Monodromy of fiber-type arrangements and orbit configuration spaces

Abstract

We prove similar theorems concerning the structure of bundles involving complements of fiber-type hyperplane arrangements and orbit configuration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we resolve two disparate conjectures. We show that the Whitehead group of the fundamental group of the complement of a fiber-type arrangement is trivial, as conjectured by Aravinda, Farrell, and Rouchon. For the orbit configuration space corresponding to the natural action of a finite cyclic group on the punctured plane, we determine the structure of the Lie algebra associated to the lower central series of the fundamental group. Our results show that this Lie algebra is isomorphic to the module of primitives in the homology of the loop space of a related orbit configuration space, as conjectured by Xicotencatl.