Cohomology of pure braid groups of exceptional cases.
type: Item
instance of: scholarly article
title: Cohomology of pure braid groups of exceptional cases.
zbMATH Open document ID: 1195.20055
author: Simona Settepanella
published in: Topology and its Applications
publication date: 2009-04-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0505566
review text: For \((W,S)\) a finite Coxeter system realized as a reflection group in \(\mathbb{R}^n\) one defines \(Y(W)\) to be the complement to the arrangement in \(\mathbb{C}^n\). The author, [in \textit{S. Settepanella}, Topology Appl. 139, No. 1-3, 37-47 (2004; Zbl 1064.20052)], has computed the cohomology of \(Y(W)\) for the series \(A_n\), \(B_n\) and \(D_n\). The present work is a natural continuation of the latter work where the cohomology of \(A(W)\) with local coefficients \(R=\mathbb{Q}[\tau,\tau^{-1}]\) (the ring of Laurent polynomials) for the finite Coxeter groups \(H_3\), \(H_4\), \(F_4\) and \(A_n\), \(n<7\), is computed. A relevant step to obtain the results are algorithms to compute the cohomologies of the flag complex \(Fl_*(W)\).
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 32S55
zbMATH DE Number: 5539525
zbMATH Keywords: pure braid groups
zbMATH Keywords: arrangements of hyperplanes
zbMATH Keywords: cohomology with local coefficients
zbMATH Keywords: Milnor fibres
zbMATH Keywords: Coxeter systems
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
OpenAlex ID: W1992419557
arXiv ID: math/0505566
cites work: Non-broken circuits of reflection groups and factorization in \(D_ n\)
cites work: Combinatorial Stratification of Complex Arrangements
cites work: Q3941610
cites work: Q4774741
cites work: Homology of iterated semidirect products of free groups
cites work: Arithmetic properties of the cohomology of braid groups
cites work: Q4487384
cites work: The Orlik-Solomon complex and Milnor fibre homology
cites work: Cohomologies of groups of braids of series C and D and certain stratifications
cites work: Homology of a local system on the complement of hyperplanes
cites work: Combinatorics and topology of complements of hyperplanes
cites work: On local systems over complements to arrangements of hyperplanes associated to Grassmann strata
cites work: The homotopy type of Artin groups
cites work: Arrangements of hyperplanes and Lie algebra homology
cites work: A stability-like theorem for cohomology of pure braid groups of the series \(A\), \(B\) and \(D\).
DOI: 10.1016/J.TOPOL.2008.12.007
description: scientific article