The cohomology ring of the boundary manifold of a combinatorial line arrangement

Abstract

We compute the integral cohomology rings of the associated $3$-manifold called the \textit{boundary manifold} for a combinatorial line arrangement, and prove that it is isomorphic to the double of the Orlik-Solomon algebra. This generalizes the Cohen-Suciu doubling formula for complex realizable case to arbitrary combinatorial line arrangements, including non-realizable ones. To handle the non-realizable case without geometric realization, we construct explicit homology cycles and compute intersection products, following the method of Doig--Horn for graph manifolds. As an application, we derive several results on the resonance variety of the boundary manifold.