Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space?

edit: after some search, I found the proof should be contained in either

GIBSON, C.G., WIRTHMULLER, K., DU PLESSIS, A.A., LOOIJENGA, E.J.N.: Topological stability of smooth mappings. Lecture Notes in Math.552, Springer-Verlag, Berlin-Heidelberg-New York, 1976

or in

LELLMAN, N.W.: Orbitraume von G-Mannigfaltigkeiten und stratifizierte Mengen. Diplomarbeit, Bonn 1975

However, for the first reference I haven't found where exactly the proof is; for the second reference, I couldn't find a copy on the internet.


1 Answer 1


Maybe of interest, in the paper "Orbispaces as differentiable stratified spaces." Lett. Math. Phys. 108 (2018), no. 3, 805–859, Crainic and Mestre prove in proposition 26 that for proper Lie groupoid $$ G\rightrightarrows M $$ $M$ and the orbit space $X=M/G$ together with the canonical stratifications, are differentiable stratified spaces. Moreover, the canonical stratifications of $M$ and $X$ are Whitney stratifications.

And Whitney stratifications induce the data satisfied by Thom-Mather stratified spaces (this was proved by Mather).

See also Watt's paper : "The differential structure of an orbifold". Rocky Mountain J. Math. 47 (2017), no. 1, 289–327.

  • $\begingroup$ Thanks. I think in the Crainic-Mestre paper the notion of Whitney stratification is extended from subsets of smooth manifolds to "differentiable spaces." It is not a priori clear to me why Whitney stratifications on differentiable spaces induce the structures required for Thom-Mather stratified space. $\endgroup$
    – UVIR
    Commented Nov 22, 2021 at 2:24

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