Real line arrangements and fundamental groups
نویسنده
چکیده
Let A be a real line arrangement in P(R), and let AC be its complexification. Let CC be the complement P (C) \ ⋃ AC. Let G be the Galois group of C/R. We construct a G-equivariant 2-dimensional strong deformation retract of CC. As an application, we give an explicit presentation of the orbifold fundamental group π1(CC//G), and deduce from it an explicit presentation of the ordinary fundamental group π1(CC). MSC 2000: 14P25, 52C30, 57M05
منابع مشابه
Π1-classification of Real Arrangements with up to Eight Lines
One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan [Fa2], and it turns out that the incidence lattice of such arrangements determines the projective fundamental group. We use actions on the set of wiring diagrams, int...
متن کاملThe Fundamental Group’s Structure of the Complement of Some Configurations of Real Line Arrangements
In this paper, we give a fully detailed exposition of computing fundamental groups of complements of line arrangements using the Moishezon-Teicher technique for computing the braid monodromy of a curve and the Van-Kampen theorem which induces a presentation of the fundamental group of the complement from the braid monodromy of the curve. For example, we treated the cases where the arrangement h...
متن کاملOn the Connection between Affine and Projective Fundamental Groups of Line Arrangements and Curves
In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a real line arrangement. We give some applications to this result.
متن کاملFundamental Groups of Some Special Quadric Arrangements
Abstract. Continuing our work on the fundamental groups of conic-line arrangements [3], we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P. The first arrangement is a union of n conics, which are tangent to each other at two common points. The second arrangement is composed of n quadrics which are tangent to each other at one common p...
متن کاملOn the Fundamental Group of the Complement of a Complex Hyperplane Arrangement
We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. In the proof we use a new invariant of the fundamental group of the complement of a line arrangement with prescribed combinatorial type with respect to isomorphisms inducing the canonical isomorphism of first homology groups.
متن کامل