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arXiv · 2202.11996

Supersolvable posets and fiber-type abelian arrangements

Christin Bibby·Emanuele Delucchi
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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.

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Christin Bibby, Emanuele Delucchi. 2022-02-24. Supersolvable posets and fiber-type abelian arrangements. https://arxiv.org/abs/2202.11996

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