Klein Stratification of Orbit Spaces: Examples

Year 2023, Volume: 16 Issue: 2, 665 - 671, 29.10.2023

Serap Gürer

https://doi.org/10.36890/iejg.1364214

Abstract

We consider the stratification of orbit spaces as defined by the local diffeomorphism action, known as the Klein stratification. We demonstrate that the Klein strata on orbit spaces with isolated point singularities are precisely the union of points where the space has the same dimension.

Keywords

Orbit Spaces, stratifications, diffeology

References

• [1] Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, New York and London, (1972).
• [2] Gürer, S., Iglesias-Zemmour, P.: Orbifolds as stratified diffeologies. Differential Geometry and its Applications. 86, (2023).
• [3] Iglesias-Zemmour, P.: Fibrations difféologiques et homotopie, State doctorate dissertation. Université de Provence, Marseille. (1985).
• [4] Iglesias, P.: Les SO(3)-variétés symplectiques et leur classification en dimension 4. Bull. Soc. Math. France. 119 (4), 371-396 (1991).
• [5] Iglesias-Zemmour, P.: Dimesion in Diffeology. Indagationes Mathematicae. 18 (4), 555-560. (2007).
• [6] Iglesias-Zemmour, P..: ODiffeology. Mathematical Surveys and Monographs. The American Mathematical Society. 185, (2013).
• [7] Iglesias-Zemmour, P.: Klein Stratification of Diffeological Spaces. Blog post. (2022). http://math.huji.ac.il/~piz/documents/ DBlog-Rmk-KSODS.pdf
• [8] Karshon, Y.,Kuriko, S.: Classification of locally standard torus manifolds up to equivariant diffeomorphism. In Preperation.
• [9] Palais, R.: On the Existence of Slices for Actions of Non-Compact Lie Groups. Annals of Mathematics Second Series. 73 (2), 295-323 (1961).
• [10] Pflaum, M.J.: Analytic and Geometric Study of Stratified Spaces. Lecture Notes in Mathematics. 1768 (1), 1-8 (2001).