Combinatorics of Milnor fibres of reflection arrangements

Abstract

Recently, Brady, Falk and Watt introduced a simplicial complex which has the homotopy type of the Milnor fibre F_Q of the reflection arrangement associated to a finite Coxeter group W. This thesis is devoted to developing a combinatorial approach to computing the integral homology groups of F_Q based on this simplicial complex. Our main result is a chain complex of free abelian groups whose integral homology is isomorphic to that of F_Q. Each chain group is isomorphic to a tensor product of the integral group ring ZW with the top reduced homology group of a rank-selected subposet of the noncrossing partition (NCP) lattice of W. Associated to the NCP lattice of W we define two isomorphic graded Z-algebras A and B, which have similarities to the Orlik-Solomon algebra and characterise the homology of the NCP lattice. The algebra A is defined in terms of generators and relations, while the algebra B is defined in a combinatorial manner which has to do with the Hurwitz actions. In particular, each element of B (or A) produces an explicit cycle of the top reduced homology group of the corresponding interval or rank-selected subposet of the NCP lattice. This permits us to calculate the homology of our chain complex computationally. The actions of both W and the monodromy may also be partly described by our chain complex. In particular, we prove that the homology of the subcomplex of W-invariant chain groups is isomorphic to the homology of F_P=F_Q/W, the Milnor fibre of the discriminant of W. This recovers the result of Brady, Falk and Watt.