Séminaire Lotharingien de Combinatoire, 91B.36 (2024), 12 pp.
Christin Bibby and Emanuele Delucchi
Supersolvable Posets and Fiber-Type Arrangements
Abstract.
We develop a theory of modularity and supersolvability for chain-finite geometric posets, extending that of Stanley for finite lattices and building a new connection between combinatorics and topology.
From a combinatorial point of view, our theory features results about factorizations of the characteristic polynomials, dovetails with established notions on geometric semilattices, and behaves well under quotients by translative group actions.
We also establish a topological counterpart in the context of toric and abelian arrangements,
akin to Terao's fibration theorem connecting bundles of hyperplane arrangements to supersolvability of their intersection lattice. From this, we obtain a combinatorially determined class of K(π,1) toric arrangements.
Moreover, we characterize combinatorially when our toric arrangement bundles are pulled back from Fadell-Neuwirth bundles of configuration spaces, and establish an analogue of Falk and Randell's formula relating the Poincar\'e polynomial to the lower central series of the fundamental group.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
The following versions are available: