Critical points and surjectivity of smooth maps

TL;DR

This paper investigates the structure of critical points in smooth maps between manifolds, revealing that either all points are critical or the critical set has high dimension, with implications for the map's surjectivity.

Contribution

It establishes a dichotomy for the critical points of smooth maps and explores consequences for the surjectivity of such mappings.

Findings

Either all points are critical points or the critical set has dimension at least n-1.

The result constrains the structure of smooth maps with non-surjective images.

Implications for the surjectivity of smooth maps are discussed.

Abstract

Let $f:M^m\to N^n$ be a smooth map between two differential manifolds with $N$ connected, $f(M)$ closed and $f(M)\neq N$. In this short note, we show that either all the points of $M$ are critical points of $f$ or the dimension the collection of all critical points of $f$ is not less than $n-1$. Some consequences of this result for surjectivity of mappings are also presented.