THE FUNDAMENTAL GROUP OF THE COMPLEMENT OF THE COMPLEXIFICATION OF A REAL ARRANGEMENT OF HYPERPLANES

Let A be an arrangement of hyperplanes (i.e., a finite set of l-codime nsion vector subspaces) in R-d. We say that the linear ordering of the hyperplanes A, H-1 < H-2 < ... < H-n, is a shellability order of A, i f there is an oriented affine line I crossing the hyperplanes of A on the given linear order. The intersection lattice L(A) is the set of al l intersections of the hyperplanes of A partially ordered by reversed inclusion. Set M(A(c)) := C-d \ boolean OR {H x C: H is an element of A}. We will prove: Suppose that there are shellability orders H-1 < H- 2 < ... < H-n, and H-1' < H-2' <' ... <'H-n', respectively, of A and A ', such that the bijective map H-i --> H-i', i is an element of [n] de termines an isomorphism of the intersection lattices L(A) and L(A'). T hen the fundamental groups pi(1)(M(A(c))) and pi(1)(M(A(c)')) are isom orphic. (C) 1998 Academic Press.