Noncrossing partitions and Milnor fibers

Abstract

For a finite real reflection group $W$ we use noncrossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$–discriminant ${\mathrm{\Delta }}_W$ and that 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 noncrossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of ${\mathrm{\Delta }}_W$.