A geometric proof of the existence of Whitney stratifications

TL;DR

This paper provides a concise geometric proof for the existence of Whitney stratifications of singular sets, simplifying previous complex proofs by avoiding advanced results and using classical theorems like Thom's transversality.

Contribution

It introduces a new geometric proof of Whitney stratification existence that relies on Thom's transversality theorem and Milnor's curve selection lemma, bypassing complex prior results.

Findings

Existence of Whitney stratifications proven geometrically

Simplifies previous proofs by avoiding advanced results

Utilizes classical theorems for a more accessible proof

Abstract

A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has a stratification. This result was extended by Lojasiewicz to real (semi)analytic sets. In this paper we present a short geometric proof of existence of stratifications based on Thom's transversality theorem and Milnor's curve selection lemma and not relying on difficult results of Lojasiewicz.