Polyhedral products and features of their homotopy theory

Abstract

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...

Characterisation of polyhedral products with finite generalised Postnikov decomposition

“…Since Z ∂∆ M i (CX, X) ≃ Σ |M i |−1 X ∧|M i | and X is assumed to be a rational homology sphere, Z K (CX, X) is of the homotopy type of a product of rational spheres of dimension at least 2. Thus (3) implies (2). Clearly, (2) implies (1), and therefore the proof is complete.…”

“…The following lemma is proved in [20, Proposition 4.2] (cf. [2]). We note that there is a terminology error in the proof of [20,Proposition 4.2]: "acyclic space" should be replaced with "aspherical space".…”

“…Polyhedral products are combinatorially defined objects and recent studies show that the interaction between topology and combinatorics through polyhedral products provides a rich source of ideas and applications [3,11,12,15,16,21]. A comprehensive survey can be found in [2].…”

“…The question of when a polyhedral product of the form Z K (X, * ) is a K(π, 1) was treated in [20,2] (see Lemma 4.6 below). The proof there does not generalise to our context.…”

A Cartan formula for the cohomology of polyhedral products and its application to the ring structure

“…Aside from the various unstable and stable splitting theorems, [1,12,11,13,14], there is an extensive history of computations of the cohomology groups and rings of various families of polyhedral products, [5,Sections 5,8 and 11], see also [15,10,6,18,19,4,8,9]. Some very early calculations of the cohomology of certain moment-angle complexes, (the case (X i , A i ) = (D 2 , S 1 ) for all i = 1, 2, .…”