Combinatorial stratifications and minimality of 2-arrangements

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 ith 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.