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.

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

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