1 J an 1 99 5 RESEARCH ANNOUNCEMENT

We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is K(π , 1) . These arrangements are the restriction of a one-parameter family of arrangements that arose in the study of tilings of certain centrally symmetric octagons. This other family is discussed as well. I. Definitions and introduction Let A be a finite set of hyperplanes (subspaces of codimension one) passing through the origin in R . The complexification of the arrangement A is the arrangement of hyperplanes in C defined by AC = {H ⊗R C |H ∈ A}. Let M(A) be the complement of AC in C d . We will say that A is K(π , 1) if the space M(A) is a K(π , 1) space; i.e., the universal covering space of M(A) is contractible and the fundamental group π1(M(A)) = π . If A is K(π , 1) , then it is known that the cohomology ring H(M(A) , Z) coincides with the group cohomology H(π , Z) . The braid arrangement A = Ad−1 is the hyperplane arrangement whose hyperplanes are defined by the linear forms {xi − xj = 0 | 1 ≤ i < j ≤ d } . In 1962 Fadell, Fox, and Neuwirth [FaN, FoN] showed that A is K(π , 1) where π is the pure braid group on d strands. Subsequently, Arnold [Ar] gave a simple presentation of the cohomology ring, thereby computing the cohomology of the pure braid group. He conjectured that there was a similar presentation of H(M(A) , Z) for an arbitrary arrangement. Brieskorn [Br] proved this conjecture in 1971 and in 1980 Orlik and Solomon [OS] used these results to give a combinatorial presentation of H(M(A) , Z) . Brieskorn also conjectured that all Coxeter arrangements are K(π , 1) . A Coxeter arrangement is the set of reflecting hyperplanes of a finite group acting on R generated by reflections (see, e.g., [Hu]). In particular the braid arrangement is Received by the editors May 31, 1994. 1991 Mathematics Subject Classification. Primary 52B30, 55P20, 20G10. c ©1995 American Mathematical Society 0273-0979/95 $1.00 + $.25 per page 1 2 P. H. EDELMAN AND VICTOR REINER the Coxeter arrangement for the symmetric group Sd permuting the coordinates in R . Brieskorn proved the latter conjecture for many Coxeter groups, and it was settled in the affirmative by Deligne [De]. In fact Deligne proved the stronger result that if an arrangement A is simplicial, i.e., every connected component of R−∪H∈AH is a union of open rays emanating from the origin whose cross-section is an open simplex, then A is K(π , 1) . (We should note that the condition of being simplicial is not generic.) In a different direction, Falk and Randell [FR] and Terao [Te] showed that a class of arrangements called supersolvable arrangements is also K(π , 1) . A common generalization of Coxeter arrangements and supersolvable arrangements is a free arrangement, which we now define. For each hyperplane H in A , let lH be the linear form in the polynomial ring S = R[x1 , . . . , xd] which vanishes on H (so that lH is uniquely defined up to a scalar multiple). The module of A-derivations D(A) is defined to be the set of all derivations θ : S → S with the property that θ(lH) is divisible by lH for all H in A . D(A) is a module over the polynomial ring S , and we say A is a free arrangement if it is a free module over S . If A is a free arrangement in R , then there exists a homogeneous basis {θ1 , θ2 , . . . , θd} for D(A) and the degrees of these polynomials (with multiplicities) only depend on A . Call this multiset of degrees the exponents of the free arrangement. In the case A is a Coxeter arrangement, these exponents coincide with the usual definition of exponents of a Coxeter group (see, e.g., [Hu, §3.20]). In 1975 Saito [Sa] conjectured that if A is a free arrangement, then it is K(π , 1) . This conjecture does not completely unify what is known about K(π , 1) arrangements because there are simplicial arrangements that are not free. Orlik and Terao [OT, p. 10] remark that this has been one of the two motivating conjectures for most of the recent work on hyperplane arrangements (the other is Terao’s conjecture that the freeness of an arrangement is dependent only on the combinatorial properties of the arrangement [OT, Conjecture 4.138]). In the next section we describe a one-parameter family of 3-dimensional arrangements that are not K(π , 1) , thus disproving Saito’s conjecture. This one-parameter family arises as a restriction of a one-parameter family of 4-dimensional arrangements that have some unusual properties as well. We will discuss this other family in the last section. II. Counterexamples to Saito’s conjecture Consider the hyperplane arrangement Aα in R 3 whose hyperplanes are defined by the linear forms {x , y , z , x− y , x− z , y − z , x− αy , x− α z , y − α z} ,