Homology of iterated semidirect products of free groups
type: Item
instance of: scholarly article
title: Homology of iterated semidirect products of free groups
zbMATH Open document ID: 0908.20033
DOI: 10.1016/S0022-4049(96)00153-3
author: Daniel C. Cohen
author: Alexander I. Suciu
published in: Journal of Pure and Applied Algebra
publication date: 1999-01-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9503002
review text: Let \(P_n\) denote the pure braid group on \(n\) strands. There is a split epimorphism \(P_{n+1}\to P_n\) whose kernel is a free group \(F_n\) on \(n\) generators. The action of \(P_n\) on \(F_n\) is given by the (restriction of the) well-known Magnus embedding of the full braid group into \(\Aut(F_n)\). In this way \(P_{n+1}\) has the structure of an iterated semidirect product of free groups. This is the situation analyzed in the paper under review. Other groups admitting such a structure include fundamental groups of complements of fiber-type arrangements, including the discriminantal arrangements of \textit{V. V. Shekhtman} and \textit{A. N. Varchenko} [Invent. Math. 106, No. 1, 139-194 (1991; Zbl 0754.17024)], and of total spaces of certain nonlinear fibrations with punctured-surface fibers. Suppose \(G\) has a finite filtration \(1=G_0\leq G_1\leq\ldots\leq G_\ell=G\) such that \(G_k\) is a split extension of \(G_{k-1}\) by a finitely-generated free group. An Eilenberg-MacLane space \(K(G,1)\) can be constructed using fibered products of bouquets of circles, with monodromy determined by the semidirect product structure. Using the bundle structure and methods of combinatorial group theory, specifically the Fox calculus, the authors construct a finite, free resolution \(C_\bullet(G)\) of \(\mathbb{Z}\) as a module over the group ring \(\mathbb{Z}[G]\). This construction yields an algorithm for computing the cohomology of \(G\), with coefficients in an arbitrary \(G\)-module. This cohomology, in case \(G=P_{n,k}:=\ker(P_n\to P_k)\), has applications to solutions of the Knizhnik-Zamolodchikov equations of conformal field theory. A cohomology vanishing theorem is deduced which strengthens previously known results. The algorithm is also used to compute the Betti numbers and algebraic monodromy of Milnor fibers associated with Coxeter arrangements (mainly of type \(A_\ell\)), and with classical discriminants, producing some previously unseen results. If \(\Gamma\) is…
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 1121692
zbMATH Keywords: group cohomology
zbMATH Keywords: braid groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: fiber-type arrangements
zbMATH Keywords: Milnor fibers
zbMATH Keywords: cohomology vanishing theorems
zbMATH Keywords: Coxeter arrangements
zbMATH Keywords: Burau and Gassner representations
reviewed by: Michael Falk
MaRDI profile type: Publication
arXiv ID: alg-geom/9503002
cites work: Milnor numbers of nonisolated Saito singularities
cites work: Gauss-Manin connection of integral of difference products
cites work: The cohomology ring of the colored braid group
cites work: Theory of braids
cites work: Braids, Links, and Mapping Class Groups. (AM-82)
cites work: Q4315615
cites work: Free resolutions for semi-direct products
cites work: Q4774741
cites work: Q4143495
cites work: Cohomology and intersection cohomology of complex hyperplane arrangements
cites work: On Milnor Fibrations of Arrangements
cites work: The homology of iterated loop spaces
cites work: Q3348085
cites work: Erratum: Cohomology of local systems on the complement of hyperplanes
cites work: Configuration Spaces.
cites work: The lower central series of a fiber-type arrangement
cites work: Q3822397
cites work: 𝛽𝐧𝐛𝐜-bases for cohomology of local systems on hyperplane complements
cites work: Free differential calculus. III: Subgroups
cites work: ALGEBRAIC EQUATIONS WITH CONTINUOUS COEFFICIENTS AND SOME PROBLEMS OF THE ALGEBRAIC THEORY OF BRAIDS
cites work: Q3993435
cites work: Q4062377
cites work: Fiber-type arrangements and factorization properties
cites work: Hecke algebra representations of braid groups and link polynomials
cites work: Série de Poincaré-Koszul associée aux groupes de tresses pures
cites work: Homology of a local system on the complement of hyperplanes
cites work: Q3800265
cites work: Q3784320
cites work: Homological representations of the Hecke algebra
cites work: Q4307570
cites work: Invariants of plane algebraic curves via representations of the braid groups
cites work: The Burau representation is not faithful for \(n\geq 6\)
cites work: Lê cycles and hypersurface singularities
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Q3933100
cites work: The Burau representation of the braid group 𝐵_{𝑛} is unfaithful for large 𝑛
cites work: Combinatorics and topology of complements of hyperplanes
cites work: Unitary reflection groups and cohomology
cites work: Q4003206
cites work: The homotopy type of Artin groups
cites work: On local systems over complements to arrangements of hyperplanes associated to Grassmann strata
cites work: Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors
cites work: Arrangements of hyperplanes and Lie algebra homology
cites work: Modular elements of lattices and topological fibration
cites work: Q4848731
cites work: General hypergeometric functions on complex Grassmannians
cites work: Cohomology of the brieskorn-orlik-solomon algebras
OpenAlex ID: W3123222025