The diffeomorphism type of small hyperplane arrangements is combinatorially determined

TL;DR

This paper proves that for small complex hyperplane arrangements with up to 7 hyperplanes, the topological and geometric properties of their complements are uniquely determined by their underlying matroid, establishing a combinatorial classification.

Contribution

It demonstrates that the diffeomorphism type of complements of small hyperplane arrangements is fully determined by their matroid, using the concept of reduced realization space.

Findings

Complement manifolds are isotopic for arrangements with same matroid up to 7 hyperplanes.

The connectedness of the reduced realization space implies combinatorial determination.

Milnor fibers and fibrations depend solely on the underlying matroid.

Abstract

It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and 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, i.e., they depend uniquely on the underlying matroid. To do this, we associate to every such matroid a topological space, that we call the reduced realization space; its connectedness, showed by means of symbolic computation, implies the desired result.