Smooth surjections and surjective restrictions

TL;DR

This paper explores conditions under which a surjective map between Banach spaces can be restricted to a subspace that preserves surjectivity, focusing on continuous, smooth, and critical value properties.

Contribution

It establishes new results on surjective restrictions for Banach space mappings, especially for continuous, uniformly open, and certain smooth functions.

Findings

Existence of surjective restrictions for continuous, uniformly open maps.

Positive results for $C^1$-smooth surjections with countable critical values.

Counterexamples showing zero-measure critical value sets are insufficient.

Abstract

Given a surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure.