Canonical stratification of definable Lie groupoids
Masato Tanabe
Journal of Singularities
volume 26 (2023), 63-75
Received: 10 February 2023. Received in revised form: 21 August 2023.
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 R, or more generally, Shiota's 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.
2020 Mathematical Subject Classification:
Primary 14P10; Secondary 32B20
Key words and phrases:
Semialgebraic sets, subanalytic sets, o-minimal category, X-category, Lie groupoids, orbit spaces, Whitney stratification, and isotopy lemma
Author(s) information:
Masato Tanabe
Department of Mathematics
Graduate school of Science
Hokkaido University
Kita 10, Nishi 8, Kita-ku
Sapporo, Hokkaido, 060-0810, Japan
email: tanabe.masato.i8@elms.hokudai.ac.jp