Abstract: Given an n-tuple of distinct complex numbers, one can “forget the last point” and obtain an (n-1)-tuple of distinct complex numbers. This is in fact a fiber bundle on configuration spaces, and the fiber of a point is homeomorphic to C with n-1 points removed. Certain complements of complex hyperplane arrangements exhibit a similar fiber bundle, and an incredible theorem of Terao relates the existence of such a fibration with a purely combinatorial property in the intersection lattice. In this talk, we will discuss a generalization of Terao’s theorem to other arrangement complements, including toric and elliptic analogues, as well as its topological consequences. Based on joint work with Emanuele Delucchi.