Milnor fiber complexes for the exceptional Shephard groups

The symmetry group of a regular real polytope is a finite Coxeter group. The intersection of the unit sphere with the reflecting hyperplanes of the corresponding Coxeter arrangement induces a simplicial triangulation of the sphere, called the Coxeter complex. A Shephard group G is the symmetry group of a regular complex polytope. Orlik has shown that there can be associated to G a real simplicial complex which possesses many properties analogous to the Coxeter complex. Let f1 be the G-invariant polynomial of minimal positive degree, and let F=f1−1(1) be its Milnor fiber. Orlik showed that there is a complex Γ which is an equivariant strong deformation retract of F, is G-stable, is stratified by the associated reflection arrangement, and satisfies specific cell-counting formulas. His proof is existential; it does not give an explicit method of constructing the complex, though Orlik and Solomon did explicitly construct complexes for one infinite family of Shephard groups. Here we describe the construction of complexes for the remaining 15 “exceptional” Shephard groups.