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

TL;DR

This paper introduces new inverse and implicit function theorems for differentiable maps with isolated critical points, revealing their discreteness and applying algebraic topology for analysis.

Contribution

It establishes optimal inverse and implicit function theorems for maps with isolated critical points, linking differentiability, discreteness, and topology.

Findings

Differentiable maps with isolated critical points are discrete.

New inverse and implicit function theorems are optimal in dimension.

Topological implicit function theorem with unique existence and continuity.

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.