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

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 $({\mathbb{R}}^{2n+1},0)\to ({\mathbb{R}}^n,0)$, $n\ge 3$, 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 $n/2\ge p\ge 2$, 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.

18 pages, 1 figure in the page 9

References in corpus (2)

• Real polynomial maps and singular open books at infinity
• Topology of real Milnor fibration for non-isolated singularities

Cited by in corpus (3)

• Fibrations of highly singular map germs
• Manifolds which admit maps with finitely many critical points into spheres of small dimensions
• On the automorphisms group of the asymptotic pants complex of an infinite surface of genus zero