New inverse and implicit function theorems for differentiable maps with isolated critical points

Abstract

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with isolated critical points are discrete maps, which means that algebraic topology methods could then be deployed to explore relevant questions. We also provide a purely topological version of implicit function theorem for continuous maps that still possesses unique existence and continuity. All new results of the paper are optimal with respect to the choice of dimensions.

Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces

“…In sharp contrast to the infinite dimensional scenario, differentiable maps with isolated critical points do have nice properties in finite dimensional spaces. Apart from the open mapping property mentioned in the Introduction, differentiable vector fields with isolated critical points on Euclidean spaces are local homeomorphisms provided the dimension of the ambient space is higher than two [3,17]. Question 8.2.…”