Quadratic relations in the cohomology of generalized pure braids and Dunkl identities
The pure braid group on \(n\) strands, \(P_n\), can be described as the fundamental group of \({\mathbf C}^n-A\) where \(A\) is the union of the hyperplanes defined by the equations \(z_j=z_k\) (\(1\leq j<k\leq n\)), with \(z_1,\dots,z_n\) being the coordinates in \({\mathbf C}^n\). Accordingly, a generalized pure braid group has been defined as \(\pi_1({\mathbf C}^n-S)\) where \(S\) is the complexification of an arrangement of hyperplanes in \({\mathbf R}^n\) such that the reflections across the hyperplanes of the arrangement generate a finite (Coxeter) group in \(O(n)\). \textit{V. I. Arnol'd} [Math. Notes 5, 138-140 (1969); translation from Mat. Zametki 5, No. 2, 227-231 (1969; Zbl 0277.55002)] described the integral cohomology ring \(H^*(P_n)=H^*({\mathbf C}^n-A)\) in two ways: in terms of generators and relations and by embedding this ring in the algebra of differential forms on \({\mathbf C}^n-A\). The latter way has been generalized to complements of arbitrary hyperplane arrangements in \({\mathbf C}^n\) by \textit{E. Brieskorn} [Sém. Bourbaki 1971/72, No. 401, Lect. Notes Math. 317, 21-44 (1973; Zbl 0277.55003)]. In the present paper the authors generalize Arnol'd's presentation of \(H^*(P_n)\) to generalized pure braid groups corresponding to arbitrary hyperplane arrangements \(S\) ``of fiber type'' (as defined by \textit{M. Falk} and \textit{R. Randell} [Invent. Math. 82, 77-88 (1985; Zbl 0574.55010)]). They prove that \(H^*(\pi_1({\mathbf C}^n-S))=H^*({\mathbf C}^n-S)\) is isomorphic to the quotient of the exterior \(\mathbf Z\)-algebra with generators \(\omega_H\), one for each hyperplane \(H\) in \(S\), by the quadratic relations \(\omega_{H_1}\wedge\omega_{H_2}+\omega_{H_2}\wedge\omega_{H_3}+ \omega_{H_3}\wedge\omega_{H_1}=0\), one for each triple \(H_1,H_2,H_3\) of hyperplanes in \(S\) with \((n-2)\)-dimensional intersection.
cohomology ring of a generalized pure braid group
scientific article; zbMATH DE number 999675