The lower central series of a fiber-type arrangement

Falk, Michael; Randell, Richard

doi:10.1007/bf01394780

Cited by 164 publications

(257 citation statements)

References 6 publications

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“…A proof is analogous to the ones in [19,15,32]; modifications in the case of the mod-p descending central series are direct, and are listed below for convenience.…”

Section: Then There Ismentioning

confidence: 99%

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On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

Cohen¹,

Wu²

2003

Categorical Decomposition Techniques in Algebraic Topology

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Abstract. The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor [25], and Habegger-Lin [17,22] on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19,20], and provides connections between these various topics.Two consequences are as follows: (1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and (2) an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW -complexes.

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“…A proof is analogous to the ones in [19,15,32]; modifications in the case of the mod-p descending central series are direct, and are listed below for convenience.…”

Section: Then There Ismentioning

confidence: 99%

“…Algebraic preliminaries are given next arising from work of [19,15,32], and the modification below for the mod-p descending central series. …”

Section: Corollary 105 the Map Of Lie Algebras Ementioning

confidence: 99%

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On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

Cohen¹,

Wu²

2003

Categorical Decomposition Techniques in Algebraic Topology

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“…This formula was established for classifying spaces of pure braids by Kohno [22], and then for complements of arbitrary fiber-type arrangements by Falk and Randell [17]. A variant of the LCS formula holds for the more general class of hypersolvable arrangements, cf [20].…”

Section: Lcs Formula For Higher Homotopy Groupsmentioning

confidence: 99%

Homotopy Lie algebras, lower central series and the Koszul property

Papadima

Suciu

2004

Geom. Topol.

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Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k -rescaling of the rational cohomology ring of X . Assume H * (X, Q) is a Koszul algebra. Then, the homotopy Lie algebra π * (ΩY ) ⊗ Q equals, up to k -rescaling, the graded rational Lie algebra associated to the lower central series of π 1 (X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H * (X, Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of π 1 (X) and the completion of [ΩS 2k+1 , ΩY ]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph. AMS Classification numbers Primary: 16S37, 20F14, 55Q15Secondary: 20F40, 52C35, 55P62, 57M25, 57Q45

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“…Since K n−1 acts trivially on the abelianization of F n , we can apply the result in [7] which claims that the associated graded sequence of Lie algebras is exact, that is,…”

Section: Vassiliev Invariants For Braids On Surfaces 237mentioning

confidence: 99%

Vassiliev invariants for braids on surfaces

González-Meneses

Paris

2003

Trans. Amer. Math. Soc.

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Abstract. We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.1. Definitions and statements 1.1. Introduction. Vassiliev knot invariants were introduced by V. A. Vassiliev ([17], [18]; see also [6], [2]), and they have been generalized to several other knotlike objects, such as links, braids, tangles, string links, knotted graphs, etc. The purpose of this paper is to consider Vassiliev invariants of braids on surfaces, and to extend some well-known results on Vassiliev invariants of Artin braids to the case of braids on surfaces.Our study of Vassiliev invariants is inspired by Papadima's work [13] on Vassiliev invariants for Artin braids with values in Z. However, the presence of the fundamental group of the surface changes the analysis substantially. Anyway, the Vassiliev theory for braids on surfaces, set forth in this paper, appears to be a natural generalization of the corresponding theory for Artin braids.1.2. Braids and singular braids on surfaces. Throughout this paper M will denote a closed, orientable surface of genus g ≥ 1, and P = {P 1 , . . . , P n } a set of n distinct points in M . Define a n-braid based at P to be a collection b = (b 1 , . . . , b n ) of disjoint smooth paths in M × [0, 1], called strings of b, such that the i-th string b i runs monotonically in t ∈ [0, 1] from the point (P i , 0) to some point (P j , 1), P j ∈ P.An isotopy in this context is a deformation through braids (which fixes the ends). Multiplication of braids is defined by concatenation, generalizing the construction of the fundamental group. The isotopy classes of braids with this multiplication form the group B n (M, P), called the braid group with n strings on M based at P. Note that the group B n (M, P) does not depend, up to isomorphism, on the set P of points, but only on the cardinality n = |P|. So we may write B n (M ) in place of B n (M, P).In the same way as Artin braid groups have been extended to singular braid monoids ([6],[1]), one can extend the braid group B n (M ) to SB n (M ), the monoid of singular braids with n strings on M . The strings of a singular braid are now

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The lower central series of a fiber-type arrangement

Cited by 164 publications

References 6 publications

On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

Homotopy Lie algebras, lower central series and the Koszul property

Vassiliev invariants for braids on surfaces

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