Supersolvable posets and fiber-type abelian arrangements

Abstract

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. This is consistent with Terao's fibration theorem connecting bundles of hyperplane arrangements to Stanley's lattice supersolvability. We obtain a combinatorially determined class of K($\pi$,1) toric and elliptic arrangements. Under a stronger combinatorial condition, we prove a factorization of the Poincar\'e polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk-Randell's formula relating the Poincar\'e polynomial to the lower central series of the fundamental group.