Kurdyka-{\L}ojasiewicz functions and mapping cylinder neighborhoods

TL;DR

This paper proves that the zero sets of Kurdyka-Łojasiewicz functions have a specific topological neighborhood structure, excluding certain wild embeddings like Alexander horned spheres from being zero loci of such functions.

Contribution

It establishes that zero loci of KŁ functions admit mapping cylinder neighborhoods, linking function properties to topological embedding constraints.

Findings

Zero loci of KŁ functions have mapping cylinder neighborhoods.

Wildly embedded 2-manifolds like Alexander horned spheres cannot be zero loci of KŁ functions.

Kurdyka-Łojasiewicz functions include subanalytic, transnormal, and Morse functions.

Abstract

Kurdyka-{\L}ojasiewicz (K{\L}) functions are real-valued functions characterized by a differential inequality involving the norm of their gradient. This class of functions is quite rich, containing objects as diverse as subanalytic, transnormal or Morse functions. We prove that the zero locus of a Kurdyka-{\L}ojasiewicz function admits a mapping cylinder neighborhood. This implies, in particular, that wildly embedded topological 2-manifolds in 3-dimensional Euclidean space, such as Alexander horned spheres, do not arise as the zero loci of K{\L} functions.