  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.
