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Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finiteCoxeter group $W$. In this paper we construct explicit bases for the tophomology groups of intervals and rank-selected subposets of $\mathcal{L}$. Wedefine a multiplicative structure on the Whitney homology of $\mathcal{L}$ interms of the basis, and the resulting algebra has similarities to theOrlik-Solomon algebra. As an application, we obtain four chain complexes whichcompute the integral homology of the Milnor fibre of the reflection arrangementof $W$, the Milnor fibre of the discriminant of $W$, the hyperplane complementof $W$ and the Artin group of type $W$, respectively. We also tabulate somecomputational results on the integral homology of the Milnor fibres.

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