Towards a combinatorial characterization of K(π, 1) arrangements

A system of half-spaces is an assignment of a half-space to each hyperplane of a real hyperplane arrangement. Such a system is said to be consistent if the intersection of the half spaces is nonempty (a chamber). Given a locally consistent system of half-spaces, using the Salvetti complex, one can construct an embedded sphere in the complexified complement. As a main result, we will show that if the system is (globally) inconsistent, then the resulting sphere gives a non-trivial element in the homotopy group, in particular, it is  on-$K(\pi, 1)$ arrangement. This construction recovers many known non-$K(\pi, 1)$ arrangements. We also formulate a conjectural characterization of $K(\pi, 1)$ arrangements. K(\pi, 1)$ arrangements. We also formulate a conjectural characterization of $K(\pi, 1)$ arrangements.

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