Combinatorial stratifications and minimality of 2-arrangements

TL;DR

This paper proves that the complement of any affine 2-arrangement in R^d is minimal, using combinatorial methods, and establishes a Lefschetz-type theorem and Alexander duality for these arrangements.

Contribution

It introduces a combinatorial approach to prove minimality and Lefschetz hyperplane theorem for affine 2-arrangements, extending previous complex hyperplane arrangement results.

Findings

Complement of affine 2-arrangements is minimal.

Established Lefschetz-type hyperplane theorem for these arrangements.

Introduced Alexander duality for combinatorial Morse functions.

Abstract

We prove that the complement of any affine 2-arrangement in R^d is minimal, that is, it is homotopy equivalent to a cell complex with as many i-cells as its i-th rational Betti number. For the proof, we provide a Lefschetz-type hyperplane theorem for complements of 2-arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca--Papadima, Hattori, Randell, and Salvetti--Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimality and the Lefschetz Hyperplane Theorem for complements of complex hyperplane arrangements.