It is known that there exist hyperplane arrangements with the same underlying matroid that admit
non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the
diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements
are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness,
shown by means of symbolic computation, implies the desired result.
