Direct Product of Free Groups as the Fundamental Group of the Complement of a Union of Lines

摘要

It is well known that the fundamental group of the complement of a complex projective algebraic curve depends on the position of its singularities. Let Σ is contained in CP~2 be a union of projective lines and let G = π_1 (CP~2 Σ). We ask under what conditions will G be independent of the position of the singularities of Σ. The purpose of this paper is to give such a condition. First, we define a topological invariant β(Σ) for Σ. To describe β, we introduce a graph Γ that lies on the arrangement of lines Σ and connects higher singularities (multiplicity ≥ 3) of Σ. This graph in general has more than one component and is not uniquely defined. However, we show that the homotopy type of Γ is independent of our choice, and we define β(Σ) to be the first Betti number of Γ. In Section 3, we prove the following theorem.