Topological line arrangements and their topological invariants

Abstract

A topological line arrangement is an arrangement of embedded spheres in the complex projective plane that topologically generalizes a complex line arrangement. In this paper, we establish foundational results on the topology of the complement of topological line arrangements. First, we prove that the cohomology ring of the complement is isomorphic to the Orlik-Solomon algebra, as for classical complex line arrangements. Since classical methods are unavailable in this setting, we use a homological method to compute the cohomology ring. We then study the homotopy type of the complement. We prove that the complement of a symplectic line arrangement has the homotopy type of a minimal CW complex. In contrast, every combinatorial type realizable by a topological line arrangement admits a realization with a non-minimal complement. Moreover, every such combinatorial type admits infinitely many realizations whose complements are pairwise non-homotopy equivalent.