New examples of Neuwirth-Stallings pairs and non-trivial real Milnor fibrations

TL;DR

This paper characterizes Neuwirth--Stallings pairs using configuration space topology and constructs polynomial map germs with non-trivial Milnor fibers, addressing longstanding questions about their topology and homotopy types.

Contribution

It provides a topological characterization of Neuwirth--Stallings pairs and constructs explicit polynomial map germs with non-trivial Milnor fibers, solving Milnor's non-triviality question.

Findings

Constructed polynomial germs with non-disk Milnor fibers.

Characterized Neuwirth--Stallings pairs via configuration spaces.

Identified conditions for Milnor fibers to be bouquets of spheres.

Abstract

We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs $(S^5, K)$ with $\dim K = 2$. As a consequence, we construct polynomial map germs $(\mathbb{R}^{6},0)\to (\mathbb{R}^{3},0)$ with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a disk, thus putting an end to Milnor's non-triviality question. Furthermore, for a polynomial map germ $(\mathbb{R}^{2n},0) \to (\mathbb{R}^{n},0)$ or $(\mathbb{R}^{2n+1},0) \to (\mathbb{R}^{n},0)$, $n \geq 3$, with an isolated singularity at the origin, we study the conditions under which the associated Milnor fiber has the homotopy type of a bouquet of spheres. We then construct, for every pair $(n, p)$ with $n/2 \geq p \geq 2$, a new example of a polynomial map germ $(\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ with an isolated singularity at the origin such that its Milnor fiber has the homotopy type of a bouquet of a positive number of spheres.