Homology, lower central series, and hyperplane arrangements

Porter, Richard D.; Suciu, Alexander I.

doi:10.1007/s40879-019-00392-x

Cited by 5 publications

(2 citation statements)

References 42 publications

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“…If A is decomposable, then ϕ 3 (G) = S∈X ϕ 3 (G S ); this equality depends only on the lattice of A [23]. The converse depends on the torsion-freeness of Lie 3 (G), which is presently not known -see [44] in this volume for recent progress.…”

Section: Definition 423mentioning

confidence: 99%

Discriminantal bundles, arrangement groups, and subdirect products of free groups

Cohen

Falk

Randell

2020

European Journal of Mathematics

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We construct bundles E k (A, F) → M over the complement M of a complex hyperplane arrangement A, depending on an integer k 1 and a set F = { f 1 ,. .. , f μ } of continuous functions f i : M → C whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence-Krammer-Bigelow representation of the pure braid group. We display such families F for rank two arrangements, reflection arrangements of types A , B , D , F 4 , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group π 1 (M). We describe the kernel of the product ρ X : G → S∈X G S of homomorphisms of a finitely-generated group G onto quotient groups G S determined by a family X of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections G → G S split in a compatible way, we show the image of ρ X is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of π 1 (M), including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina-Brady groups, and a dichotomy for a decomposable arrangement A: either π 1 (M) has a conjugation-free presentation or it is not residually nilpotent.

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Section: Definition 423mentioning

confidence: 99%

Discriminantal bundles, arrangement groups, and subdirect products of free groups

Cohen

Falk

Randell

2020

European Journal of Mathematics

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“…We conclude with a construction of a family of spaces whose homotopy types cannot be distinguished by the usual cup products and triple Massey products, yet can be distinguished using our restricted Massey products. The theory of generalized Massey products continues the program initiated in [25] and is developed more fully in [27], along with further applications.…”

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confidence: 99%

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Porter,

Suciu

2021

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Motivated by the construction of Steenrod cup-i products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.

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Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Porter

Suciu

2022

Topology and its Applications

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Homology, lower central series, and hyperplane arrangements

Cited by 5 publications

References 42 publications

Discriminantal bundles, arrangement groups, and subdirect products of free groups

Discriminantal bundles, arrangement groups, and subdirect products of free groups

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

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