Online Talk

The Milnor fiber F of a weighted-homogeneous polynomial f is a smooth affine variety whose topology encodes subtle information about the singularity of f at the origin. A basic question is whether F is formal, in the sense of rational homotopy theory. A realization theorem of Fernández de Bobadilla connects this question to polyhedral products: for every finite simplicial complex K, the moment-angle complex  \mathcal{Z}_K is homotopy equivalent to the Milnor fiber of an explicit  weighted-homogeneous polynomial \Phi_K with trivial monodromy. Combined with the triple Massey products of Baskakov and Denham–Suciu, this yielded non-formal Milnor fibers that are simply connected, but never more than 2-connected. Using instead the Grbić–Linton constructions of n-fold Massey products in moment-angle complexes, I will show that Milnor fibers can be non-formal and arbitrarily highly connected. For an explicit family of such examples, I will also determine the exact formality degree of the Milnor fiber and show that the Kato–Matsumoto connectivity bound is sharp.