Boundary manifolds of line arrangements

Eriko Hironaka

This paper contains a description of the homotopy type of the complement of a real line arrangement in the complex plane in terms of the fat graph of the line arrangement. A real line arrangement is a complex line arrangement defined by real equations. The fat graph is the point/line incidence graph of the line arrangement, together with orderings of the edges emanating from each vertex.

The homotopy type of the boundary manifold of the line arrangement is presented as a graph manifold over the incidence graph and is shown to depend only on the incidence graph. The complement of a real line arrangement in the complex plane is shown to be homotopy equivalent to the graph manifold representing the boundary manifold with a continuous image of the incidence graph contracted to a point. The mapping of the incidence graph into the graph manifold can be defined solely in terms of the fat graph of the line arrangement.