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 .
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 [ U = Ui , i∈I If i 6= j, Ui ∩ Uj 6= ∅ ⇒ Ui ⊂ Uj .
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 7 / 24 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 [ 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 ).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 7 / 24 The manifolds Sk are called the strata of the stratification.
If S is a stratified space then there is a family of disjoint locally closed smooth manifolds Sk , k ∈ IS such that for every k ∈ IS
Sk ∩ U = Ui , for some i ∈ IU .
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 8 / 24 If S is a stratified space then there is a family of disjoint locally closed smooth manifolds Sk , k ∈ IS such that for every k ∈ IS
Sk ∩ U = Ui , for some i ∈ IU .
The manifolds Sk are called the strata of the stratification.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 8 / 24 Local: A proper Lie group action admits slices and tubes (Palais). Then If x ∈ M, then there is a neighborhood U of [x] in M/G such that
U ' S/Gx
where S is a linear slice for the G-action at x and Gx is the stabilizer of x which has a linear representation on S.
Since the action is proper Gx is compact, therefore using Invariant theory (Hilbert, Schwartz, Tarski-Seidenberg,...) U is a semi-algebraic Whitney stratified space (isotropy stratification)
Stratifications for Lie group actions
Let G be a Lie group, M a smooth manifold and
G × M → M
a proper smooth action. Let π : M → M/G be the projection.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 9 / 24 If x ∈ M, then there is a neighborhood U of [x] in M/G such that
U ' S/Gx
where S is a linear slice for the G-action at x and Gx is the stabilizer of x which has a linear representation on S.
Since the action is proper Gx is compact, therefore using Invariant theory (Hilbert, Schwartz, Tarski-Seidenberg,...) U is a semi-algebraic Whitney stratified space (isotropy stratification)
Stratifications for Lie group actions
Let G be a Lie group, M a smooth manifold and
G × M → M
a proper smooth action. Let π : M → M/G be the projection.
Local: A proper Lie group action admits slices and tubes (Palais). Then
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 9 / 24 Since the action is proper Gx is compact, therefore using Invariant theory (Hilbert, Schwartz, Tarski-Seidenberg,...) U is a semi-algebraic Whitney stratified space (isotropy stratification)
Stratifications for Lie group actions
Let G be a Lie group, M a smooth manifold and
G × M → M
a proper smooth action. Let π : M → M/G be the projection.
Local: A proper Lie group action admits slices and tubes (Palais). Then If x ∈ M, then there is a neighborhood U of [x] in M/G such that
U ' S/Gx
where S is a linear slice for the G-action at x and Gx is the stabilizer of x which has a linear representation on S.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 9 / 24 Stratifications for Lie group actions
Let G be a Lie group, M a smooth manifold and
G × M → M
a proper smooth action. Let π : M → M/G be the projection.
Local: A proper Lie group action admits slices and tubes (Palais). Then If x ∈ M, then there is a neighborhood U of [x] in M/G such that
U ' S/Gx
where S is a linear slice for the G-action at x and Gx is the stabilizer of x which has a linear representation on S.
Since the action is proper Gx is compact, therefore using Invariant theory (Hilbert, Schwartz, Tarski-Seidenberg,...) U is a semi-algebraic Whitney stratified space (isotropy stratification)
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 9 / 24 The connected components of M(H) are submanifolds of M for every H ⊂ G. [ M = M(H) is a locally finite disjoint partition. (H)
The connected components of π(M(H)) are the smooth strata of the isotropy stratification of M/G.
Global: Let
M(H) = {x ∈ M : Gx is conjugate to H} (orbit types).
Then
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 10 / 24 [ M = M(H) is a locally finite disjoint partition. (H)
The connected components of π(M(H)) are the smooth strata of the isotropy stratification of M/G.
Global: Let
M(H) = {x ∈ M : Gx is conjugate to H} (orbit types).
Then
The connected components of M(H) are submanifolds of M for every H ⊂ G.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 10 / 24 The connected components of π(M(H)) are the smooth strata of the isotropy stratification of M/G.
Global: Let
M(H) = {x ∈ M : Gx is conjugate to H} (orbit types).
Then
The connected components of M(H) are submanifolds of M for every H ⊂ G. [ M = M(H) is a locally finite disjoint partition. (H)
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 10 / 24 Global: Let
M(H) = {x ∈ M : Gx is conjugate to H} (orbit types).
Then
The connected components of M(H) are submanifolds of M for every H ⊂ G. [ M = M(H) is a locally finite disjoint partition. (H)
The connected components of π(M(H)) are the smooth strata of the isotropy stratification of M/G.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 10 / 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 11 / 24 Therefore we cannot define orbit types M(H). We need a different approach
Tube theorem + Foliation theory
Let s, t : G ⇒ M a Lie groupoid.
The analogous construction to the Lie group case cannot be used for studying M/G since the stabilizers
−1 −1 Gx = s (x) ∩ t (x) cannot be compared by conjugation at points lying in different orbits.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 12 / 24 Let s, t : G ⇒ M a Lie groupoid.
The analogous construction to the Lie group case cannot be used for studying M/G since the stabilizers
−1 −1 Gx = s (x) ∩ t (x) cannot be compared by conjugation at points lying in different orbits.
Therefore we cannot define orbit types M(H). We need a different approach
Tube theorem + Foliation theory
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 12 / 24 Theorem (Weinstein, Zung) Let G ⇒ M be a source locally trivial proper groupoid and x ∈ M with orbit O. Then there is an a action of GO on NO = TOM/T O, with associated action groupoid
GO n NO ⇒ NO and
G is locally isomorphic to GO n NO.
Tube theorem for proper groupoids
We will assume the following conditions for the Lie groupoid s, t : G ⇒ M: (s, t): G → M × M is a proper map. (proper groupoid) s is locally trivial. (source local triviality) Every orbit of G is of finite type.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 13 / 24 Then there is an a action of GO on NO = TOM/T O, with associated action groupoid
GO n NO ⇒ NO and
G is locally isomorphic to GO n NO.
Tube theorem for proper groupoids
We will assume the following conditions for the Lie groupoid s, t : G ⇒ M: (s, t): G → M × M is a proper map. (proper groupoid) s is locally trivial. (source local triviality) Every orbit of G is of finite type.
Theorem (Weinstein, Zung) Let G ⇒ M be a source locally trivial proper groupoid and x ∈ M with orbit O.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 13 / 24 Tube theorem for proper groupoids
We will assume the following conditions for the Lie groupoid s, t : G ⇒ M: (s, t): G → M × M is a proper map. (proper groupoid) s is locally trivial. (source local triviality) Every orbit of G is of finite type.
Theorem (Weinstein, Zung) Let G ⇒ M be a source locally trivial proper groupoid and x ∈ M with orbit O. Then there is an a action of GO on NO = TOM/T O, with associated action groupoid
GO n NO ⇒ NO and
G is locally isomorphic to GO n NO.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 13 / 24 Theorem Let G ⇒ M be a source locally trivial proper groupoid, and x ∈ M.
Then the action of GO on NO restricts to a representation of Gx on Nx O −1 −1 (Gx = s (x) ∩ t (x) is the stabilizer, a compact Lie group) with associated action groupoid
Gx × Nx O ⇒ Nx O, and
G is locally Morita equivalent to Gx × Nx O.
Morita equivalence
Using the Tube theorem we can prove that every proper groupoid is locally Morita equivalent to an action groupoid for a representation of a compact group on a vector space.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 14 / 24 Then the action of GO on NO restricts to a representation of Gx on Nx O −1 −1 (Gx = s (x) ∩ t (x) is the stabilizer, a compact Lie group) with associated action groupoid
Gx × Nx O ⇒ Nx O, and
G is locally Morita equivalent to Gx × Nx O.
Morita equivalence
Using the Tube theorem we can prove that every proper groupoid is locally Morita equivalent to an action groupoid for a representation of a compact group on a vector space. Theorem Let G ⇒ M be a source locally trivial proper groupoid, and x ∈ M.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 14 / 24 G is locally Morita equivalent to Gx × Nx O.
Morita equivalence
Using the Tube theorem we can prove that every proper groupoid is locally Morita equivalent to an action groupoid for a representation of a compact group on a vector space. Theorem Let G ⇒ M be a source locally trivial proper groupoid, and x ∈ M.
Then the action of GO on NO restricts to a representation of Gx on Nx O −1 −1 (Gx = s (x) ∩ t (x) is the stabilizer, a compact Lie group) with associated action groupoid
Gx × Nx O ⇒ Nx O, and
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 14 / 24 Morita equivalence
Using the Tube theorem we can prove that every proper groupoid is locally Morita equivalent to an action groupoid for a representation of a compact group on a vector space. Theorem Let G ⇒ M be a source locally trivial proper groupoid, and x ∈ M.
Then the action of GO on NO restricts to a representation of Gx on Nx O −1 −1 (Gx = s (x) ∩ t (x) is the stabilizer, a compact Lie group) with associated action groupoid
Gx × Nx O ⇒ Nx O, and
G is locally Morita equivalent to Gx × Nx O.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 14 / 24 2 Form the bi-bundle
−1 GO n NO s (x) × Nx O Gx × Nx O U π1 iii UUU π2 iiii UUUU iiii UUUU iii UUUU tiiii UU* NO Nx O
with π1(g, v) = g · v and−π1 2(g, v) = v. −1 (Gx × Nx O) × (s (x) × Nx O) → (s (x) × Nx O) −1 3 These two actions are free,−1 they0 commute and the−10 momentum0 map of (GO n NO) × (s (g(x, )v)×· N(gx,Ov)) →7→ ((sgg (x), g× N· vx)O) one is the orbit map of the other. (g 0, v 0) · (g, v) 7→ (g 0g, v),
Proof.
1 By the Tube theorem, G is locally isomorphic to GO n NO
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 15 / 24 with π1(g, v) = g · v and−π1 2(g, v) = v. −1 (Gx × Nx O) × (s (x) × Nx O) → (s (x) × Nx O) −1 3 These two actions are free,−1 they0 commute and the−10 momentum0 map of (GO n NO) × (s (g(x, )v)×· N(gx,Ov)) →7→ ((sgg (x), g× N· vx)O) one is the orbit map of the other. (g 0, v 0) · (g, v) 7→ (g 0g, v),
Proof.
1 By the Tube theorem, G is locally isomorphic to GO n NO 2 Form the bi-bundle
−1 GO n NO s (x) × Nx O Gx × Nx O U π1 iii UUU π2 iiii UUUU iiii UUUU iii UUUU tiiii UU* NO Nx O
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 15 / 24 −1 −1 (Gx × Nx O) × (s (x) × Nx O) → (s (x) × Nx O) −1 3 These two actions are free,(g they0, v) · commute(g, v) 7→ and(gg the0 momentum, g 0 · v) map of one is the orbit map of the other.
Proof.
1 By the Tube theorem, G is locally isomorphic to GO n NO 2 Form the bi-bundle
−1 GO n NO s (x) × Nx O Gx × Nx O U π1 iii UUU π2 iiii UUUU iiii UUUU iii UUUU tiiii UU* NO Nx O
with π1(g, v) = g · v and π2(g, v) = v.
−1 −1 (GO n NO) × (s (x) × Nx O) → (s (x) × Nx O) (g 0, v 0) · (g, v) 7→ (g 0g, v),
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 15 / 24 with π1(g, v) = g · v and π2(g, v) = v.
3 These two actions are free,−1 they commute and the−1 momentum map of (GO n NO) × (s (x) × Nx O) → (s (x) × Nx O) one is the orbit map of the other. (g 0, v 0) · (g, v) 7→ (g 0g, v),
Proof.
1 By the Tube theorem, G is locally isomorphic to GO n NO 2 Form the bi-bundle
−1 GO n NO s (x) × Nx O Gx × Nx O U π1 iii UUU π2 iiii UUUU iiii UUUU iii UUUU tiiii UU* NO Nx O
−1 −1 (Gx × Nx O) × (s (x) × Nx O) → (s (x) × Nx O) (g 0, v) · (g, v) 7→ (gg 0−1, g 0 · v)
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 15 / 24 with π1(g, v) = g · v and−π1 2(g, v) = v. −1 (Gx × Nx O) × (s (x) × Nx O) → (s (x) × Nx O) −1 0 −10−1 0 (GO n NO) × (s (g(x, )v)×· N(gx,Ov)) →7→ ((sgg (x), g× N· vx)O) (g 0, v 0) · (g, v) 7→ (g 0g, v),
Proof.
1 By the Tube theorem, G is locally isomorphic to GO n NO 2 Form the bi-bundle
−1 GO n NO s (x) × Nx O Gx × Nx O U π1 iii UUU π2 iiii UUUU iiii UUUU iii UUUU tiiii UU* NO Nx O
3 These two actions are free, they commute and the momentum map of one is the orbit map of the other.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 15 / 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 16 / 24 Therefore locally the orbit space of G is a quotient for a representation of a compact Lie group.
In particular Theorem The orbit space for a source locally trivial proper groupoid is a locally semi-algebraic Whitney stratified space.
Since G is locally Morita equivalent to Gx × Nx O then near [x]
M/G ∼ Nx O/Gx (local homeomorphism).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 17 / 24 In particular Theorem The orbit space for a source locally trivial proper groupoid is a locally semi-algebraic Whitney stratified space.
Since G is locally Morita equivalent to Gx × Nx O then near [x]
M/G ∼ Nx O/Gx (local homeomorphism).
Therefore locally the orbit space of G is a quotient for a representation of a compact Lie group.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 17 / 24 Since G is locally Morita equivalent to Gx × Nx O then near [x]
M/G ∼ Nx O/Gx (local homeomorphism).
Therefore locally the orbit space of G is a quotient for a representation of a compact Lie group.
In particular Theorem The orbit space for a source locally trivial proper groupoid is a locally semi-algebraic Whitney stratified space.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 17 / 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 18 / 24 Gx A(Nx O) = {X + Y : X ∈ X (Nx O), Y tangent to orbits}. It is well-known that Lemma
A(Nx O) defines a singular integrable distribution,
its leaves are Gx -invariant,
the strata of Nx O/Gx are the projection of the leaves of A(Nx O).
With this Lemma and the Morita equivalence we can prove: Theorem Let G ⇒ M be a source locally trivial proper groupoid. Then Gbas(M) induces a singular integrable distribution,
the strata of M/G are the projections of the leaves of Gbas(M).
Define the following families of vector fields G G Gbas(M) = {X ∈ X(M): Xf ∈ C (M) ∀f ∈ C (M)},
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 19 / 24 It is well-known that Lemma
A(Nx O) defines a singular integrable distribution,
its leaves are Gx -invariant,
the strata of Nx O/Gx are the projection of the leaves of A(Nx O).
With this Lemma and the Morita equivalence we can prove: Theorem Let G ⇒ M be a source locally trivial proper groupoid. Then Gbas(M) induces a singular integrable distribution,
the strata of M/G are the projections of the leaves of Gbas(M).
Define the following families of vector fields G G Gbas(M) = {X ∈ X(M): Xf ∈ C (M) ∀f ∈ C (M)}, Gx A(Nx O) = {X + Y : X ∈ X (Nx O), Y tangent to orbits}.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 19 / 24 With this Lemma and the Morita equivalence we can prove: Theorem Let G ⇒ M be a source locally trivial proper groupoid. Then Gbas(M) induces a singular integrable distribution,
the strata of M/G are the projections of the leaves of Gbas(M).
Define the following families of vector fields G G Gbas(M) = {X ∈ X(M): Xf ∈ C (M) ∀f ∈ C (M)}, Gx A(Nx O) = {X + Y : X ∈ X (Nx O), Y tangent to orbits}. It is well-known that Lemma
A(Nx O) defines a singular integrable distribution,
its leaves are Gx -invariant,
the strata of Nx O/Gx are the projection of the leaves of A(Nx O).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 19 / 24 Define the following families of vector fields G G Gbas(M) = {X ∈ X(M): Xf ∈ C (M) ∀f ∈ C (M)}, Gx A(Nx O) = {X + Y : X ∈ X (Nx O), Y tangent to orbits}. It is well-known that Lemma
A(Nx O) defines a singular integrable distribution,
its leaves are Gx -invariant,
the strata of Nx O/Gx are the projection of the leaves of A(Nx O).
With this Lemma and the Morita equivalence we can prove: Theorem Let G ⇒ M be a source locally trivial proper groupoid. Then Gbas(M) induces a singular integrable distribution,
the strata of M/G are the projections of the leaves of Gbas(M).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 19 / 24 2 Morita equivalence ⇒ Gbas(M)U ←→ A(Nx O) (1:1). 3 by the general properties of stratifications and the Lemma, for each stratum Si of M/G,
Si ∩ U/GU is the projection of a leaf of Gbas(M)U
4 by the maximality property of the leaves of a foliation, we have globally Si is the projection of a leaf of Gbas(M)
Proof. 1 For each x ∈ M there is a saturated neighborhood U of O Morita equivalent to Gx × Nx O
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 20 / 24 3 by the general properties of stratifications and the Lemma, for each stratum Si of M/G,
Si ∩ U/GU is the projection of a leaf of Gbas(M)U
4 by the maximality property of the leaves of a foliation, we have globally Si is the projection of a leaf of Gbas(M)
Proof. 1 For each x ∈ M there is a saturated neighborhood U of O Morita equivalent to Gx × Nx O
2 Morita equivalence ⇒ Gbas(M)U ←→ A(Nx O) (1:1).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 20 / 24 4 by the maximality property of the leaves of a foliation, we have globally Si is the projection of a leaf of Gbas(M)
Proof. 1 For each x ∈ M there is a saturated neighborhood U of O Morita equivalent to Gx × Nx O
2 Morita equivalence ⇒ Gbas(M)U ←→ A(Nx O) (1:1). 3 by the general properties of stratifications and the Lemma, for each stratum Si of M/G,
Si ∩ U/GU is the projection of a leaf of Gbas(M)U
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 20 / 24 Proof. 1 For each x ∈ M there is a saturated neighborhood U of O Morita equivalent to Gx × Nx O
2 Morita equivalence ⇒ Gbas(M)U ←→ A(Nx O) (1:1). 3 by the general properties of stratifications and the Lemma, for each stratum Si of M/G,
Si ∩ U/GU is the projection of a leaf of Gbas(M)U
4 by the maximality property of the leaves of a foliation, we have globally Si is the projection of a leaf of Gbas(M)
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 20 / 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 21 / 24 The action algebroid of this action is π : g × M → M with projection π(ξ, x) = x and anchor ρA(ξ, x) = a(ξ)(x). If the vector fields a(ξ) are not complete, this action does not integrate to a Lie group action. Therefore there is no quotient M/G. However the action algebroid is integrable to a Lie groupoid G ⇒ M. Therefore we can define the quotient of a non-complete action as M/G. By our result this is a Whitney stratified space
Which are the strata of M/G?
1. Non-complete actions
Let a : g → X(M) be a Lie algebra action.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 22 / 24 If the vector fields a(ξ) are not complete, this action does not integrate to a Lie group action. Therefore there is no quotient M/G. However the action algebroid is integrable to a Lie groupoid G ⇒ M. Therefore we can define the quotient of a non-complete action as M/G. By our result this is a Whitney stratified space
Which are the strata of M/G?
1. Non-complete actions
Let a : g → X(M) be a Lie algebra action. The action algebroid of this action is π : g × M → M with projection π(ξ, x) = x and anchor ρA(ξ, x) = a(ξ)(x).
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 22 / 24 However the action algebroid is integrable to a Lie groupoid G ⇒ M. Therefore we can define the quotient of a non-complete action as M/G. By our result this is a Whitney stratified space
Which are the strata of M/G?
1. Non-complete actions
Let a : g → X(M) be a Lie algebra action. The action algebroid of this action is π : g × M → M with projection π(ξ, x) = x and anchor ρA(ξ, x) = a(ξ)(x). If the vector fields a(ξ) are not complete, this action does not integrate to a Lie group action. Therefore there is no quotient M/G.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 22 / 24 1. Non-complete actions
Let a : g → X(M) be a Lie algebra action. The action algebroid of this action is π : g × M → M with projection π(ξ, x) = x and anchor ρA(ξ, x) = a(ξ)(x). If the vector fields a(ξ) are not complete, this action does not integrate to a Lie group action. Therefore there is no quotient M/G. However the action algebroid is integrable to a Lie groupoid G ⇒ M. Therefore we can define the quotient of a non-complete action as M/G. By our result this is a Whitney stratified space
Which are the strata of M/G?
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 22 / 24 There is a (Hamiltonian) action of the symplectic groupoid of Σ(P) on M with momentum map J. The orbit space for this action is the same as the orbit space M/G where G is the action groupoid
Σ(P) n M ⇒ M
Is M/G a Whitney-Poisson stratified space?, Which are the strata?
2. Hamiltonian groupoid actions
Let (M, ω) be a symplectic manifold, (P, {·, ·}) an integrable Poisson manifold and J : M → P a Poisson map.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 23 / 24 The orbit space for this action is the same as the orbit space M/G where G is the action groupoid
Σ(P) n M ⇒ M
Is M/G a Whitney-Poisson stratified space?, Which are the strata?
2. Hamiltonian groupoid actions
Let (M, ω) be a symplectic manifold, (P, {·, ·}) an integrable Poisson manifold and J : M → P a Poisson map. There is a (Hamiltonian) action of the symplectic groupoid of Σ(P) on M with momentum map J.
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 23 / 24 2. Hamiltonian groupoid actions
Let (M, ω) be a symplectic manifold, (P, {·, ·}) an integrable Poisson manifold and J : M → P a Poisson map. There is a (Hamiltonian) action of the symplectic groupoid of Σ(P) on M with momentum map J. The orbit space for this action is the same as the orbit space M/G where G is the action groupoid
Σ(P) n M ⇒ M
Is M/G a Whitney-Poisson stratified space?, Which are the strata?
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 23 / 24 THE END
Miguel Rodr´ıguez-Olmos (EPFL) The Orbit Space of a Proper Groupoid 24 / 24