  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.
