The Orbit Space of a Proper Groupoid Miguel Rodr´ıguez-Olmos EPFL, Switzerland Joint work with Oana Dragulete (EPFL) Rui Loja Fernandes (IST) Tudor S. Ratiu (EPFL) Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 1 / 24 If the action is free, M/G is a smooth manifold in the quotient topology. If the action is not free, M/G is a locally semi-algebraic space endowed with a canonical Whitney stratification. This correspond to a particular case of a groupoid: The action groupoid: G × M ⇒ M and M/G is precisely the orbit space of this action groupoid. Motivation Let G be a Lie group, M a smooth manifold and G × M → M a proper smooth action. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 2 / 24 If the action is not free, M/G is a locally semi-algebraic space endowed with a canonical Whitney stratification. This correspond to a particular case of a groupoid: The action groupoid: G × M ⇒ M and M/G is precisely the orbit space of this action groupoid. Motivation Let G be a Lie group, M a smooth manifold and G × M → M a proper smooth action. If the action is free, M/G is a smooth manifold in the quotient topology. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 2 / 24 This correspond to a particular case of a groupoid: The action groupoid: G × M ⇒ M and M/G is precisely the orbit space of this action groupoid. Motivation Let G be a Lie group, M a smooth manifold and G × M → M a proper smooth action. If the action is free, M/G is a smooth manifold in the quotient topology. If the action is not free, M/G is a locally semi-algebraic space endowed with a canonical Whitney stratification. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 2 / 24 and M/G is precisely the orbit space of this action groupoid. Motivation Let G be a Lie group, M a smooth manifold and G × M → M a proper smooth action. If the action is free, M/G is a smooth manifold in the quotient topology. If the action is not free, M/G is a locally semi-algebraic space endowed with a canonical Whitney stratification. This correspond to a particular case of a groupoid: The action groupoid: G × M ⇒ M Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 2 / 24 Motivation Let G be a Lie group, M a smooth manifold and G × M → M a proper smooth action. If the action is free, M/G is a smooth manifold in the quotient topology. If the action is not free, M/G is a locally semi-algebraic space endowed with a canonical Whitney stratification. This correspond to a particular case of a groupoid: The action groupoid: G × M ⇒ M and M/G is precisely the orbit space of this action groupoid. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 2 / 24 Let −1 Ox = t(s (x)) be the orbit through x ∈ M, and define the equivalence class x ∼ y if Ox = Oy for x, y ∈ M. Then the orbit space of G is M/G := M/ ∼. More generally, let s, t : G ⇒ M be a groupoid. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 3 / 24 and define the equivalence class x ∼ y if Ox = Oy for x, y ∈ M. Then the orbit space of G is M/G := M/ ∼. More generally, let s, t : G ⇒ M be a groupoid. Let −1 Ox = t(s (x)) be the orbit through x ∈ M, Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 3 / 24 Then the orbit space of G is M/G := M/ ∼. More generally, let s, t : G ⇒ M be a groupoid. Let −1 Ox = t(s (x)) be the orbit through x ∈ M, and define the equivalence class x ∼ y if Ox = Oy for x, y ∈ M. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 3 / 24 More generally, let s, t : G ⇒ M be a groupoid. Let −1 Ox = t(s (x)) be the orbit through x ∈ M, and define the equivalence class x ∼ y if Ox = Oy for x, y ∈ M. Then the orbit space of G is M/G := M/ ∼. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 3 / 24 s(g, x) = x t(g, x) = g · x and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 t(g, x) = g · x and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M s(g, x) = x Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M s(g, x) = x t(g, x) = g · x Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M s(g, x) = x t(g, x) = g · x and then the orbit space M/G is exactly M/G. Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 Is it a Whitney stratified space? In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M s(g, x) = x t(g, x) = g · x and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 In that case, what is the global description of the strata? In the case of the action groupoid, G = G × M s(g, x) = x t(g, x) = g · x and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 In the case of the action groupoid, G = G × M s(g, x) = x t(g, x) = g · x and then the orbit space M/G is exactly M/G. QUESTION: What is the structure of M/G for a general (not action) groupoid? In particular Is it a Whitney stratified space? In that case, what is the global description of the strata? Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 4 / 24 2 Proper Lie Groupoids 3 Orbit Space of a Proper Groupoid: Local 4 Orbit Space of a Proper Groupoid: Global 5 Applications Outline 1 Proper Lie Group Actions Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 5 / 24 3 Orbit Space of a Proper Groupoid: Local 4 Orbit Space of a Proper Groupoid: Global 5 Applications Outline 1 Proper Lie Group Actions 2 Proper Lie Groupoids Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 5 / 24 4 Orbit Space of a Proper Groupoid: Global 5 Applications Outline 1 Proper Lie Group Actions 2 Proper Lie Groupoids 3 Orbit Space of a Proper Groupoid: Local Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 5 / 24 5 Applications Outline 1 Proper Lie Group Actions 2 Proper Lie Groupoids 3 Orbit Space of a Proper Groupoid: Local 4 Orbit Space of a Proper Groupoid: Global Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 5 / 24 Outline 1 Proper Lie Group Actions 2 Proper Lie Groupoids 3 Orbit Space of a Proper Groupoid: Local 4 Orbit Space of a Proper Groupoid: Global 5 Applications Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 5 / 24 Outline 1 Proper Lie Group Actions 2 Proper Lie Groupoids 3 Orbit Space of a Proper Groupoid: Local 4 Orbit Space of a Proper Groupoid: Global 5 Applications Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 6 / 24 [ U = Ui , i∈I If i 6= j, Ui ∩ Uj 6= ∅ ⇒ Ui ⊂ Uj . The stratification is called Whitney if for every pair Ui ⊂ Uj , Uj 3 {xk }k∈N → x ∈ Ui ⇒ Txk Uj → V > Tx Ui . N (this requires an embedding of U in R ). Stratifications Definition A topological space S is a stratified space if for every x ∈ S there exists a neighborhood U and a finite family of disjoint locally closed smooth manifolds Ui ⊂ U, i ∈ I such that Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 7 / 24 The stratification is called Whitney if for every pair Ui ⊂ Uj , Uj 3 {xk }k∈N → x ∈ Ui ⇒ Txk Uj → V > Tx Ui .