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For a finite real reflection group, W, we use non-crossing partitions of type W to construct a finite cell complex with the homotopy type of the Milnor fiber of the associated W–discriminant, Δ_W, and another with the homotopy type of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of Δ_W.
Brady, T., Falk, M. J., & Watt, C. (2018). Noncrossing partitions and milnor fibers. Algebraic and Geometric Topology, 18(7), 3821-3838. DOI: 10.2140/agt.2018.18.3821