Topology of Complex Reflection Arrangements

Let V be a finite dimensional complex vector space and W ⊂ GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. A classical conjecture predicts that V reg is a K(π, 1) space. When W is a complexified real reflection group, the conjecture follows from a theorem of Deligne, [20]. Our main result validates the conjecture for duality (or, equivalently, well-generated) complex reflection groups. This includes the complexified real case (but our proof is new) and new cases not previously known. We also address a number of questions about π1(W\\V ), the braid group of W . Let V be a finite dimensional complex vector space and W ⊂ GL(V ) be an irreducible complex reflection group (all reflection groups considered here are assumed to be finite). Let d1 ≤ · · · ≤ dn be the degrees of W . Let d1 ≥ · · · ≥ dn = 0 be the codegrees of W . We say that W is a duality group if di + d ∗ i = dn for all i (by analogy with the real case, we say that dn is the Coxeter number of W and we often denote it by h). We say that W is well-generated if it may be generated by n reflections. Orlik-Solomon observed, by inspecting the classification of Shephard-Todd, that W is a duality group ⇔ W is well-generated. Let V reg be the complement in V of the reflecting hyperplanes. In the case when W is a type A reflection group, Fadell and Neuwirth proved in the early 1960’s that V reg is a K(π, 1) (this is an elementary use of fibration exact sequences, [22]). Brieskorn conjectured in 1971, [13], that the K(π, 1) property holds when W is a complexified real reflection group. It is not clear who first stated the conjecture for arbitrary complex reflection groups. The conjecture may be found in Orlik-Terao’s book: Conjecture 0.1 ([29], p. 163 & p. 259). The universal cover of V reg is contractible. The complexified real case (i.e., Brieskorn’s conjecture) was quickly settled by Deligne, [20]. The rank 2 case is trivial. The case of the infinite family was solved in 1983 by Nakamura, [26] (here again, the monomiality of the group allows an efficient use of fibrations). A few other cases immediately follow from the observation by Orlik-Solomon, [28], that certain orbifolds of non-real complex reflection groups are isomorphic to orbifolds of complexified real reflection groups. Combining all previously known results, the conjecture remained open for six exceptional types: G24, G27, G29, G31, G33 and G34. Our main result is: Theorem 0.2. Assume that W is a well-generated complex reflection group. The universal cover of V reg is contractible. Five of the six open cases are duality groups: G24, G27, G29, G33 and G34. The theorem also applies to the complexified real case, for which we obtain a new proof, not relying