Éclatements quasi homogènes
In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into , where is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in . Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class with smooth boundary.
Equivalence and zero sets of certain maps on infinite dimensional spaces are studied using an approach similar to the deformation lemma from the singularity theory.
We show under some assumptions that a differentiable function can be transformed globally to a polynomial or a rational function by some diffeomorphism. One of the assumptions is that the function is proper, the number of critical points is finite, and the Milnor number of the germ at each critical point is finite.