An equivariant discrete model for complexified arrangement complements

TL;DR

This paper introduces a new equivariant discrete model for complexified arrangement complements, revealing new homotopy equivalences and expanding understanding of fundamental groups beyond classical arrangements.

Contribution

It constructs a partial order on topes of an oriented matroid and shows the geometric realization has the same homotopy type as the Salvetti complex, with new group-theoretic insights.

Findings

|Q| is homotopy equivalent to the arrangement complement

|Q(M, e)| models the decone complement of an arrangement

Fundamental groups of these complexes are more diverse than classical arrangement groups

Abstract

We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of linear hyperplanes in real space, the Z_4 action corresponds to the diagonal action of C* on the complement M of the complexification of A: |Q| is equivariantly homotopy-equivalent to M under the identification of Z_4 with {+1, i, -1, -i}, and |Q(M, e)| is homotopy-equivalent to the complement of the decone of A relative to the hyperplane corresponding to e. All constructions and arguments are carried out at the level of the underlying posets. If a group G acts on the set of topes of M preserving adjacency, then G acts simplicially on |Q|. We also show that the class of fundamental groups of such complexes is strictly larger than the class of fundamental groups of complements of complex hyperplane arrangements. Specifically, the group of the non-Pappus arrange- ment is not isomorphic to any realizable arrangement group.