The lower central series of a fiber-type arrangement
An \(\ell\)-arrangement is a finite collection of hyperplanes in \({\mathbb{C}}^{\ell}\) (supposed here through the origin). The complement M of the union of the hyperplanes is a topological space with fundamental group \(\Pi_ 1(M)=G\). Let \(G=G_ 1\), \(G_{n+1}=[G_ n,G]\) for \(n\geq 1\) be the lower central series and \(\phi_ n=rank G_ n\). The author proves that for arrangements of ''fiber type'' the Poincaré series of M satisfies the relation \(P_ M(-t)=\prod^{\infty}_{j=1}(1-t^ j)^{\phi_ j}\). This result is a generalisation of a result of \textit{T. Kohno} [ibid. 82, 57-75 (1985; see the review above)].
complement of a union of complex hyperplanes
scientific article; zbMATH DE number 3917256