On the fundamental group of certain polyhedral products

Stafa, Mentor

doi:10.1016/j.jpaa.2014.09.001

Cited by 12 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When establishing the minimality of the generator set, we replace L K = (R, Z) K by the polyhedral product (EG, G) K , which is the classifying space for the group G(K) by [St,Theorem 1.1] or [PV,Theorem 3.2]. First assume that each G k is a finite group.…”

Section: Generalisation To Graph Productsmentioning

confidence: 99%

On the commutator subgroup of a right-angled Artin group

Panov

Veryovkin

2019

Journal of Algebra

View full text Add to dashboard Cite

show abstract

Section: Generalisation To Graph Productsmentioning

confidence: 99%

On the commutator subgroup of a right-angled Artin group

Panov

Veryovkin

2019

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Moreover, if K = K 0 is the 0-skeleton of K, then it follows from Example 2.3 that (BG, 1) K 0 = BG 1 ∨ · · · ∨ BG n . The homotopy fibre (I, F ) K 0 has the homotopy type of a finite wedge of circles (I, F ) K 0 ≃ ρ K 0 S 1 , as shown in [24], where…”

Section: Structure Of Papermentioning

confidence: 99%

“…The homotopy fibre (I, F ) K 0 has the homotopy type of a finite wedge of circles (I, F ) K 0 ≃ ρ K 0 S 1 , as shown in [24], where (4)…”

Section: Polyhedral Products and Related Fibrationsmentioning

confidence: 99%

Polyhedral products, flag complexes and monodromy representations

Stafa

2018

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups Aut(Fn) and outer automorphisms of free groups Out(Fn), respectively, as well as faithful representations of products of finite groups into the linear groups SL(n, Z) and GL(n, Z). These faithful representations are realized as monodromy representations. 1 2. Polyhedral products and related fibrations Moment-angle complexes, originally invented by Davis and Januszckiewicz [8], appeared also in the work of Buchstaber and Panov [5] in the context of toric topology. Polyhedral products are a generalization of moment-angle complexes and were introduced and popularized by work of Bahri, Bendersky, Cohen and Gitler [2], and are the main objects of study in toric topology; see the more recent monograph by Buchstaber and Panov [6]. Definition 2.1. Let (X, A) denote a sequence of pointed CW -pairs {(X i , A i )} n i=1 and [n] denote the sequence of integers {1, 2, . . . , n}.

show abstract

“…The important case Z(K; (X, * )) is discussed in Section 10. It arises in algebraic combinatorics, the study of free groups and monodromy representations, geometric group theory, right-angled Coxeter and Artin groups and asphericity, In this section also, we use a recent result of T. Panov and S. Theriault [113] to give a short proof that if K is a flag simplicial complex then, for a discrete group G, Z(K; (BG, * )) is an Eilenberg-Mac Lane space, a known result, [43,53,122].…”

Section: Introductionmentioning

confidence: 99%

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

View full text Add to dashboard Cite

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...

show abstract

On the fundamental group of certain polyhedral products

Cited by 12 publications

References 11 publications

On the commutator subgroup of a right-angled Artin group

On the commutator subgroup of a right-angled Artin group

Polyhedral products, flag complexes and monodromy representations

Polyhedral products and features of their homotopy theory

Contact Info

Product

Resources

About