On the K(π,1)-problem for restrictions of complex reflection arrangements

Abstract

Let W⊂GL(V) be a complex reflection group and A(W) the set of the mirrors of the complex reflections in W. It is known that the complement X(A(W)) of the reflection arrangement A(W) is a K(π,1) space. For Y an intersection of hyperplanes in A(W), let X(A(W)Y) be the complement in Y of the hyperplanes in A(W) not containing Y. We hope that X(A(W)Y) is always a K(π,1). We prove it in case of the monomial groups W=G(r,p,ℓ). Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this K(π,1) property remains to be proved.