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

DOI10.1016/J.AIM.2023.108897arXiv2206.05721MaRDI QIDQ6401824

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Publication date: 12 June 2022

Abstract: Let $mathcalL$ 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 $mathcalL$. We define a multiplicative structure on the Whitney homology of $mathcalL$ 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.

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