Analysis and topology of hyperplane complements: The generalized Witt formula (Q1198467)

Let \(A\) be a finite set, let \(\text{Lib}(A)\) be the free Lie algebra over \(A\) and let \(F(A)\) be the free group over \(A\). The dimensions of the homogeneous components of \(\text{Lib}(A)\) which are the ranks of the factors of the lower central series of \(F(A)\) are given by the classical Witt formula. This formula can be interpreted as a relation between cohomology, lower central series of the fundamental group and holonomy Lie algebra of the complement of a set of \(\text{card}(A)\) points in \({\mathbf C}\). Let \(\mathcal A\) be a complexified Coxeter arrangement of hyperplanes or a fiber-type arrangement of hyperplanes. We can establish a similar relation called generalized Witt formula or LCS formula. We consider the complex \(R\), due to Aomoto. If this complex is acyclic, the LCS formula is satisfied. We introduce a filtration on this complex; if the spectral sequence satisfies \(E^{pq}_ 1=0\) for \(p+q\neq 0\), the complex is acyclic.