Canonical stratification of definable Lie groupoids

TL;DR

This paper establishes a canonical Whitney stratification for definable Lie groupoids within tame topology frameworks, generalizing classical results on algebraic group actions by employing Shiota's isotopy and approximation theorems.

Contribution

It introduces a precise tame topology counterpart to the canonical stratification of Lie groupoids, extending and refining prior algebraic group action results.

Findings

Existence of a canonical Whitney stratification for definable Lie groupoids.

Stratification invariance under groupoid actions.

Use of Shiota's theorems to refine classical proofs.

Abstract

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over $\mathbb{R}$, or more generally, Shiota's $\mathfrak{X}$-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J. N. Mather and V. A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.