Milnor fibration theorem for differentiable maps

Abstract

AbstractIn Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps $$f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)$$ f : ( R n , 0 ) → ( R k , 0 ) , where $$n\ge k\ge 2$$ n ≥ k ≥ 2 , with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class $$C^{\ell }$$ C ℓ , $$\ell \ge 2$$ ℓ ≥ 2 , with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class $$C^{\ell }$$ C ℓ between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.