Monodromy of fiber-type arrangements and orbit configuration spaces (Q2716430)

Using techniques developed for the calculation of the braid monodromy of a complex line arrangement, the author gives a detailed description of the monodromy of a strictly linear fibration in terms of pure braids. The prototypical example is the map of configuration spaces \(F(\mathbb{R}^2,n+1)\to F(\mathbb{R}^2,n)\) given by forgetting the \((n+1)\)-st point, with fiber a plane with \(n\) punctures. Here the monodromy is given by the Artin representation, restricted to the pure braid group. A general strictly linear fibration of complex hyperplane complements is shown to be the pullback of this bundle by the ``root map'' associated with the fibration, yielding formulas for the monodromy action. The method is illustrated by calculating the monodromies and associated fundamental group presentation for the (fiber-type) reflection arrangement of type \(B_n\).NEWLINENEWLINENEWLINE\textit{C. S. Aravinda, F. T. Farrell}, and \textit{S. K. Roushon} [Asian J. Math. 4, No. 2, 337-344 (2000; Zbl 0980.19001)] showed that the pure braid group has trivial Whitehead group, and conjectured that the same holds for any fiber-type arrangement group. The author gives an inductive proof of this conjecture.NEWLINENEWLINENEWLINEAn analogous bundle structure theory is developed for orbit configuration spaces \(F_\Gamma(M,n)\), for \(\Gamma\) a group acting freely on a manifold \(M\). Here \(F_\Gamma(M,n)\) consists of those configurations of \(n\) (distinct) points in \(M\) whose \(\Gamma\)-orbits are disjoint.NEWLINENEWLINENEWLINEIn case \(\Gamma\) is a cyclic group acting on \(\mathbb{C}^*\), the orbit configuration space coincides with the complement of a fiber-type arrangement, the unitary reflection arrangement associated with the full monomial group. Its fundamental group, the ``monomial pure braid group,'' is analyzed in detail. Geometric ``monomial braids'' are defined, which serve as natural generators, subject to relations similar to the classical (pure) braid relations. A presentation for the fundamental group, involving some unspecified conjugations, is obtained from the monodromy calculations. By comparing presentations, it is shown that the graded Lie algebra associated with the lower central series of this group is isomorphic, up to regrading, to the Lie algebra of primitive elements in the homology of the loop space of the orbit configuration space \(F_\Gamma(\mathbb{C}^k-\{0\},n)\), where \(\Gamma\) is a cyclic group acting diagonally on \(\mathbb{C}^k-\{0\}\). This observation affirms a conjecture of \textit{M. Xicoténcatl} [Thesis, Univ. Rochester (1997)]. This result was subsequently generalized by the author, \textit{F. R.~Cohen}, and \textit{Xicoténcatl} [Int. Math. Res. Not. 2003, No. 29, 1591-1621 (2003)] and further by \textit{S. Papadima} and \textit{A. I. Suciu} [preprint 2001, \texttt{http://math.arxiv.org/math.AT/0110303}].