One of the fundamental problems in the topological study of polynomial functions

- Zariski and Van Kampen, in the early 1930's, gave an algorithm for finding a finite presentation for p₁(C^l-V(f)).

- Libgober (1994) extended the Zariski-Van Kampen method to give information about p_k(C^l-V(f))?Q for k ³ 1 when V(f) is irreducible.

- Otherwise, much less is known.

We concentrate on the case where f=?_i f_i and deg(f_i)=1, i.e., V(f) is associated to a hyperplane arrangement.The cohomology of the complement is determined by combinatorics. As we noticed, this is not true for the fundamental group and very few is known about higher homotopy groups. We now focus on the following problem.Problem: Characterize the arrangements such that the complement is a K(p,1)-space.Such arrangements are called K(p,1)- arrangements.