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

الأصل · EN

We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs (S⁵, K) with K = 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 ≥ 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.