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

TL;DR

This paper constructs explicitly non-trivial embedded spheres in the complexified complement of real hyperplane arrangements by analyzing local and global consistency of half-space systems, revealing conditions for homotopical non-triviality.

Contribution

It introduces the notion of locally consistent half-space systems and provides a method to determine the homotopical triviality of embedded spheres in hyperplane arrangement complements.

Findings

Homotopically non-trivial spheres correspond to globally inconsistent half-space systems.

A criterion for triviality based on local versus global consistency.

Explicit computation of twisted intersection numbers confirms non-triviality.

Abstract

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.