On Commuting and Noncommuting Complexes

Pakianathan, Jonathan; Yalcin, Ergun

doi:10.1006/jabr.1999.8501

Cited by 7 publications

(5 citation statements)

References 3 publications

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“…Shareshian [154] and Shareshian and Wachs [156] use Theorems 5.5.2 and 5.5.4, to derive information about the homology of the Quillen complex A p (S n ) from a hypergraph version of the matching complex. Another application of Theorems 5.5.2 and 5.5.4 can be found in [131]. Here we discuss the original example that led to the general fiber theorem, namely the colored chessboard complex [201].…”

Section: Inflations Of Simplicial Complexesmentioning

confidence: 99%

Poset topology: Tools and applications

Wachs¹

2007

IAS/Park City Mathematics Series

192

215

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Basic definitions, results, and examples Order complexes and face posetsWe begin by defining the order complex of a poset and the face poset of a simplicial complex. These constructions enable us to view posets and simplicial complexes as essentially the same topological object. We shall assume throughout these lectures that all posets and simplicial complexes are finite, unless otherwise stated.An abstract simplicial complex ∆ on finite vertex set V is a nonempty collection of subsets of V such that

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Section: Inflations Of Simplicial Complexesmentioning

confidence: 99%

Poset topology: Tools and applications

Wachs¹

2007

IAS/Park City Mathematics Series

192

215

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“…This definition appears in[18], page 415. A related notion, that of a centralizer abelian or CA group, refers to the situation when commutativity is transitive on all non-trivial elements in the group.…”

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confidence: 99%

Commuting elements, simplicial spaces and filtrations of classifying spaces

Ádem¹,

Cohen²,

Giese³

2011

Math. Proc. Camb. Phil. Soc.

165

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Abstract. Let G denote a topological group. In this article the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n-tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit-Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non-central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.

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“…A homological criterion has been given in [18] by J.Pakianathan and E.Yalcin for the existence of maximal non-commuting sets in groups by using non-commuting and commuting simplicial complexes. First a structural result of the simplicial complex as wedge of a base complex and suspension spaces has been observed due to A.Björner et al in [5] and then especially in one of the cases of the non-commuting structure, where every centralizer is at least two, a homological criterion has been given.…”

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confidence: 99%