], [2] (and in the unpublished notes of N. Strickland). The family of CW-pairs (X,A) to be investigated”