Journal of Geometry and Physics

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Journal of Geometry and Physics

journal homepage:www.elsevier.com/locate/jgp

Stability of Lie groupoid C ^∗ -algebras

^✩

Claire Debord

^a,∗

, Georges Skandalis

^b

aLaboratoire de Mathématiques, UMR 6620 - CNRS, Université Blaise Pascal, Campus des Cézeaux, BP 80026, F-63171 Aubière cedex, France

bUniversité Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Paris 06, CNRS, IMJ-PRG, UFR de Mathématiques, CP 7012 - Bâtiment Sophie Germain 5 rue Thomas Mann, 75205 Paris CEDEX 13, France

a r t i c l e i n f o

Article history:

Received 4 March 2015 Accepted 5 March 2016 Available online 18 March 2016

MSC:

primary 58H05 secondary 46L89 58J22

Keywords:

Lie groupoids StableC^∗-algebras Singular foliations

a b s t r a c t

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C^∗-algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C^∗-algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.

©2016 Elsevier B.V. All rights reserved.

1. Introduction: Statement of the theorem and the steps of the proof

The aim of this paper is to generalize Theorem 1 of [1] stating that theC^∗-algebra of any foliation (of nonzero dimension!) is stable.

Theorem 1. Let G be a Lie groupoid with

σ

^{-compact G(0)}. Assume that at every x

∈

G⁽⁰⁾the anchor

♮

x

:

g_x

→

T_xG⁽⁰⁾is nonzero.

Then C^∗

(

^G

)

^{is stable.}

In other words,C^∗

(

^G

)

is stable wheneverGhas no orbit of dimension 0. We refer to [2] for the general definition of groupoid C^∗-algebras.

The converse is also true ifGiss-connected. Indeed, ifGiss-connected and the anchor atxis the zero map, then the orbit ofxis reduced tox. ThereforeC^∗

(

^G

)

has a character: the trivial representation of the groupG^x_x.

Since the reducedC^∗-algebraC_r^∗

(

^G

)

^ofGis a quotient ofC^∗

(

^G

)

, it follows that it is also stable whenGhas no orbit of dimension 0.

Here however, the converse may fail for the reducedC^∗-algebra: the reducedC^∗-algebra of the groupPSL₂

(

R

)

^{is stable!}

Our proof is not very different from the one of [1] and based on Kasparov’s stabilization theorem [3]. Note that, unlike in [1], we do not assume the spaceG⁽⁰⁾to be compact—but this is actually a rather minor point.

The proof is as follows.

✩The authors were partially supported by ANR-14-CE25-0012-01.

∗Corresponding author.

E-mail addresses:[email protected](C. Debord),[email protected](G. Skandalis).

http://dx.doi.org/10.1016/j.geomphys.2016.03.011 0393-0440/©2016 Elsevier B.V. All rights reserved.

1. Letx

∈

G⁽⁰⁾. There is a sectionY of the algebroidA

(

^G

)

whose image under the anchor is a vector fieldXsatisfying X

(

^x

) ̸=

0. Taking a local exponentiation ofXwe obtain a relatively compact open neighborhoodW diffeomorphic to U

×

_RwhereXis proportional to the vector field along theRlines

{

u

} ×

_R.

This step will be clarified in Section2.1

We thus choose a locally finite cover

(

^Wn

)

by relatively compact open subsets and diffeomorphismsf_n

:

U_n

×

_R

→

W_n such thatW_n^′

=

f_n

(

^Un^′

×

_R

)

^{coverG(0)withU}n^′relatively compact inU_n. Letp_n

:

W_n

→

U_nbe the composition off_n⁻¹with the projectionU_n

×

_R

→

U_n.

2. One may then construct a locally finite family of open subsetsV_jofG⁽⁰⁾such that:

•

EveryV_jsits in aW_n(j)and its intersection with each linef_n(j)

(

^u

×

_R

)

is an (open) interval. More precisely,f_n(j)

(

^Vj

)

^{is of} the form

{ (

^x

,

^t

) ∈

U_n(j)

×

_R

; ϕ

j⁻

(

^x

) <

^t

< ϕ

j⁺

(

^x

) }

where

ϕ

⁻j

, ϕ

j⁺

:

U_n(j)

→

_Rare smooth and

ϕ

j⁺

− ϕ

⁻j is nonnegative with compact support.

•

TheV_jare pairwise disjoint and locally finite: every compact subset ofMmeets only finitely manyV_j’s.

•

For everyn, thep_n

(

^Vj

∩

W_n

)

^coverUn^′: we haveU_n^′

⊂



j;n(^j)=np_n

(

^Vj

∩

W_n

)

^. The details of the constructions of theV_j’s are given in Section2.2.

3. Let then q_j be the characteristic function of V_j. We prove thatq_j is a multiplier of C^∗

(

^G

)

. By local finiteness, the characteristic functionq

=



q_jofV

=



V_jis also a multiplier ofC^∗

(

^G

)

^. See section Section3.3.

4. We show thatqC^∗

(

^G

)

is a full Hilbert submodule ofC^∗

(

^G

)

^(seeCorollary 9—Section3.2).

5. Considering a natural diffeomorphismV_j

≃

p_n

(

^Vj

) ×]

0

,

¹

[

, it follows that the HilbertC^∗

(

^G

)

^-modulesqjC^∗

(

^G

)

^andqC∗

(

^G

)

are stable.

6. Using Kasparov’s stabilization Theorem [3], it follows thatC^∗

(

^G

)

^{is stable.}

This follows fromCorollary 6—see Section3.4.

In Section4, we prove an analogous theorem for singular foliations in the sense of [4]. We prove:

Theorem 2.The C^∗-algebra of a singular foliation (as defined in[4]) which has no leaves reduced to a single point is stable.

The main steps of the proof are the same as forTheorem 1. Using vector fields along the foliation, we construct the same small open subsetsV_j. Note that, in proving that the characteristic functions of theseV_jare multipliers ofC^∗

(

^M

,

^F

)

, we chose to take a somewhat different path in order to shed a new light to it. This led us to construct groupoid homomorphisms between singular foliation groupoids (Section4.2). Of course, we could have used the same kind of proof as for the Lie groupoid case.

2. Geometric constructions

2.1. Nonzero vector fields in the algebroid

LetMbe a smooth (open) manifold,xa point ofM, and letX

∈

_X

(

^M

)

be a smooth vector field with compact support on Msuch thatX

(

^x

) ̸=

0. Denote byΨX

= (

ΨX^t

)

t∈_Rthe flow ofX. One can find a codimension one submanifoldUofMand a neighborhoodIof 0 inRsuch that the restriction ofΨXtoU

×

Iis a diffeomorphism onto an open (tubular) neighborhoodW ofUinM. In other words,Uis a codimension one submanifold ofMwhich containsxand which is transverse to the integral curves ofX.

IfGis a Lie groupoid andYa section with compact support of its Lie algebroid such that the vector fieldX

:= ♮(

^Y

)

^{does not} vanish on a pointx

∈

G⁽⁰⁾, letZ_Ybe the associated right invariant vector field onGandΨZYits flow. We haver

◦

_ΨZ^t

Y

=

_Ψ^t

◦

r.

Applying the construction above, one finds a codimension one submanifoldUofG⁽⁰⁾and a neighborhoodIof 0 inRsuch that the composition map

U

×

I

−→

^ΨZY G

−→

^r G⁽⁰⁾

is precisely the restriction of the flowΨXofXtoU

×

Iand thus a diffeomorphism onto an open neighborhoodWofxinG⁽⁰⁾. Note thats

◦

_ΨZY is the projectionU

×

I

→

U.

Now the following maps are diffeomorphisms:

G^U

×

I

→

G^W

(γ ,

^t

) →

_ΨZY

(

^r

(γ ),

^t

)γ

^{and G}

U

U

×

I

×

I

→

G^W_W

(γ ,

^t

, λ) →

_ΨZY

(

^r

(γ ),

^t

)γ

ΨZY

(

^s

(γ ), λ)

⁻¹

.

2.2. Construction of the familyV_j

In this section, we explain the construction of theV_j’s.

The construction above yields a locally finite cover

(

^Wn

)

by relatively compact open subsets and diffeomorphisms f_n

:

U_n

×

_R

→

W_nsuch thatW_n^′

=

f_n

(

^Un^′

×

_R

)

^{coverG(0)withU}n^′relatively compact inU_n. We will often identifyW_n andU_n

×

_Runderfn.

Letp_n

:

W_n

→

U_nbe the composition off_n⁻¹with the projectionU_n

×

_R

→

U_n. As

(

^Wn

)

is locally finite andG⁽⁰⁾is

σ

-compact, the set of indices is countable; we identify it withN.

•

First, choose a (Riemannian) metricdonG⁽⁰⁾.

•

For everyn

∈

_N, let

ε

n

>

0 be small enough so that, for everym

∈

_Nsuch thatW_n

∩

W_m

̸= ∅

, and everyx

∈

U_m^′, d

(

^fm

(

^x

, −

1

),

^fm

(

^x

,

¹

)) > ε

n. Thus ifW_n

∩

W_m

̸= ∅

, for everyx

∈

U_m^′: diam

(

^fm

( {

x

} ×

_R

)) > ε

n.

•

Letn

∈

_N. Assume that we have already constructed a finite setJ_nwith a map,j

→

n

(

^j

)

^fromJntoNwithn

(

^j

) <

^{nand a} family of open sets

(

^Vj

)

j∈Jnwith pairwise disjoint closures satisfying,

∀ ℓ <

ⁿ

,

^U_ℓ^′

⊂



j;n(j)=ℓ

p_ℓ

(

^Vj

),

^diam

(

^Vj

) ≤ ε

n(j)

.

By the diameter assumption, for everyx

∈

U_n^′, the setf_n

( {

x

} ×

_R

)

is not contained in aV_jwithj

∈

J_n; thereforef_n

( {

x

} ×

_R

)

is not contained in

j∈JnV_j(by connectedness ofR).

By compactness, we may then construct a finite cover

(

^Aℓ

)

ℓ∈J′

nofU_n^′by open subsets ofU_nand open intervalsI_ℓinRsuch thatf_n

(

^Aℓ

×

I_ℓ

) ∩



j∈JnV_j

= ∅

and diam

(

^fn

(

^Aℓ

×

I_ℓ

)) ≤ ε

n.

Replacing theI_ℓ’s by smaller intervals, we now further assume that theI_ℓwith

ℓ ∈

J_n^′ are pairwise disjoint. One then constructs for each

ℓ ∈

_Jn^′two smooth functions

ϕ

^±_ℓ

:

_Un

→

_{Iℓsuch that}

ϕ

⁺_ℓ

≥ ϕ

_ℓ⁻and such thatA_ℓ

= {

_x

∈

_Un

; ϕ

_ℓ⁻

(

^x

) <

ϕ

_ℓ⁺

(

^x

) }

.

Let thenJ_n+1be the disjoint union ofJ_nendJ_n^′. For

ℓ ∈

J_n^′, putV_ℓ

=

f_n

( { (

^x

,

^t

) ∈

U_n

×

_R

; ϕ

_ℓ⁻

(

^x

) <

^t

< ϕ

_ℓ⁺

(

^x

) } )

^{and define} n

(ℓ) =

n.

•

We will also use the diffeomorphismh_j

:

A_j

× ]

0

,

¹

[→

V_jgiven by h_j

(

^x

,

^t

) =

f_n(j)

x

, (

¹

−

t

)ϕ

⁻j

(

^x

) +

t

ϕ

⁺_ℓ

(

^x

)



.

⁽✧)

•

We thus construct the family

(

^Vj

)

inductively. Every compact set meets only finitely manyW_n’s and therefore finitely manyV_j’s sinceV_j

⊂

_Wn(j)and

{

_j

;

_n

(

^j

) =

_n

}

_{is finite.}

3. Hilbert modules

3.1. Stable Hilbert modules; full Hilbert modules

We start by briefly recalling some now classical general facts on Hilbert modules. The basic reference for them is [3].

See also e.g. [5,6].

LetAbe a separableC^∗-algebra.

Definition 3. A separable HilbertA-moduleEis said to be Stable if it is isomorphic to

ℓ

²

(

N

) ⊗

E.

Full if the vector span of the set of products

⟨

x

|

y

⟩

withx

,

^y

∈

Eis dense inA.

We have:

Proposition 4. 1. Let J be a countable set and

(

^Ej

)

j∈Ja family of (separable) stable Hilbert A modules, then

j∈_JE_jis stable.

2. A separable Hilbert A-module E is stable if and only if the C^∗-algebraK

(

^E

)

is stable (see e.g.[7,Facts 4.7]).

Let us also recall Kasparov’s stabilization Theorem. PutHA

= ℓ

²

(

N

) ⊗

A.

Theorem 5 (Kasparov’s Stabilization Theorem). For every separable Hilbert A-module E, the Hilbert A-modulesHAand E

⊕

_HA are isomorphic.

The following statement is an immediate consequence of Kasparov’s stabilization theorem. We outline a proof for completeness.

Corollary 6. 1. A separable full and stable Hilbert A-module is isomorphic toHA(cf.[6,Prop. 7.4, p. 73]).

2. If E is a separable Hilbert A-module and p

∈

_L

(

^E

)

is a projection such that pE is full and stable, then E is isomorphic toHA. Proof. 1. PutB

=

_K

(

^E

)

^andE∗

=

_K

(

^E

,

^A

)

considered as a HilbertBmodule. SinceEis full, we findK

(

^E∗

) =

E^∗

⊗

_BE

=

A.

SinceEis stable, the HilbertB-moduleBis isomorphic toHB, therefore, using Kasparov’s stabilization theorem, we find an isomorphismu

:

B

→

B

⊕ (ℓ

²

(

N

) ⊗

E^∗

)

. We thus obtain an isomorphism of HilbertA-modulesu

⊗

_B1_Ebetween E

=

B

⊗

_BEandE

⊕ (ℓ

²

(

N

) ⊗

E^∗

⊗

_BE

) ≃

_HA.

2. WriteE

=

pE

⊕ (

¹

−

p

)

^E

≃

_HA

⊕ (

¹

−

p

)

^E

≃

_HA.

3.2. Hilbert module associated with a transverse map

We now recall the rather well known construction of generalized smooth transverse mapsM

→

Gfrom a manifoldM to a smooth groupoidG. This construction is useful in steps 4 and 5 of the proof of our theorem as explained in Section1.

Note that we actually only use it for a (locally closed) submanifoldM

⊂

G⁽⁰⁾transverse toG.

Definition 7. LetM be a smooth manifold and Ga smooth groupoid with algebroid Aand anchor

♮

. A smooth map f

:

M

→

G⁽⁰⁾is said to betransversetoGif for everyx

∈

M,df_x

(

^TxM

) + ♮

f(x)A_f(x)

=

T_f(x)G⁽⁰⁾.

Thegraphof a smooth mapf

:

M

→

G⁽⁰⁾is the setΓ^f

=

M

×

_G(0)G

= { (

^x

, γ ) ∈

M

×

G

;

f

(

^x

) =

r

(γ ) }

.

Equivalently,f is transverse if and only if the source map s_f is a submersion, wheres_f

:

_Γ^f

→

G⁽⁰⁾ is the map

(

^x

, γ ) →

s

(γ )

^.

We will in fact only use the case wheref

:

M

→

G⁽⁰⁾is the inclusion of a (locally closed) submanifoldMofG⁽⁰⁾. Then, Γ^f

=

G^M

= { γ ∈

G

;

r

(γ ) ∈

M

}

.

For the sake of completeness, we recall also the notion of ageneralized transverse map.

LetGbe a smooth groupoid,Ma smooth manifold. A generalized morphismf

:

M

→

Gis given either by:

A cocycle : An open cover

(

^Ui

)

i∈IofM, smooth mapsf_i

:

U_i

→

G⁽⁰⁾and smooth mapsf_ij

:

U_i

∩

U_j

→

Gsatisfyingr

◦

f_ij

=

f_i, s

◦

f_ij

=

f_jand, for allx

∈

U_i

∩

U_j

∩

U_k,f_ik

(

^x

) =

f_ij

(

^x

)

^fjk

(

^x

)

^.

The graph off : A setΓ^{f which is a}G-principal bundle overM. We therefore are given mapsr_f

:

_Γ^f

→

_{Mwhich is a} smooth surjective submersion ands_f

:

_Γ^f

→

G⁽⁰⁾with a right action ofGwhich is a smooth map

(

^x

, γ ) →

x

γ

fromΓ^f

×

_G(0)G

= { (

^x

, γ ) ∈

_Γ^f

×

G

;

s_f

(

^x

) =

r

(γ ) }

toΓ^f. This action is assumed to be proper and free with quotientM:

Proper means that the map

(

^x

, γ ) → (

^x

,

^x

γ )

is proper fromΓ^f

×

_G(0)GtoΓ^f

×

_Γ^f. Free means that the map

(

^x

, γ ) → (

^x

,

^x

γ )

is injective.

The quotient isM means that for a pair

(

^x

,

^y

) ∈

_Γ^f

×

_Γ^fthere exists

γ ∈

Gwithx

γ =

yif and onlyr_f

(

^x

) =

r_f

(

^y

)

^. Note that by freeness, this

γ

is unique. We will denote it byx⁻¹y.

These three conditions altogether mean that the map

(

^x

, γ ) → (

^x

,

^x

γ )

is a diffeomorphism fromΓ^f

×

_G(0)Gonto Γ^f

×

_MΓ^f

= { (

^x

,

^y

) ∈

_Γ^f

×

_Γ^f

;

r_f

(

^x

) =

r_f

(

^y

) }

—whose inverse is

(

^x

,

^y

) → (

^x

,

^x−1y

)

^.

Given a cocycle

(

^fi

)

^,

(

^fi,j

)

, we obtain the graphΓ^fby gluing the graphsΓ^fithanks to thef_i,j’s. Conversely, we obtain a cocycle out ofΓ^fby means of local sections of the submersionr_f

:

_Γ^f

→

M.

The generalized morphismfistransversetoG(or a submersion fromMto the ‘bad manifold’G⁽⁰⁾

/

^{G) ifs}f

:

_Γ^f

→

G⁽⁰⁾is a submersion. This is equivalent to saying in the cocycle vision that the mapsf_iare transverse toG.

Iff is transverse toG, the map

(

^x

, γ ) → γ

is a submersion fromΓ^f

×

_G(0)GintoG, whence the map

(

^x

,

^y

) →

_x⁻¹_{yis a} submersion fromΓ^f

×

_MΓ^ftoG.

It then defines a Hilbert-C^∗

(

^G

)

^-moduleC∗

(

Γ^f

)

: this is the completion ofC_c

(

Γ^f

)

with respect to theC^∗

(

^G

)

-valued inner product given (using 1

/

2-densities) by

⟨

g

|

h

⟩ (γ ) =



x∈_Γ^f;sf(x)=r(γ )

g

(

^x

)

^h

(

^x

γ ).

Let us state the following easy fact which is important for our constructions:

Proposition 8. Let f

:

M

→

G be a generalized transverse morphism with graphΓ^f. The module C^∗

(

Γ^f

)

is full if and only if s_f

:

_Γ^f

→

G⁽⁰⁾is onto.

Proof. The image of the submersions_f

:

_Γ^f

→

G⁽⁰⁾is an open subsetUofG⁽⁰⁾. Using the action ofGonΓ^f, we deduce that Uis saturated inG⁽⁰⁾—i.e. ifr

(γ ) ∈

U, there existsx

∈

_Γ^f withr

(γ ) =

s_f

(

^x

)

^{; thens}

(γ ) =

s_f

(

^x

γ ) ∈

U.

The image of the map

(

^x

,

^y

) →

x⁻¹yfrom (defined onΓ^f

×

_MΓ^f) isG^U_U.

It follows that the inner products

⟨

g

|

h

⟩

withg

,

^h

∈

C_c^∞

(

Γ^f

)

, span a dense subset ofC_c^∞

(

^GUU

)

^.

IfU

=

G⁽⁰⁾, these scalar products span a dense subspace inC^∗

(

^G

)

^{; ifU}

̸=

G⁽⁰⁾, they all sit in the kernel of a regular representation associated to any pointx

̸∈

U.

Note that ifMis a submanifold ofG⁽⁰⁾, this condition means thatMmeets all theGorbits.

Let us state this proposition in the precise way we will need to use it:

Corollary 9. Let G be smooth and V an open subset of G⁽⁰⁾. Consider C₀

(

^V

)

as sitting in C₀

(

^G(0)

)

and therefore in the multiplier algebra of C^∗

(

^G

)

. Assume that every orbit of G has a nonempty intersection with V . Then C₀

(

^V

)

^C∗

(

^G

)

is a full submodule of C^∗

(

^G

)

^.

3.3. The characteristic function ofV_jis a projection

We prove here that the characteristic functionq_jofV_jis a multiplier ofC^∗

(

^G

)

^,i.e.thatC0

(

^Vj

)

^C∗

(

^G

)

^isorthocomplemented inC^∗

(

^G

)

^.

Indeed, putn

=

n

(

^j

)

. We may writeq_j

= ϑ

jq_j

ϑ

jwhere

ϑ

jis a smooth real valued function onG⁽⁰⁾with support inW_n which is equal to 1 onV_j. It is then enough to prove thatq_j

∈

_L

(

^C∗

(

^GWn

))

^viewing

ϑ

jas an elementT_ϑj

∈

_L

(

^C∗

(

^G

) ;

C^∗

(

^GWn

))

and writeq_j

=

T_ϑ^∗

jq_jT_ϑj.

Using the identification between W_n and U_n

×

_R coming from the diffeomorphism f_n, we may write C^∗

(

^GWn

) = (

^C0

(

^Un

) ⊗

L²

(

R

)) ⊗

_C0(Un)C^∗

(

^GUn

)

^.

The characteristic function ofV_jis given by a (

∗

-)strongly continuous map fromU_ntoL

(

^L2

(

R

))

, and therefore is an elementQ_j

∈

_L

(

^C0

(

^Un

) ⊗

L²

(

R

))

. Thereforeq_j

=

Q_j

⊗

_C0(Un)1 is inL

(

^C∗

(

^GWn

))

^.

Note also thatQ_jis the

∗

-strong limit of a sequence

(θ

k

)

of smooth functions onU_n

×

_R: put

θ

k

(

^x

,

^t

) = φ(

^k

(

^t

− ϕ

⁻

(

^x

)))φ(

^k

(ϕ

⁺

(

^x

) −

t

))

^where

φ :

_R

→ [

0

,

¹

]

is a continuous function vanishing fort

≤

0 and equal to 1 fort

≥

1.

It follows that the range ofq_jis (the closure of)C₀

(

^Vj

)

^C∗

(

^G

)

. It is the HilbertC^∗

(

^G

)

^-moduleC∗

(

^GVj

)

corresponding to the inclusionV_j

→

G⁽⁰⁾.

Now the projectionsq_jare pairwise orthogonal and, by local finiteness, the sum

q_jis strictly convergent. Indeed, the sum

q_j

ξ

has only finitely many nonzero terms for

ξ ∈

C_c

(

^G

)

^. 3.4. Stability

Stability ofq_jC^∗

(

^G

)

. Using the diffeomorphism h_j

:

A_j

×]

0

,

¹

[→

V_j (see Section 2.2, formula (_✧)), we deduce a diffeomorphism G^Vj

≃

G^Aj

×]

0

,

¹

[

, whence an isomorphism of Hilbert C^∗

(

^G

)

^-modulesC∗

(

^GVj

) ≃

C^∗

(

^GAj

) ⊗

L²

( ]

0

,

¹

[ )

. Therefore, the HilbertC^∗-moduleq_jC^∗

(

^G

)

^{is stable.}

Stability ofqC^∗

(

^G

)

^{. Sinceq}

=



q_j(

∗

-strong convergence), it follows thatqC^∗

(

^G

) =



jq_jC^∗

(

^G

)

. ThereforeqC^∗

(

^G

)

^is stable too.

Conclusion. It then follows fromCorollary 6that the HilbertC^∗

(

^G

)

^-moduleC∗

(

^G

)

is isomorphic toHC∗(^G). TheC^∗-algebra C^∗

(

^G

) =

_K

(

^C∗

(

^G

)) =

_K

(

^HC∗(G)

)

^{is stable.}

This ends the proof ofTheorem 1.

4. Stability of theC^∗-algebra of a singular foliation 4.1. The holonomy groupoid of a singular foliation

TheC^∗-algebra of a singular foliation was defined in [4]. Let us briefly recall a few facts and constructions from [4].

Recall that a foliation on a manifoldMis defined in [4] to be a (locally) finitely generated submoduleF, stable by Lie brackets, of theC^∞

(

^M

)

^-moduleXc

(

^M

)

of smooth vector fields onMwith compact support.

Abi-submersionofF is the data of

(

^N

,

^rN

,

^sN

)

^whereNis a smooth manifold,r_N

,

^sN

:

N

→

Mare smooth submersions such that:

r_N⁻¹

(

^F

) =

s⁻_N¹

(

^F

)

^{and s−}N¹

(

^F

) =

C_c^∞

(

^N

;

kerds_N

) +

C_c^∞

(

^N

;

kerdr_N

).

¹

The inverse of

(

^N

,

^rN

,

^sN

)

^is

(

^N

,

^sN

,

^rN

)

^{and if}

(

^T

,

^rT

,

^sT

)

is another bi-submersion forF thecomposition is given by

(

^N

,

^rN

,

^sN

) ◦ (

^T

,

^rT

,

^sT

) := (

^N

×

_sN,rTT

,

^rN

◦

p_N

,

^sT

◦

p_T

) ,

^wherepNandp_Tare the natural projections respectively ofN

×

_sN,rTT onNand onT.

Amorphismfrom

(

^N

,

^rN

,

^sN

)

^to

(

^T

,

^rT

,

^sT

)

is a smooth maph

:

N

→

Tsuch thats_T

◦

h

=

s_Nandr_T

◦

h

=

r_Nand it islocal when it is defined only on an open subset ofN.

Finally a bi-submersion can berestricted: ifUis an open subset ofN,

(

^U

,

^rU

,

^sU

)

is again a bi-submersion, wherer_Uand s_Uare the restriction ofr_Nands_NtoU.

ForxinM, we define thefiber ofF atxto be the quotientFx

=

_F

/

^IxF. LetX

= (

^Xi

)

i∈[[1,n]]

∈

_Fⁿbe such that Xx

= ( [

X_i

]

_x

)

i∈[[1,n]]is a basis ofFx. For any

ξ = (ξ

i

)

i∈[[1,n]]

∈

_Rⁿ, we consider the vector fieldX_ξ

:=

n

i=1

ξ

iX_iand we denote byΨ_ξ^sits flow at times. We consider the two smooth submersions fromM

×

_RⁿtoM:

(

^sX

,

^rX

) :

M

×

_Rⁿ

−→

M

×

M

; (

^x

, ξ) → (

^x

,

Ψ_ξ¹

(

^x

)).

According to Propositions 2.10 and 3.11 of [4], one can find an open neighborhoodWof

(

^x

,

⁰

)

^inM

×

_Rⁿsuch that

(

^W

,

^rW

,

^sW

)

is a bi-submersion, where the mapr_W and s_W are the restriction to W of the mapsr_X and s_X defined above. Such a bi-submersion is called apath holonomy bi-submersion minimal at x.

1 Ifh:N→Mis a smooth submersionh⁻¹(^F)is the vector space generated by tangent vector fieldsfZwheref∈C_c^∞(^N)^andZis a smooth tangent vector field onNwhich is projectable bydhand such thatdh(Z)belongs toF.

Notice, that any restriction around

(

^x

,

⁰

)

of a path holonomy bi-submersion minimal atx, is again a path holonomy bi-submersion minimal atx.

LetU

= (

^Ui

,

^ri

,

^si

)

i∈Ibe a family of bi-submersions ofF. A bi-submersion

(

^U

,

^rU

,

^sU

)

^{isadaptedtoUif for anyu}

∈

U there is a local morphism aroundufromUto aU_ifor somei.

A familyU

= (

^Ui

,

^ri

,

^si

)

i∈Iof bi-submersions ofF which satisfiesM

=



i∈Is_i

(

^Ui

)

and the inverse of any element inU is adapted toUtogether with the composition of any two elements ofUis anatlas.

Thepath holonomy atlasis the family of bi-submersions ofF generated by the path holonomy bi-submersions.

Thegroupoid of the atlasUis the quotientG

(

^U

) = ⊔

_i∈IU_i

/ ∼

whereU_i

∋

u

∼ v ∈

U_jif and only if there is a local morphism fromU_itoU_jsendinguon

v

^{. When}

(

^U

,

^rU

,

^sU

)

^{belongs toUandu}

∈

U, let us denote by

[

U

,

^rU

,

^sU

]

_uits image in G

(

^U

)

. The structural morphisms ofG

(

^U

)

are given by:

source and range: s

( [

U

,

^rU

,

^sU

]

_u

) =

s_U

(

^u

),

^r

( [

U

,

^rU

,

^sU

]

_u

) =

r_U

(

^u

)

^, inverse:

[

U

,

^rU

,

^sU

]

⁻_u¹

= [

U

,

^sU

,

^rU

]

_u,

product:

[

U

,

^rU

,

^sU

]

_u

· [

V

,

^rV

,

^sV

]

_v

= [

U

×

_sU,rVV

,

^rU

◦

p_U

,

^sV

◦

p_V

]

_(u,v)whens_U

(

^u

) =

r_V

(v)

^.

The groupoid of an atlas is endowed with the quotient topology which is quite bad, in particular the dimension of the fibers may change.

Theholonomy groupoidofF is the groupoid of the path holonomy atlas.

4.2. Subfoliations

AsubfoliationF1of a foliationF2is a submodule ofF2which is a foliationi.e.it is locally finitely generated and stable by Lie brackets.

In this section, we fix a foliationF2and a subfoliationF1ofF2. 4.2.1. The atlas of compatible bi-submersions

Definition 10. A bi-submersion

(

^U

,

^r

,

^s

)

^ofF1is said to becompatiblewithF2ifr⁻¹

(

^F2

) =

s⁻¹

(

^F2

)

^.

Proposition 11. 1. For every x₀

∈

M, there is a bi-submersion

(

^W

,

^r

,

^s

)

^{of F}1compatible withF2such that x₀

∈

s

(

^W

)

^. 2. If f

: (

^U

,

^rU

,

^sU

) → (

^V

,

^rV

,

^sV

)

is a morphism of bi-submersions forF1and

(

^V

,

^rV

,

^sV

)

is compatible withF2, then

(

^U

,

^rV

,

^sV

)

is compatible withF1.

3. The bi-submersions ofF1compatible withF2form an atlas.²

Proof. 1. Fix vector fields X₁

, . . . ,

^Xn which generate F1 in a neighborhood of x₀. For y

= (

^y1

, . . . ,

^yn

) ∈

_Rⁿ_{, put}

ϕ

y

=

exp

(



y_iX_i

) ∈

expF. OnRⁿ

×

M, puts₀

(

^y

,

^x

) =

xandt₀

(

^y

,

^x

) = ϕ

y

(

^x

)

. It is proved in [4, section 2.3] that there is a neighborhoodWof

(

⁰

,

^x0

)

ⁱⁿRⁿ

×

Msuch that

(

^W

,

^t

,

^s

)

is a bi-submersion forF1wheresandtare the restrictions of s₀andt₀.

LetYbe the vector field onRⁿ

×

Mgiven byY

(

^y

,

^x

) = (

⁰

,



y_iX_i

) ∈

_Rⁿ

×

T_xM

=

T_(y,x)

(

Rⁿ

×

M

)

^{. SinceY}

∈

s⁻¹F1

⊂

s⁻¹F2, it follows thats⁻¹F2is invariant under expY. We find thats⁻¹

(

^F2

) = (

^s

◦

expY

)

⁻¹

(

^F2

)

as desired.

2. Asr_U

=

r_V

◦

fands_U

=

s_V

◦

f, we findr_U⁻¹

(

^F

) =

f⁻¹

(

^rV⁻¹

(

^F

)) =

f⁻¹

(

^s−V¹

(

^F

)) =

s⁻_U¹

(

^F

)

^. 3. Let

(

^U

,

^rU

,

^sU

)

^and

(

^V

,

^rV

,

^sV

)

be bi-submersions (forF1) compatible withF2.

(a) Obviously, the inverse

(

^U

,

^sU

,

^rU

)

^of

(

^U

,

^rU

,

^sU

)

is compatible withF2.

(b) Recall that the composition

(

^W

,

^rW

,

^sW

)

^of

(

^U

,

^rU

,

^sU

)

^and

(

^V

,

^rV

,

^sV

)

is constructed as follows: defineW

= { (

^u

, v) ∈

U

×

V

;

s_U

(

^u

) =

r_V

(v) }

and for

(

^u

, v) ∈

W, sets_W

(

^u

, v) =

s_V

(v)

^andrW

(

^u

, v) =

r_U

(

^u

)

^{. Asp}U

: (

^u

, v) →

uand p_V

(

^u

, v) → v

are submersions onW, ands_U

◦

p_U

=

r_V

◦

p_V, we find

r_W⁻¹

(

^F2

) =

p⁻_U¹

(

^rU⁻¹

(

^F2

)) =

p⁻_U¹

(

^s−U¹

(

^F2

))

=

p⁻_V¹

(

^rV⁻¹

(

^F2

)) =

p⁻_V¹

(

^s−V¹

(

^F2

))

=

s⁻_W¹

(

^F2

).

Proposition 12. The composition of a bi-submersion of F1compatible withF2with a bi-submersion of F2is a bi-submersion of F2.

Proof. Let

(

^U

,

^rU

,

^sU

)

be a bi-submersion ofF1compatible withF2and

(

^V

,

^rV

,

^sV

)

be a bi-submersion forF2. As previously letW

= { (

^u

, v) ∈

U

×

V

;

s_U

(

^u

) =

r_V

(v) }

and denote byp_Uandp_V the projections onUandV. We sets_W

=

s_V

◦

p_V, r_W

=

r_U

◦

p_Uand

α =

s_U

◦

p_U

=

r_V

◦

p_V.

2 To make it a set, take only bi-submersions defined on open subsets ofR^Nfor allN. Note that a bi-submersion(^U,^r,^s)^ofF1is then adapted to this atlas if and only ifr⁻¹F2=s⁻¹F2.