Milnor fibre homology complexes

TL;DR

This paper develops algebraic tools to compute the homology and cohomology of Milnor fibres associated with finite Coxeter groups, introducing new algebraic structures and revealing connections to known algebras.

Contribution

It presents a new algebraic presentation of the non-crossing algebra, introduces a larger algebra ilde{A}, and connects it to the Orlik-Solomon and Fomin-Kirillov algebras, enabling simplified homology computations.

Findings

Constructed chain complexes for Milnor fibre homology.

Defined a new algebra ilde{A} related to existing algebras.

Computed multiplicities of representations in homology groups.

Abstract

Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as ``the non-crossing algebra'', which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $\widetilde{A}$, which seems to be ``dual'' to the Fomin-Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between $\widetilde{A}$ and the Orlik-Solomon algebra, in analogy with the fact that the Fomin-Kirillov algebra contains the coinvariant algebra of $W$. This analysis is applied to compute the multiplicities $\langle \rho, H^k(F,\mathbb{C})\rangle_W$ and $\langle \rho, H^k(M,\mathbb{C})\rangle_W$, where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre associated to $W$ and $\rho$ is a representation of $W$.