Bar complex, configuration spaces and finite type invariants for braids
type: Item
instance of: scholarly article
title: Bar complex, configuration spaces and finite type invariants for braids
zbMATH Open document ID: 1181.57016
author: Toshitake Kohno
published in: Topology and its Applications
publication date: 2009-12-03T00:00:00Z
review text: This note presents a collection of refinements of the author's work on configuration spaces and finite type invariants for braid groups. In [\textit{T. Kohno}, Topology Appl. 118, No.~1--2, 147-157 (2002; Zbl 0995.32018)], the author defined the bar-complex \(B^\ast (A)\) of the Orlik-Solomon algebra \(A\) of an arrangement of complex hyperplanes and related it to Chen's theory of iterated integrals. The advantage of this approach is that it is algebraic. The present paper shows that the bar-complex is acylic, meaning that its \(j\)th cohomology vanishes for \(j>0\). Next, \(H^0(B^\ast (A))\) is identified with the space of finite type invariants for braids, in parallel with the identification of the space of finite type invariants for braids with the \(0\)th cohomology of a space of iterated integrals of logarithmic forms in [\textit{T. Kohno}, Falk, Michael (ed.) et al., Arrangements -- Tokyo 1998. Proceedings of a workshop on mathematics related to arrangements of hyperplanes, Tokyo, Japan, July 13-18, 1998. In honor of the 60th birthyear of Peter Orlik. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 27, 157--168 (2000; Zbl 0978.57006)]. Next, a universal holonomy homomorphism from the braid group to the space of horizontal chord diagrams is made over \(\mathbb{Q}\), making use of the existence of a rational associator. The author had previously carried this out over \(\mathbb{C}\). Finally, using a theorem of Cohen and Gitler, Section 6 proves the existence of a ``rescaling isomorphism'' of Hopf algebras from the homology of the based loop space of the configuration space of \(n\) points in \(\mathbb{R}^m\) to the space of horizontal chord diagrams.
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 32S22
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55R80
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 5641738
zbMATH Keywords: bar complex
zbMATH Keywords: configuration space
zbMATH Keywords: iterated integral
zbMATH Keywords: braid group
zbMATH Keywords: finite type invariant
zbMATH Keywords: Drinfel'd associator
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: Orlik-Solomon algebra
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.057
OpenAlex ID: W2056347712
cites work: On the acyclicity of free cobar constructions. I, II
cites work: Q4382048
cites work: Iterated path integrals
cites work: On loop spaces of configuration spaces
cites work: Q5749433
cites work: Q3351551
cites work: The lower central series of a fiber-type arrangement
cites work: Q4328355
cites work: Vassiliev invariants of braids and iterated integrals
cites work: Bar complex of the Orlik-Solomon algebra
cites work: Q4273352
cites work: Q4884990
cites work: Unitary reflection groups and cohomology
cites work: Homotopy Lie algebras; lower central series and the Koszul property
cites work: Combinatorial formulas for cohomology of knot spaces
DOI: 10.1016/J.TOPOL.2009.04.057
Recommended article: Q4799508
Recommended article: Vassiliev invariants of braids and iterated integrals
Recommended article: On the cohomology of spaces of links and braids via configuration space integrals
Recommended article: Q4731834
Recommended article: Hyperplane arrangements, local system homology and iterated integrals.
Recommended article: Bar complex of the Orlik-Solomon algebra
Recommended article: On the braid groups for the Möbius band
Recommended article: Q3825098
Recommended article: Crossing-changeable braids from chromatic configuration spaces
Recommended article: On loop spaces of configuration spaces
description: scientific article; zbMATH DE number 5641738