Non-crossing partitions and Milnor fibers

TL;DR

This paper constructs cell complexes from non-crossing partitions of finite real reflection groups to study Milnor fibers, revealing their homotopy types, monodromy actions, and homology through shellability.

Contribution

It introduces a novel approach linking non-crossing partitions to Milnor fibers, providing explicit complexes and homological computations for reflection group arrangements.

Findings

Constructed finite cell complexes homotopy equivalent to Milnor fibers.

Established natural cyclic group actions representing monodromy.

Derived chain complexes computing integral homology of Milnor fibers.

Abstract

For a finite real reflection group $W$ we use non-crossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$-discriminant $\Delta_W$ 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 non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of $\Delta_W$.