Homotopy types of complements of 2-arrangements in R4

We study the homotopy types of complements of arrangements of n transverse planes in R4, obtaining a complete classification for n⩽6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of a 2-arrangement in R4 is not determined by the cohomology ring, thereby answering a question of Ziegler. The invariants that we use are derived from the characteristic varieties of the complement. The nature of these varieties illustrates the difference between real and complex arrangements.