Bar complex, configuration spaces and finite type invariants for braids
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.
scientific article; zbMATH DE number 5641738
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