Bordifications of hyperplane arrangements and their curve complexes. Issue 2 (21st March 2021)

• Record Type: Journal Article
• Authors: Davis, Michael W.; Huang, Jingyin
• Is Part Of: Journal of topology. Volume 14:Issue 2(2021)
• Page Start: 419
• Page End: 459
• Publication Date: 2021-03-21
• Subjects: 20F36 -- 20F65 -- 57S30 (primary) -- 20F55 -- 32S22 -- 52B12 -- 57N65 (secondary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12184
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: The complement of an arrangement of hyperplanes in C n has a natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex C, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex C plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that C has the homotopy type of a wedge of spheres. Our results apply in particular to spherical Artin groups, where the associated arrangement is a reflection arrangement of a finite Coxeter group.