  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.
