We use the topology of configuration spaces to give a characterization of Neuwirth-Stallings pairs (S5,K) with dimK = 2. As a consequence, we construct polynomial map germs (R6, 0) → (R3, 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 (R2n, 0) → (Rn, 0) or (R2n+1, 0) → (Rn, 0), n 7ge; 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 ≥ p ≥ 2, a new example of a polynomial map germ (Rn, 0) → (Rp, 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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology