A construction of homotopically non-trivial embedded spheres for hyperplane arrangements

We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the half-spaces. We then prove that the sphere is homotopically trivial if and only if the system of half-spaces is globally consistent. To prove its non-triviality, we compute the twisted intersection number of the sphere with a specific, explicitly constructed twisted Borel-Moore cycle.