Minimal stratifications for line arrangements and positive homogeneous presentations for fundamental groups

TL;DR

This paper introduces minimal stratifications for real hyperplane arrangements in two dimensions, providing explicit descriptions and showing that their fundamental groups admit positive homogeneous presentations, advancing understanding of their topological structure.

Contribution

It defines minimal stratifications as semialgebraic sets for real arrangements and demonstrates that their fundamental groups have positive homogeneous presentations, offering new insights into their topology.

Findings

Stratifications partition the complement into contractible pieces matching Betti numbers.

Fundamental groups of real arrangements have positive homogeneous presentations.

Explicit descriptions of minimal stratifications as semialgebraic sets.

Abstract

The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the "dual" objects so called minimal stratifications. The strata are explicitly described as semialgebraic sets. The stratification induces a partition of the complement into a disjoint union of contractible spaces, which is minimal in the sense that the number of codimension $k$ pieces equals the $k$-th Betti number. We also discuss presentations for the fundamental group associated to the minimal stratification. In particular, we show that the fundamental groups of complements of a real arrangements have positive homogeneous presentations.