Deficient and multiple points of maps into a manifold

TL;DR

This paper investigates the properties of deficient and multiple points of maps into manifolds, providing dimension estimates and density results, along with examples illustrating the structure of these point sets.

Contribution

It offers new estimates on the dimension of deficient points and analyzes the density of multiple points, including examples where their complements are dense.

Findings

Dimension estimates for deficient points.

Density results for multiple points.

Examples with dense complements of multiple points.

Abstract

For a map $f:X \to M$ into a manifold $M$, we study the sets of deficient and multiple points of $f$. In case of the set of deficient points, we estimate its dimension. For multiple points, we study its density in $X$, and we also provide examples where its complement is dense.