On the homology of the noncrossing partition lattice and the Milnor fibre

TL;DR

This paper constructs explicit bases for the top homology of noncrossing partition lattices related to Coxeter groups and applies these to compute the homology of Milnor fibres and related spaces, revealing algebraic structures similar to Orlik-Solomon algebras.

Contribution

It introduces explicit bases for homology groups of noncrossing partition lattices and defines a multiplicative structure, linking combinatorics with algebraic topology of Milnor fibres.

Findings

Explicit bases for homology groups of noncrossing partition lattices.

A new algebraic structure on Whitney homology resembling Orlik-Solomon algebra.

Computational results on the integral homology of Milnor fibres.

Abstract

Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperplane complement of $W$ and the Artin group of type $W$, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.