The fundamental group of the complement of an arrangement of complex hyperplanes

Arrangements of Hyperplanes

“…This work is reviewed in the survey article by Falk and Randell [81]. The problem was solved for arbitrary complex arrangements by W. Arvola [13]. The problem was solved for arbitrary complex arrangements by W. Arvola [13].…”

“…Their main result asserts that a presentation may be obtained from the underlying real arrangement of A, which we refer to as the real graph r(A). This section is a modified version of his paper [14]. He constructed a generalization of the real graph called the admissible graph, and he proved that a presentation for 7r1(M) may be obtained from an admissible graph.…”

Generalized μ-Invariants for Links and Hyperplane Arrangements

“…The result is proved in [1] only for the case where rkA 3. If rkA < 3, then J 0 = and the result is trivial, since G f1g if rkA 1, and G < 1 F X (with 1-marking given by the classes of the x i ) if rkA 2.…”

“…Theorem 1.4 (Arvola [1]). There is r fr j g j P J , where r j P G 2 F X for all j, such that Gr < 1 G (where the H 1 M-group Gr is constructed as in Example 1.2) and having the property that…”

“…Here x i and l i correspond respectively to geometrically de®ned standard meridian-longitude circle pairs on small 2-tori neighbouring the component K i ; see § 1 for details. In the other case, Arvola showed [1] that G= G k G is the quotient of F X =G k F X by relations r j 1, with j 1; . .…”