  We study the homotopy types of complements of arrangements of n transverse
planes in R^4, 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 R^4 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.
