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

2024/05/30 by Yoshinaga, Masahiko · 1 citation
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2405.20010

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.

Cited by

• Yoshinaga's criterion and the Varchenko--Gelfand algebra

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