Generalized μ-Invariants for Links and Hyperplane Arrangements

Some analogs of Zariski’s Theorem on nodal line arrangements

“…The fundamental group of an ordered projective arrangement is always in this paper endowed with this geometric 1-marking. [32].…”

“…Remark 1.5 If G and G ′ are two finitely presented groups, there is no algorithm to decide whether they are or not isomorphic. However, if G and G ′ have in addition only commutator relations (which is the case for π 1 (M (A))), then there is an algorithmic obstruction theory for an analogous 1,2-marked isomorphism problem, see Papadima [32,Section 4]. Moreover, this reduces to the above ≈ 1 -problem, when G and G ′ are fundamental groups of projective arrangements with the same combinatorics, see [32,Remark 4.7].…”

“…Moreover, it is known that A ≈ B implies π 1 (M (A)) ≈ 1 π 1 (M (B)) with φ = f ♯ ; see Lemma 2.7 from [32]. Remark 1.5 If G and G ′ are two finitely presented groups, there is no algorithm to decide whether they are or not isomorphic.…”

“…However, if G and G ′ have in addition only commutator relations (which is the case for π 1 (M (A))), then there is an algorithmic obstruction theory for an analogous 1,2-marked isomorphism problem, see Papadima [32,Section 4]. Moreover, this reduces to the above ≈ 1 -problem, when G and G ′ are fundamental groups of projective arrangements with the same combinatorics, see [32,Remark 4.7]. The second obstruction in this sense plays a key role in Rybnikov's work [36].…”

“…The first definition was systematically investigated by Papadima in [32], in analogy with classical knot theory. Definition 1.2 Two arrangements A = {H 0 , ..., H n } and B = {H 0 , ..., H n } in the same ambient projective space P N are said to have the same oriented topological type (notation A ≈ B ) if there is an orientation-preserving homeomorphism f : P N → P N , inducing for all k = 0, ..., n orientation-preserving homeomorphisms f |H k :…”