The fundamental group of the complement of the complexification of a real arrangement of hyperplanes
Let \({\mathcal A}\) be a finite collection of lines in \({\mathbb{C}}P^2\). The homotopy type of the complement \(M={\mathbb{C}}P^2 - \bigcup {\mathcal A}\) is determined by the braid monodromy homomorphism \(\phi: F_{s} \to B_{n}\). This homomorphism is defined using a generic projection \({\mathbb{C}}P^{2} \to {\mathbb{C}}P^{1}\). The rank \(s\) of the free group \(F_{s}\) is the number of singular points of the curve \(\bigcup {\mathcal A}\). The monodromy of the projection along a loop \(x\) around the image of one of these singular points in \({\mathbb{C}}P^1\) yields a braid \(\phi(x)\) on \(n\) strands, \(n=| {\mathcal A}| \), realized as a self-diffeomorphism of \({\mathbb{C}}^2\). The fundamental group of \(M\) depends only on the braid monodromy group \(G\), the image of \(\phi\), considered as a subgroup of \(\Aut (F_n)\). Indeed, \(\pi_1(M)\) is the largest quotient of \(F_n\) on which \(G\) acts trivially. The paper under review is concerned with the dependence of \(G\) on the underlying combinatorial structure of \(\mathcal A\). The arrangement \(\mathcal A\) is assumed to be a complexified arrangement defined by real equations. Then the topology of \(M\) is determined by the oriented matroid of the real arrangement, via the associated Salvetti complex. The author defines a combinatorial braid monodromy group \(G_c\) using the Salvetti complex, and shows that it may replace \(G\) in the braid monodromy presentation of \(\pi_1(M).\) The oriented matroid, and the underlying matroid, are determined by a linear order of the lines of \(\mathcal A\). This order is assumed to be a ``realizable shelling order,'' determined by a generic line traversing the lines of the real arrangement. With this additional structure, it is asserted that \(G_c\), and thus the fundamental group of \(M\), depends only on the underlying unoriented matroid. The result applies to arrangements of arbitrary rank, since the fundamental group is determined by a 2-dimensional generic linear section, by the Zariski-Lefschetz Theorem. This last result represents a partial solution to a major open problem in the theory of arrangements. But the proof relies on an assertion which is certainly false. It is stated in Remark 2.2 that a braid \(\beta \in B_n\), identified as an automorphism of \(F_n\), is determined up to inversion by the group \(\langle x_1, \ldots, x_n \mid x_{i}=\beta(x_i)\), \(1 \leq i \leq n \rangle.\) This would imply for instance that knots and links are determined, up to ambient homeomorphism, by their groups. It is possible that the proof can be accomplished without using the erroneous remark. In any case the paper presents an interesting and detailed approach to the study of the braid monodromy of line arrangements via oriented matroids.
scientific article; zbMATH DE number 1221606
scientific article