Highly connected non-formal Milnor fibers via polyhedral products

TL;DR

This paper combines the realization theorem and Massey product constructions to produce highly connected, non-formal Milnor fibers associated with weighted-homogeneous polynomials, expanding previous connectivity limitations.

Contribution

It introduces a method to generate arbitrarily highly connected, non-formal Milnor fibers using polyhedral products and Massey products, surpassing earlier connectivity constraints.

Findings

Constructed Milnor fibers with arbitrary high connectivity.

Produced non-formal Milnor fibers using Massey products.

Extended previous results from 2-connected to highly connected fibers.

Abstract

We show that the realization theorem of Fern\'andez de Bobadilla, which identifies the Milnor fiber of a weighted-homogeneous polynomial with the complement of a germ of analytic set, can be combined with the systematic Massey product constructions of Grbi\'c-Linton for moment-angle complexes $\mathcal{Z}_K = \mathcal{Z}_K(D^2, S^1)$ to produce weighted-homogeneous polynomials whose Milnor fibers are arbitrarily highly connected and non-formal. The original application of this strategy, due to Fern\'andez de Bobadilla, used the Denham-Suciu classification of lowest-degree triple Massey products and yielded only $2$-connected non-formal Milnor fibers. The Grbi\'c-Linton framework, which constructs non-trivial $n$-fold Massey products in $H^*(\mathcal{Z}_K;\mathbb{Z})$ for arbitrary $n$ and in arbitrary cohomological degrees, removes this connectivity restriction entirely.