Homotopy types of complements of 2-arrangements in R4

Abstract

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.1991 Mathematics Subject Classification. Primary 57M05, 57M25, 52B30; Secondary 14M12, 20F36.

Untitled

“…More precisely, we consider arrangements of transverse planes through the origin of R 4 . If G is the group of such a 2-arrangement, the varieties V d (G, C) need not be unions of translated subtori, as shown in [32], and also here, in Example 10.3. Furthermore, the tangent cone at the origin to V d (G, C) may not coincide with the resonance variety R d (G, C), as we point out in Remark 10.4.…”

“…Furthermore, the tangent cone at the origin to V d (G, C) may not coincide with the resonance variety R d (G, C), as we point out in Remark 10.4. Finally, using the metabelian Hall invariants δ S 3 and δ A 4 , we recover the homotopy-type classification of complements of 2-arrangements of n ≤ 6 planes in R 4 (first established in [32]), and extend it to horizontal arrangements of n = 7 planes.…”

“…For n = 6, there are 4 nonhorizontal arrangements: L, M, and their mirror images. These arrangements were introduced by Mazurovskiȋ in [34]; further details about them can be found in [32]. For n = 7, there are 13 non-horizontal arrangements, see [2].…”

“…The (complex) characteristic varieties of 2-arrangement groups G = G(A) were studied in [32]. Note that V 1 (G, C) is the hypersurface in (C * ) n defined by the Alexander polynomial of the link L(A); see Penne [36] for another way to compute this polynomial.…”

“…The fundamental group of the complement, G(A) = π 1 (X(A)), has the structure of a semidirect product of free groups: G(A) = F n−1 ⋊ ξ 2 Z, where ξ is a certain pure braid in P n−1 , determined by β, see [32]. Moreover, X(A) is an Eilenberg-MacLane space K(G, 1).…”