Thin-thick decomposition for real definable isolated singularities

TL;DR

This paper introduces a new blow-spherical invariant called the thin-thick decomposition for isolated singularity germs, generalizing previous concepts from complex surface singularities to a broader setting.

Contribution

It defines the thin-thick decomposition as a natural blow-spherical invariant, extending the classical thin-thick decomposition to real definable isolated singularities.

Findings

Defines blow-spherical equivalence and tangent links.

Introduces the thin-thick decomposition as a blow-spherical invariant.

Generalizes the thin-thick decomposition from complex surface singularities.

Abstract

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thin-thick decomposition of an isolated singularity germ - which happens to be a natural blow-spherical invariant. This decomposition is a generalization of the thin-thick decomposition of normal complex surface singularity germs introduced in [7]