Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus

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Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus

by Claire Debord and Georges Skandalis

Université Clermont Auvergne LMBP, UMR 6620 - CNRS Campus des Cézeaux, 3, Place Vasarely TSA 60026 CS 60026 63178 Aubière cedex, France

claire.debord@math.univ-bpclermont.fr Universit´e Paris Diderot, Sorbonne Paris Cit´e

Sorbonne Universités, UPMC Paris 06, CNRS, IMJ-PRG UFR de Mathématiques,CP7012- Bâtiment Sophie Germain 5 rue Thomas Mann, 75205 Paris CEDEX 13, France

skandalis@math.univ-paris-diderot.fr

Abstract

We present natural and general ways of building Lie groupoids, by using the classical proce- dures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions ofC^∗-algebras and to full index problems. We compute the corresponding K-theory maps. Finally, the blowup of a manifold sitting in a transverse way in the space of objects of a Lie groupoid leads to a calculus which is quite similar to the Boutet de Monvel calculus for manifolds with boundary.

1 Introduction

Let G ⇒ M be a Lie groupoid. The Lie groupoid G comes with its natural pseudodifferential calculus. For example

• if the groupoid G is just the pair groupoid M ×M, the associated calculus is the ordinary (pseudo)differential calculus onM;

• if the groupoid G is a family groupoid M ×_B M associated with a fibration p : M → B, the associated (pseudo)differential operators are families of operators acting on the fibers ofp (those of [3]);

• if the groupoidG is the holonomy groupoid of a foliation, the associated (pseudo)differential operators are longitudinal operators as defined by Connes in [6];

• if the groupoidGis the monodromy groupoidi.e. the groupoid of homotopy classes (with fixed endpoints) of paths in a (compact) manifoldM, the associated (pseudo)differential operators are theπ1(M)-invariant operators on the universal cover ofM...

The groupoidGdefines therefore a class of partial differential equations.

Our study will focus here on the corresponding index problems on M. The index takes place naturally in theK-theory of theC^∗-algebra of G.

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Let thenV be a submanifold ofM. We will considerV as bringing a singularity into the problem:

it forces operators of G to “slow down” near V, at least in the normal directions. Inside V, they should only propagate along a sub-Lie-groupoid Γ⇒V ofG.

This behavior is very nicely encoded by a groupoid SBlupr,s(G,Γ) obtained by using a blow-up construction of the inclusion Γ→G.

The blowup construction (Blup) and the deformation to the normal cone (DNC) are well known constructions in algebraic geometry as well as in differential geometry. Let X be a submanifold of a manifoldY. Denote by N_X^Y the normal bundle.

• Thedeformation to the normal coneofX inY is a smooth manifoldDN C(Y, X) obtained by naturally gluingN_X^Y × {0} withY ×R^∗.

• Theblowup ofX inY is a smooth manifoldBlup(Y, X) whereX is inflated to the projective space PN_X^Y. It is obtained by gluing Y \X with PN_X^Y in a natural way. We will mainly consider its variant thespherical blowup SBlup(Y, X) (which is a manifold with boundary) in which the sphere bundleSN_X^Y replaces the projective bundlePN_X^Y.

The first use of deformation groupoids in connection with index theory appeared in [8]. A. Connes showed there that the analytic index on a compact manifoldM can be described using a groupoid, called the “tangent groupoid”. This groupoid was obtained as a deformation to the normal cone of the diagonal inclusion ofM into the pair groupoid M×M.

Since Connes’ construction, deformation groupoids were used by many authors in various contexts.

• This idea of Connes was extended in [19] by considering the same construction of a deformation to the normal cone for smooth immersions which are groupoid morphisms. The groupoid obtained was used in order to define the wrong way functoriality for immersions of foliations ([19, section 3]). An analogous construction for submersions of foliations was also given in a remark ([19, remark 3.19]).

• In [31, 34] Monthubert-Pierrot and Nistor-Weinstein-Xu considered the deformation to the normal cone of the inclusion G⁽⁰⁾ → G of the space of units of a smooth groupoid G. This generalization of Connes’ tangent groupoid was called the adiabatic groupoid ofGand denoted byG_ad. It was shown that this adiabatic groupoid still encodes the analytic index associated withG.

• Many other important articles use this idea of deformation groupoids. We will briefly discuss some of them in the sequel of the paper. It is certainly out of the scope of the present paper to review them all...

Let us briefly present the objectives of our paper.

The groupoids DN C(G,Γ) and SBlup_r,s(G,Γ).

In the present paper, we give a systematic construction of deformation to the normal cone groupoids and define the blowup deformations of groupoids. Our constructions of deformation and blowup groupoids recover all the ones discussed above.

More precisely, we use the functoriality of these two constructions (Blup andDN C) and note that any smooth subgroupoid Γ⇒V of a Lie groupoidG⇒M gives rise to a deformation to the normal cone Lie groupoid DN C(G,Γ) ⇒ DN C(M, V) and to a blowup Lie groupoid Blupr,s(G,Γ) ⇒ Blup(M, V).

We will be mainly interested here to the restrictionDN C₊(G,Γ) toR+of the deformation groupoid DN C(G,Γ) and to a variant ofBlupr,s(G,Γ) which is thespherical blowup Lie groupoid SBlupr,s(G,Γ)⇒ SBlup(M, V).

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Notation 1.1. We will use the following notation:

IfE is a real vector bundle over a manifold (or over a locally compact space)M, the corresponding projective bundleP(E) is the bundle overM whose fiber over a pointxofM is the projective space P(Ex). The bundle P(E) is simply the quotient of E\M by the natural action ofR^∗ by dilation.

The quotient of E\M under the action of R^∗+ by dilation is the (total space of the) sphere bundle S(E).

By construction, both these groupoids are the union of an open subgroupoid, their “regular part”, and a “singular” one which is a closed subgroupoid.

• The regular part of the groupoidDN C₊(G,Γ) is the direct productG×R^∗+ (whereR^∗+ is just a space);

• its “singular part” is the closed subgroupoid N_Γ^G ⇒ N_M^V which is the normal bundle of the inclusion Γ→Gendowed with a Lie groupoid structure over N_M^V. This normal groupoid is a VB groupoid (in the sense of Pradinescf. [35, 22]).

In the same way, the spherical blowup groupoidSBlupr,s(G,Γ) is the disjoint union of

• an open subgroupoid which is the restrictionG^M^˚_˚

M ofG to ˚M =M\V;

• a boundary which is a groupoid SN_Γ^G ⇒ SN_V^M which is fibered over Γ that is a spherical bundle groupoid over Γ.

The spherical blowup groupoid naturally encodes evolution along G which is constrained to fix V = Γ⁽⁰⁾ and evolve along Γ on it. When looking at the corresponding index problems, we are naturally led to consider natural exact sequences and the correspondingKK-elements.

It turns out, that the corresponding index problems forDN C+(G,Γ) are somewhat easier to handle, but in fact, in most cases equivalent to those ofSBlupr,s(G,Γ). Both of them are particular cases of connecting maps and full index problems associated to a saturated open subset in a Lie groupoid.

We will therefore just need to apply the results of [13] in order to handle them.

Connecting maps and index maps

Connecting maps. The decomposition of SBlup_r,s(G,Γ) and DN C₊(G,Γ) gives rise to exact sequences of C^∗-algebras that we wish to “compute”:

0−→C^∗(G^M_M^˚_˚)−→C^∗(SBlupr,s(G,Γ))−→C^∗(SN_Γ^G)−→0 E_SBlup^∂ and

0−→C^∗(G×R^∗+)−→C^∗(DN C+(G,Γ))−→C^∗(N_Γ^G)−→0 E_{DN C}^∂ ₊ Full index maps. Denote by Ψ^∗(DN C₊(G,Γ)) and Ψ^∗(SBlup_r,s(G,Γ)) the C^∗-algebra of order 0 pseudodifferential operators on the Lie groupoids DN C₊(G,Γ) and SBlup_r,s(G,Γ) respectively.

The above decomposition of groupoids give rise to extensions of groupoidC^∗-algebras of pseudodifferential type

0−→C^∗(G^M_M^˚_˚)−→Ψ^∗(SBlup_r,s(G,Γ))−−−→Σ^σ^{f ull} _SBlup(G,Γ)−→0 E_SBlup^ind^f and

0−→C^∗(G×R^∗+)−→Ψ^∗(DN C₊(G,Γ))−−−→^σ^{f ull} Σ_{DN C}₊(G,Γ)−→0 E_{DN C}^ind^f ₊ where Σ_{DN C}₊(G,Γ) and Σ_SBlup(G,Γ) are called the full symbol algebra, and the morphisms σ_{f ull} thefull symbol maps.

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The full symbol maps. The full symbol algebras are naturally fibered products (see [13,§4]):

ΣSBlup(G,Γ) =C(SA^∗SBlupr,s(G,Γ))×_C(

SA^∗SN_Γ^G)Ψ^∗(SN_Γ^G) and

Σ_{DN C}₊(G,Γ) =C(SA^∗DN C₊(G,Γ))×_C(

SA^∗N_Γ^G)Ψ^∗(N_Γ^G).

Thus, the full symbol maps have two components:

• The usual commutative symbol map. They are morphisms:

Ψ^∗(DN C+(G,Γ))→C(SA^∗DN C+(G,Γ)) and Ψ^∗(SBlupr,s(G,Γ))→C(SA^∗SBlupr,s(G,Γ)).

The commutative symbol takes its values in the algebra of continuous fonctions on the sphere bundle of the algebroid of the Lie groupoids (with boundary)DN C₊(G,Γ)) andSBlup_r,s(G,Γ).

• The restriction to the boundary:

σ∂ : Ψ^∗(SBlupr,s(G,Γ))→Ψ^∗(SN_Γ^G) and Ψ^∗(DN C+(G,Γ))→Ψ^∗(N_Γ^G) .

Associated KK-elements. Assume that the groupoid Γ is amenable. Then the groupoids N_Γ^G andSN_Γ^Gare also amenable, and exact sequencesE_SBlup^∂ andE^∂_{DN C}₊give rise to connecting elements

∂_SBlup^G,Γ ∈KK¹(C^∗(SN_Γ^G), C^∗(G^M_M^˚_˚)) and∂_{DN C}^G,Γ

+ ∈KK¹(C^∗(N_Γ^G), C^∗(G×R^∗+)) (cf. [20]). Also, the full symbolC^∗-algebras Σ_SBlup(G,Γ) and ΣDN C+(G,Γ) are nuclear and we also get KK-elements indf^G,Γ_SBlup ∈KK¹(Σ_SBlup(G,Γ), C^∗(G^M_M^˚_˚)) andindf^G,Γ_{DN C}₊ ∈KK¹(ΣDN C+(G,Γ), C^∗(G×R^∗+)).

If Γ is not amenable, these constructions can be carried over in E-theory (of maximal groupoid C^∗-algebras).

Connes-Thom elements. We will establish the following facts.

a) There is a natural Connes-Thom element β ∈ KK¹(C^∗(SBlupr,s(G,Γ)), C^∗(DN C+(G,Γ))).

This element restricts to very natural elements β⁰ ∈ KK¹(C^∗(G^M_M^˚_˚), C^∗(G×R^∗+)) and β⁰⁰ ∈ KK¹(C^∗(SN_Γ^G), C^∗(N_Γ^G)).

These elements extend to elements βΨ ∈ KK¹(Ψ^∗(SBlupr,s(G,Γ)),Ψ^∗(DN C+(G,Γ))) and βΣ∈KK¹(Σ_SBlup(G,Γ)),ΣDN C+(G,Γ))).

We have∂_SBlup^G,Γ ⊗β⁰ =−β⁰⁰⊗∂_{DN C}^G,Γ

+ (cf. facts 5.1 and 5.2) andindf^G,Γ_SBlup⊗β⁰ =−β_Σ⊗indf^G,Γ_{DN C}₊ (fact 5.3).

b) IfM \V meets all the orbits of G, thenβ⁰ is KK-invertible. Therefore, in that case, ∂_{DN C}^G,Γ

+

determines∂_SBlup^G,Γ andindf^G,Γ_{DN C}₊ determines indf^G,Γ_SBlup.

c) We will say that V is AG-small if the transverse action of G on V is nowhere 0, i.e. if for everyx ∈V, the image by the anchor of the algebroid AGof G is not contained in TxV (cf.

definition 5.5). In that case, β⁰, β⁰⁰, β_Σ are KK-invertible: the connecting elements ∂_{DN C}^G,Γ

+

and ∂_SBlup^G,Γ determine each other, and the full index maps indf^G,Γ_{DN C}₊ and indf^G,Γ_SBlup determine each other

Computation. If Γ = V, then C^∗(N_V^G) is KK-equivalent to C₀(N_V^G) using a Connes-Thom isomorphism ([7]) and the element ∂_{DN C}^G,Γ

+ is the Kasparov product of the inclusion of N_V^G in the algebroid AG = N_M^G of G (using a tubular neighborhood) and the index element indG ∈ KK(C₀(A^∗G), C^∗(G)) of the groupoidG(prop. 5.11.d). Of course, ifV isAG-small, we obtain the analogous result for∂^G,Γ_SBlup.

The computation of the corresponding full index is also obtained in the same way in prop. 5.11.e).

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A Boutet de Monvel type calculus.

Let H be a Lie groupoid. In [11], extending ideas of Aastrup, Melo, Monthubert and Schrohe [1], we studied thegauge adiabatic groupoid Hga: the crossed product of the adiabatic groupoid of H by the natural action ofR^∗+. We constructed a bimodule EH giving a Morita equivalence between the algebra of order 0 pseudodifferential operators on H and a natural ideal in the convolution C^∗-algebraC^∗(Hga) of this gauge adiabatic groupoid.

The gauge adiabatic groupoidHga is in fact a blowup groupoid, namelySBlupr,s(H×(R×R), H⁽⁰⁾) (restricted to the clopen subsetH⁽⁰⁾×R+ ofSBlup(H⁽⁰⁾×R, H⁽⁰⁾) =H⁽⁰⁾×(R−tR+)).

Let nowG⇒M be a Lie groupoid and letV be a submanifoldM which is transverse to the action of G (see def. 2.1). We construct a Poisson-trace bimodule: it is aC^∗(SBlupr,s(G, V))−Ψ^∗(G^V_V) bimoduleEP T(G, V), which is a full Ψ^∗(G^V_V) Hilbert module. When Gis the direct product of G^V_V with the pair groupoid R×R the Poisson-trace bimodule coincide with E_G^V

V constructed in [11].

In the general case, thanks to a convenient (spherical) blowup construction, we construct a linking space between the groupoidsSBlup_r,s(G, V) and (G^V_V)_ga=SBlup_r,s(G^V_V ×(R×R), V). This linking space defines aC^∗(SBlup_r,s(G, V))−C^∗((G^V_V)_ga) bimoduleE(G, V) which is a Morita equivalence of groupoids whenV meets all the orbits ofG. The Poisson-trace -bimodule is then the composition ofE_G^V

V with E(G, V).

Denote by Ψ^∗_BM(G;V) theBoutet de Monvel type algebra consisting of matricesR=

Φ P T Q

with Φ ∈ Ψ^∗(SBlup_r,s(G, V)), P ∈ EP T(G, V), T ∈ EP T^∗ (G, V) and Q ∈ Ψ^∗(G^V_V), and C_BM^∗ (G;V) its ideal - where Φ∈C^∗(SBlupr,s(G, V)). This algebra has obvious similarities with the one involved in the Boutet de Monvel calculus for manifolds with boundary [4]. We will examine its relationship with these two algebras in a forthcoming paper.

We still have two natural symbol maps: the classical symbolσc: Ψ^∗_BM(G, V)→C(SA^∗G) given by σc

Φ P T Q

=σc(Φ) and the boundary symbol rV which is restriction to the boundary.

We have an exact sequence:

0→C^∗(G^MtV_M^˚_˚`

V)→Ψ^∗_BM(G;V)−−−→^σ^BM Σ_BM(G, V)→0

where Σ_BM(G, V) = Ψ^∗_BM(G;V)/C^∗(G^MtV_MtV^˚_˚ ) andσ_BM is defined using both σ_c and r_V.

We may note that Ψ^∗(SBlup_r,s(G, V)) identifies with the full hereditary subalgebra of Ψ^∗_BM(G, V) consisting of elements of the form

Φ 0 0 0

. We thus obtain Boutet de Monvel type index theorems for the connecting map of this exact sequence - as well as for the corresponding relative K-theory.

The paper is organized as follows:

• In section 2 we recall some classical facts, constructions and notation involving groupoids.

• In section 3 we review two geometric constructions: deformation to the normal cone and blowup, and their functorial properties.

• In section 4, using this functoriality, we study deformation to the normal cone and blowup in the Lie groupoid context. We outline examples which recover groupoids constructed previously by several authors.

• In section 5, applying the results obtained in [13], we compute the connecting maps and index maps of the groupoids constructed in section 4.

• In section 6, we describe the above mentioned Boutet de Monvel type calculus.

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• The present paper is the second part of the article that appeared on the arXiv (arXiv:1705.09588).

Since this paper was quite long and addressed a large variety of situations, we decided to split it into two pieces hoping to make it easier to read. The first part is [13].

Our constructions involved a large amount of notation, that we tried to choose as coherent as possible. We found it however helpful to list several items of the notation introduced in [13]

and the one introduced here in an index at the end of the paper.

Acknowledgements. We would like to thank Vito Zenobi for his careful reading and for pointing out quite a few typos in an earlier version of the manuscript.

2 Transversality and Morita equivalence of groupoids

2.1 Some notation LetG

r,s

⇒G⁽⁰⁾ be a groupoid with sources, ranger and space of unitsG⁽⁰⁾. For any mapsf :A→ G⁽⁰⁾ and g:B →G⁽⁰⁾, define

G^f ={(a, x)∈A×G; r(x) =f(a)}, G_g ={(x, b)∈G×B; s(x) =g(b)}

and

G^f_g ={(a, x, b)∈A×G×B; r(x) =f(a), s(x) =g(b)}.

In particular forA, B ⊂G⁽⁰⁾, we put G^A ={x ∈G; r(x)∈ A} and G_A ={x∈ G; s(x) ∈A}; we also put G^B_A =G_A∩G^B.

2.2 Transversality

Let us recall the following definition (see e.g. [38] for details):

Definition 2.1. LetG

r,s

⇒M be a Lie groupoid with set of objects G⁽⁰⁾=M and Lie algebroidAG with anchor map \. Let V be a manifold. A smooth map f :V → M is said to be transverse to (the action of the groupoid)G if for everyx∈V,dfx(TxV) +\_f(x)A_f(x)G=T_f_(x)M.

An equivalent condition is that the map (γ, y)7→r(γ) defined on the fibered product G_f =G×

s,f

V is a submersion fromG_f toM.

A submanifoldV of M istransverse toGif the inclusionV →M is transverse to G- equivalently, if for everyx∈V, the compositionq_x =p_x◦\_x :A_xG→(N_V^M)_x =T_xM/T_xV is onto.

Remark 2.2. Let V be a (locally) closed submanifold of M transverse to a groupoid G

r,s

⇒ M.

Denote by N_V^M the (total space) of the normal bundle of V in M. Upon arguing locally, we can assume thatV is compact.

By the transversality assumption the anchor\:AG_|V →T M_|V induces a surjective bundle morphism AG|V → N_V^M. Choosing a subbundle W⁰ of the restriction AG|V such that W⁰ → N_V^M is an isomorphism and using an exponential map, we thus obtain a submanifold W ⊂ G such that r : W → M is a diffeomorphism onto an open neighborhood of V in M and s is a submersion from W onto V. Replacing W by a an open subspace, we may assume that r(W) is a tubular neighborhood of V in M, diffeomorphic to N_V^M. The map W ×_V G^V_V ×_V W → G defined by (γ₁, γ₂, γ₃) 7→ γ₁ ◦γ₂ ◦γ₃⁻¹ is a diffeomorphism and a groupoid isomorphism from the pull back groupoid (see next section) (G^V_V)^s_s=W ×_V G^V_V ×_V W onto the open subgroupoid G^r(W_r(W⁾₎ of G.

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2.3 Pull back

If f : V → M is transverse to a Lie groupoid G

r,s

⇒ M, then G^f_f is a submanifold of V ×G×V naturally equipped with a structure of Lie groupoidG^f_f ⇒V. It is called thepull back groupoid.

If fi :Vi → M are transverse to G(for i= 1,2) then we obtain a Lie groupoidG^f_f¹^tf²

1tf2 ⇒ V1tV2. Thelinking manifold G^f_f¹

2 is a clopen submanifold. We denote byC^∗(G^f_f¹

2) the closure inC^∗(G^f_f¹^tf²

1tf2) of the space of functions (half densities) with support inG^f_f¹

2; it is aC^∗(G^f_f¹

1)−C^∗(G^f_f²

2) bimodule.

Fact 2.3. The bimoduleC^∗(G^f_f¹

2) is full if all the G-orbits meeting f2(V2) meet also f1(V1).

2.4 Morita equivalence Two Lie groupoidsG1

r,s

⇒M1 andG2 r,s

⇒M2 areMorita equivalent if there exists a groupoidG

r,s

⇒M and smooth maps f_i :M_i → M transverse toG such that the pull back groupoids G^f_fⁱ

i identify to Gi and fi(Mi) meets all the orbits of G.

Equivalently, a Morita equivalence is given by a linking manifold X with extra data: surjective smooth submersionsr :X→G⁽⁰⁾₁ ands:X →G⁽⁰⁾₂ and compositionsG₁×_s,rX→X,X×_s,rG₂→ X,X×_r,rX →G₂ and X×_s,sX →G₁ with natural associativity conditions (see [32] for details).

In the above situation, X is the manifold G^f_f¹

2 and the extra data are the range and source maps and the composition rules of the groupoidG^f_f¹^tf²

1tf2 ⇒M₁tM₂ (see [32]).

If the map r : X → G⁽⁰⁾₁ is surjective but s : X → G⁽⁰⁾₂ is not necessarily surjective, then G₁ is Morita equivalent to the restriction ofG₂ to the open saturated subspaces(X). We say thatG₁ is sub-Morita equivalent to G2.

2.5 Remarks on possible singularity

About corners We wish to emphasize a remark already made in [13]:

Many manifolds and groupoids that occur in our constructions have boundaries or corners. In fact all the groupoids we consider sit naturally inside Lie groupoids without boundaries as restrictions to closedsaturated subsets. This means that we consider subgroupoidsG^V_V =G_V of a Lie groupoid G

r,s

⇒ G⁽⁰⁾ whereV is a closed subset ofG⁽⁰⁾. Such groupoids, have a natural algebroid, adiabatic deformation, pseudodifferential calculus,etc. that are restrictions toV andG_V of the corresponding objects onG⁽⁰⁾ and G. We chose to give our definitions and constructions for Lie groupoids for the clarity of the exposition. The case of a longitudinally smooth groupoid over a manifold with corners is a straightforward generalization using a convenient restriction.

About non-Haudorffness Our groupoids need not be Hausdorff. Precisely, for G ⇒ G⁽⁰⁾, the manifold G may be a non-Haudorff manifold, but G⁽⁰⁾ will always be assumed to be Haudorff. Of course a non Hausdorff manifold is locally Hausdorff.

3 Two classical geometric constructions: Blowup and deformation to the normal cone

One of the main objects in our study is a Lie groupoidG based on a groupoid restricted to a half space. This corresponds to the inclusion of a hypersurface V of G⁽⁰⁾ into G and gives rise to the

“gauge adiabatic groupoid”gaG. The construction ofgaGis in fact a particular case of the blowup construction corresponding to the inclusion of a Lie subgroupoid into a groupoid. In this section, we will explain this general construction. We will give a more detailed description in the case of an inclusionV →Gwhen V is a submanifold ofG⁽⁰⁾.

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LetY be a manifold andX a locally closed submanifold (the same constructions hold if we are given an injective immersionX→Y). Denote byN_X^Y the (total space) of the normal bundle of X inY. 3.1 Deformation to the normal cone

The deformation to the normal coneDN C(Y, X) is obtained by gluingN_X^Y × {0}withY ×R^∗. The smooth structure ofDN C(Y, X) is described by use of any exponential map θ :U⁰ → U which is a diffeomorphism from an open neighborhoodU⁰ of the 0-section in N_X^Y to an open neighborhood U of X. The map θ is required to satisfy θ(x,0) = x for all x ∈ X and p_x ◦dθ_x = p⁰_x where px:TxY →(N_X^Y)x = (TxY)/(TxX) andp⁰_x :TxN_X^Y '(N_X^Y)x⊕(TxX)→(N_X^Y)x are the projections.

The manifold structure ofDN C(Y, X) is described by the requirement that:

a) the inclusionY ×R^∗ →DN C(Y, X) and

b) the map Θ : Ω⁰ ={((x, ξ), λ)∈N_X^Y×R; (x, λξ)∈U⁰} →DN C(Y, X) defined by Θ((x, ξ),0) = ((x, ξ),0) and Θ((x, ξ), λ) = (θ(x, λξ), λ)∈Y ×R^∗ ifλ6= 0.

are diffeomorphisms onto open subsets ofDN C(Y, X).

It is easily shown thatDN C(Y, X) has indeed a smooth structure satisfying these requirements and that this smooth structure does not depend on the choice ofθ. (See for example [5] for a detailed description of this structure).

In other words,DN C(Y, X) is obtained by gluingY ×R^∗ with Ω⁰ by means of the diffeomorphism Θ : Ω⁰∩(N_X^Y ×R^∗)→U ×R^∗.

Let us recall the following facts which are essential in our construction.

Definition 3.1. The gauge action ofR^∗. The groupR^∗ acts onDN C(Y, X) byλ.(w, t) = (w, λt) and λ.((x, ξ),0) = ((x, λ⁻¹ξ),0) with λ, t∈R^∗,w∈Y,x∈X andξ ∈(N_X^Y)x

.

Remarks 3.2. a) Since some natural Lie groupoids are non Hausdorff, we may have to consider non Hausdorff manifolds. In that case, the usual properness condition is replaced by local properness (cf. [13, Remark 2.5]). This means that every point has a neighborhood invariant under the action, on which the action is proper.

b) The gauge action is easily seen to be free and (locally) proper on the open subsetDN C(Y, X)\ X×R. Indeed, for (x, ξ, t)∈Ω⁰⊂N_X^Y×Rthe gauge action is given byλ.(x, ξ, t) = (x, λ⁻¹ξ, λt) under the map Θ⁻¹.

Definition 3.3. Functoriality. Given a commutative diagram of smooth maps X ^//

fX

Y

fY

X⁰ ^//Y⁰

where the horizontal arrows are inclusions of submanifolds, we naturally obtain a smooth map DN C(f) : DN C(Y, X) → DN C(Y⁰, X⁰). This map is defined by DN C(f)(y, λ) = (f_Y(y), λ) for y∈Y and λ∈R∗ and DN C(f)(x, ξ,0) = (f_X(x), f_N(ξ),0) for x ∈X and ξ ∈(N_X^Y)_x =T_xY /T_xX where fN : Nx → (N_X^Y⁰⁰)_f_X_(x) = T_f_X_(x)Y⁰/T_f_X_(x)X⁰ is the linear map induced by the differential (df_Y)x atx. This map is of course equivariant with respect to the gauge action ofR^∗.

Remarks 3.4. Let us make a few remarks concerning the DNC construction.

a) The map equal to identity on X×R^∗ and sending X× {0} to the zero section of N_X^Y leads to an embedding ofX×Rinto DN C(Y, X), we may often identify X×R with its image in DN C(Y, X). As DN C(X, X) =X×R, this corresponds to the fonctoriality of DN C for the diagram

X ^//

X

X ^//Y

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b) We have a natural smooth map π : DN C(Y, X) → Y ×R defined by π(y, λ) = (y, λ) (for y∈Y and λ∈R^∗) and π((x, ξ),0) = (x,0) (for x∈X ⊂Y and ξ ∈(N_X^Y)x a normal vector).

This corresponds to the fonctoriality ofDN C for the diagram X ^//

Y

Y ^//Y

c) To see that the smooth structure on DN C(Y, X) is well defined and establish functoriality, one may also note that the following maps are smooth:

• the mapπ :DN C(Y, X)→Y ×R defined above;

• given a smooth functionf :Y →Rwhose restriction toXis 0, the mapF_f :DN C(Y, X)→ R defined by Ff(y, λ) = f(y)

λ (for y ∈ Y and λ ∈ R^∗) and Ff(x, px(ξ),0) =dfx(ξ) for x ∈ X and ξ ∈ T_xY where p_x : T_xY → (N_X^Y)_x = T_xY /T_xX is the quotient map (note thatdfx vanishes onTxX).

These maps describe the smooth structure ofDN C(Y, X). Indeed given a manifoldZ, a map g:Z →DN C(Y, X) is smooth if and only ifπ◦g and the mapsFf◦gare smooth. Actually, a finite number of those give rise to an immersionDN C(Y, X)→Y ×R×R^k (at least locally - if we do not assumeX to be compact).

d) If Y₁ is an open subset of Y₂ such that X ⊂ Y₁, then DN C(Y₁, X) is an open subset of DN C(Y₂, X) and DN C(Y₂, X) is the union of the open subsets DN C(Y₁, X) and Y₂×R^∗. This reduces to the case whenY1 is a tubular neighborhood - and therefore to the case where Y is (diffeomorphic to) the total space of a real vector bundle over X. In that case one gets DN C(Y, X) = Y ×R and the gauge action of R^∗ on DN C(Y, X) =Y ×R is given by λ.((x, ξ), t) = ((x, λ⁻¹ξ), λt) (withλ∈R^∗,t∈R,x∈X and ξ∈Yx).

e) More generally, letE be (the total space of) a real vector bundle over Y. ThenDN C(E, X) identifies with the total space of the pull back vector bundle ˆπ^∗(E) over DN C(Y, X), where ˆ

π is the composition ofπ:DN C(Y, X)→Y ×R(remark b) with the projection Y ×R→Y. The gauge action ofR^∗ isλ.(w, ξ) = (λ.w, λ⁻¹ξ) for w∈DN C(Y, X) and ξ∈E_ˆ_π(w).

f) LetX₁be a (locally closed) smooth submanifold of a smooth manifoldY₁and letf :Y₂ →Y₁be a smooth map transverse toX1. Put X2 =f⁻¹(X1). Then the normal bundleN_X^Y²

2 identifies with the pull back of N_X^Y¹

1 by the restriction X₂ → X₁ of f. It follows that DN C(Y₂, X₂) identifies with the fibered productDN C(Y1, X1)×_Y₁ Y2.

g) More generally, let Y, Y₁, Y₂ be smooth manifolds and f_i :Y_i →Y be smooth maps. Assume thatf1 is transverse tof2. Let X⊂Y and Xi ⊂Yi be (locally closed) smooth submanifolds.

Assume thatf_i(X_i)⊂X and that the restrictionsg_i :X_i → X of f_i are transverse also. We thus have a diagram

X1 _

g1 //X_{_}

X2 _

g2

oo

Y₁ ^f¹ ^//Y ^oo ^f² Y₂

Then the maps DN C(f_i) : DN C(Y_i, X_i) → DN C(Y, X) are transverse and the deformation to the normal cone of fibered productsDN C(Y1×_Y Y2, X1×_X X2) identifies with the fibered productDN C(Y₁, X₁)×_{DN C(Y,X)}DN C(Y₂, X₂).

Note that construction f) is the particular caseX=Y =Y₁ of our construction here.

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Notation 3.5. For every locally closed subsetT ofRcontaining 0 and with the notation of remark 3.4.b below, we define :

DN C^T(Y, X) =Y ×(T \ {0})∪N_X^Y × {0}=π⁻¹(Y ×T)

It is therestriction ofDN C(Y, X) toT. We putDN C₊(Y, X) =DN C^R⁺(Y, X) =Y ×R^∗₊∪N_X^Y × {0}.

3.2 Blowup constructions

The blowupBlup(Y, X) is a smooth manifold which is a union ofY\Xwith the (total space)P(N_X^Y) of the projective space of the normal bundleN_X^Y ofXinY. We will also use the “spherical version”

SBlup(Y, X) ofBlup(Y, X) which is a manifold with boundary obtained by gluing Y \X with the (total space of the) sphere bundle S(N_X^Y). We have an obvious smooth onto map SBlup(Y, X)→ Blup(Y, X) with fibers 1 or 2 points. These spaces are of course similar and we will often give details in our constructions to the one of them which is the most convenient for our purposes.

We may view Blup(Y, X) as the quotient space of a submanifold of the deformation to the normal coneDN C(Y, X) under the gauge action ofR^∗.

Recall that the groupR^∗ acts onDN C(Y, X) byλ.(w, t) = (w, λt) andλ.((x, ξ),0) = ((x, λ⁻¹ξ),0) (withλ, t∈R^∗, w∈Y,x ∈X and ξ∈(N_X^Y)_x). According to remark 3.2.b), this action is free and (locally) proper on the open subsetDN C(Y, X)\X×R.

Definition 3.6. We put

Blup(Y, X) =

DN C(Y, X)\X×R

/R^∗ and

SBlup(Y, X) =

DN C+(Y, X)\X×R+

/R^∗₊.

Remark 3.7. With the notation of section 3.1, Blup(Y, X) is thus obtained by gluing Y \X = ((Y \X)×R^∗)/R^∗, with (Ω⁰\(X×R))/R^∗ using the map Θ which is equivariant with respect to the gauge action ofR^∗.

Choose a euclidean metric on N_X^Y. Let S = {((x, ξ), λ) ∈ Ω⁰; kξk = 1} and τ the involution of S given by ((x, ξ), λ) 7→ ((x,−ξ),−λ). The map Θ induces a diffeomorphism of S/τ with an open neighborhoodΩ ofe P(N_X^Y) in Blup(Y, X).

Since ˆπ :DN C(Y, X)→Y is invariant by the gauge action ofR^∗, we obtain a natural smooth map

˜

π : Blup(Y, X) → Y whose restriction to Y \X is the identity and whose restriction to P(N_X^Y) is the canonical projectionP(N_X^Y)→X⊂Y. This map is easily seen to be proper.

Remark 3.8. Note that, according to remark 3.4.e), DN C(Y, X) canonically identifies with the open subsetBlup(Y ×R, X× {0})\Blup(Y × {0}, X× {0}) ofBlup(Y ×R, X× {0}). Thus, since the mapBlup(Y ×R, X× {0}) →Y ×R is proper, one may think at Blup(Y ×R, X× {0}) as a

“local compactification” ofDN C(Y, X).

Example 3.9. In the case where Y is a real vector bundle over X, Blup(Y, X) identifies non canonically with an open submanifold of the bundle of projective spacesP(Y ×R) over X. Indeed, in that case DN C(Y, X) = Y ×R; choose a euclidian structure on the bundle Y. Consider the smooth involution Φ from (Y \X)×R onto itself which to (x, ξ, t) associates (x, ξ

kξk², t) (for x ∈ X, ξ ∈Yx, t∈R). This map transforms the gauge action of R^∗ on DN C(Y, X) into the action of R^∗by dilations on the vector bundleY×RoverXand thus defines a diffeomorphism ofBlup(Y, X) into its image which is the open setP(Y×R)\X whereX embeds intoP(Y×R) by mappingx∈X to the line{(x,0, t), t∈R}.

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Remark 3.10. Since we will apply this construction to morphisms of groupoids that need not be proper, we have to relax properness as in [13, Remark 2.2]: we will say that f :Y → X is locally proper if every point in X has a neighborhood V such that the restriction f⁻¹(V) → V of f is proper. In particular, ifY is a non Hausdorff manifold andX is a locally closed submanifold ofY, then the map Blup(Y ×R, X× {0})→Y ×R is locally proper

Functoriality

Definition 3.11 (Functoriality). Let

X ^//

fX

Y

fY

X⁰ ^//Y⁰

be a commutative diagram of smooth maps, where the horizontal arrows are inclusions of closed submanifolds. Let U_f = DN C(Y, X)\DN C(f)⁻¹(X⁰×R) be the inverse image by DN C(f) of the complement in DN C(Y⁰, X⁰) of the subset X⁰×R. We thus obtain a smooth map Blup(f) : Blup_f(Y, X)→ Blup(Y⁰, X⁰) where Blup_f(Y, X)⊂Blup(Y, X) is the quotient ofU_f by the gauge action ofR^∗.

In particular,

a) IfX ⊂Y₁are (locally) closed submanifolds of a manifoldY₂, thenBlup(Y₁, X) is a submanifold ofBlup(Y2, X).

b) Also, if Y₁ is an open subset of Y₂ such that X ⊂Y₁, then Blup(Y₁, X) is an open subset of Blup(Y₂, X) andBlup(Y₂, X) is the union of the open subsetsBlup(Y₁, X) and Y₂\X. This reduces to the case whenY1 is a tubular neighborhood.

Fibered products

LetX₁ be a (locally closed) smooth submanifold of a smooth manifold Y₁ and let f :Y₂→Y₁ be a smooth map transverse to X1. PutX2 =f⁻¹(X1). Recall from remark 3.4.f that in this situation DN C(Y2, X2) identifies with the fibered productDN C(Y1, X1)×_Y₁Y2. ThusBlup(Y2, X2) identifies with the fibered productBlup(Y₁, X₁)×_Y₁Y₂.

4 Constructions of groupoids

4.1 Normal groupoids, deformation groupoids and blowup groupoids 4.1.1 Definitions

Let Γ be a closed Lie subgroupoid of a Lie groupoidG. Using functoriality (cf. Definition 3.11) of theDN C and Blupconstruction we may construct a deformation and a blowup groupoid.

a) The normal bundle N_Γ^G carries a Lie groupoid structure with objects N_Γ^G(0)⁽⁰⁾. We denote by N_Γ^G ⇒N_Γ^G(0)⁽⁰⁾ this groupoid.

b) The manifold DN C(G,Γ) is naturally a Lie groupoid (unlike what was asserted in remark 3.19 of [19]). Its unit space is DN C(G⁽⁰⁾,Γ⁽⁰⁾); its source and range maps are DN C(s) and DN C(r); the space of composable arrows identifies withDN C(G⁽²⁾,Γ⁽²⁾) and its product with DN C(m) wherem denotes both products G⁽²⁾→Gand Γ⁽²⁾ →Γ.

c) The subset DN C(G,^ Γ) = Ur ∩Us of DN C(G,Γ) consisting of elements whose image by DN C(r) and DN C(s) is not in G⁽⁰⁾₁ ×R is an open subgroupoid of DN C(G,Γ): it is the restriction ofDN C(G,Γ) to the open subspaceDN C(G⁽⁰⁾, G⁽⁰⁾₁ )\G⁽⁰⁾₁ ×R.

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d) The groupR^∗ acts onDN C(G,Γ) via the gauge action by groupoid morphisms. Its action on DN C(G,^ Γ) is (locally) proper. Therefore the open subset Blupr,s(G,Γ) = DN C(G,^ Γ)/R^∗ of Blup(G,Γ) inherits a groupoid structure as well: its space of units is Blup(G⁽⁰⁾₂ , G⁽⁰⁾₁ ); its source and range maps areBlup(s) and Blup(r) and the product is Blup(m).

e) In the same way, we define the groupoid SBlup_r,s(G,Γ). It is the quotient of the restriction DN C^₊(G,Γ) ofDN C(G,^ Γ) toR+ by the action of R^∗+.

f) The singular part of SBlupr,s(G,Γ),i.e. its restriction to the boundary SN_V^M is the spherical normal groupoid SN_Γ^G. It is the quotient by the action of R^∗+ of the restriction ofN_Γ^G⇒N_V^M to the open subset N_V^M \V of its objects.

An analogous result about the groupoid structure on Blup_r,s(G,Γ) in the case of Γ⁽⁰⁾ being a hypersurface ofG⁽⁰⁾ can be found in [15, Theorem 2.8] (cf. also [14]).

4.1.2 Algebroid and anchor

The (total space of the) Lie algebroid AΓ is a closed submanifold (and a subbundle) of AG. The Lie algebroid of DN C(G,Γ) is DN C(AG,AΓ). Its anchor map is DN C(\G) : DN C(AG,AΓ) → DN C(T G⁽⁰⁾, TΓ⁽⁰⁾).

The groupoid DN C(G,Γ) is the union of its open subgroupoid G×R^∗ with its closed Lie subgroupoid N_Γ^G. The algebroid of G×R^∗ is AG×R^∗ and the anchor is just the map \G ×id : AG×R^∗ →T(G⁽⁰⁾×R^∗+).

4.1.3 Stability under Morita equivalence

LetG₁ ⇒G⁽⁰⁾₁ andG₂ ⇒G⁽⁰⁾₂ be Lie groupoids, Γ₁ ⊂G₁ and Γ₂⊂G₂ Lie subgroupoids. AMorita equivalence of the pair (Γ₁⊂G₁) with the pair (Γ₂ ⊂G₂) is given by a pair (X⊂Y) where Y is a linking manifold which is a Morita equivalence betweenG1 and G2 and X⊂Y is a submanifold of Y such that the mapsr, sand products ofY (see page 8) restrict to a Morita equivalenceXbetween Γ₁ and Γ₂.

Then, by functoriality,

• DN C(Y, X) is a Morita equivalence between DN C(G₁,Γ₁) andDN C(G₂,Γ₂),

• DN C₊(Y, X) is a Morita equivalence betweenDN C₊(G₁,Γ₁) andDN C₊(G₂,Γ₂),

• Blup_r,s(Y, X) is a Morita equivalence between Blup_r,s(G₁,Γ₁) and Blup_r,s(G₂,Γ₂),

• SBlupr,s(Y, X) is a Morita equivalence between SBlupr,s(G1,Γ1) and SBlupr,s(G2,Γ2)...

Note that if Y and X are sub-Morita equivalences, the above linking spaces are also sub-Morita equivalences.

4.1.4 Groupoids on manifolds with boundary

LetM be a manifold and V an hypersurface in M and suppose thatV cutsM into two manifolds with boundaryM =M−∪M₊ withV =M−∩M₊. Then by considering a tubular neighborhood of V inM,DN C(M, V) =M×R^∗∪ N_V^M × {0}identifies with M×R, the quotientDN C(M, V^ )/R^∗+

identifies with two copies ofM andSBlup(M, V) identifies with the disjoint unionM−tM₊. Under this last identification, the class under the gauge action of a normal vector inN_V^M\V× {0}pointing in the direction ofM+ is an element ofV ⊂M+.

LetM_b be manifold with boundaryV. Apiece of Lie groupoid is the restriction G=Ge^M_M^b

b toM_b of a Lie groupoid Ge⇒M whereM is a neighborhood of M_b and Ge is a groupoid without boundary.

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With the above notation, since V is of codimension 1 in M, SBlup(M, V) = Mb tM− where M−=M\M˚ is the complement inM of the interior ˚M =M_b\V ofM_b inM.

Let then Γ⇒V be a Lie subgroupoid ofG.e

We may constructSBlupr,s(G,e Γ) and consider its restriction to the open subsetMbofSBlup(M, V).

We thus obtain a longitudinally smooth groupoid that will be denotedSBlupr,s(G,Γ).

Note that the groupoid SBlup_r,s(G,Γ)⇒M_b is the restriction to M_b of a Lie groupoidG⇒M for whichM_b is saturated. IndeedSBlup_r,s(G,Γ) is an open subgroupoid ofSBlup_r,s(G,e Γ)⇒M_btM₋ which is a piece of the Lie groupoidDN C(^ G,e Γ)/R^∗+⇒DN C(M, V^ )/R^∗+'MtM. We may then letG be the restriction of DN C(M, V^ )/R^∗+ to one of the copies ofM.

In this way, we may treat by induction a finite number of boundary components and in particular groupoids on manifolds with (embeded) corners.

Remarks 4.1. a) Let us highlight that we donot assumeV to be saturated forG. In particular the boundary V can happen to be transverse to the groupoid G. In that casee G is in fact a manifold with corners. The blowup construction will changeGein such a way that V becomes a saturated subset.

b) IfMis a manifold with boundaryV andG=M×M is the pair groupoid, thenSBlup_r,s(G, V) is in fact the groupoid associated with the 0 calculus in the sense of Mazzeo (cf. [23, 28, 25]), i.e. the canonical pseudodifferential calculs associated with SBlupr,s(G, V) is the Mazzeo- Melrose’s 0-calculus. Indeed, the sections of the algebroid of SBlup_r,s(G, V) are exactly the vector fields ofM vanishing at the boundary V,i.e. those generating the 0-calculus.

c) In a recent paper [33], an alternative description ofSBlup_r,s(G, V) is given under the name of edge modification forG along the “AG-tame manifold” V, thus in particular V is transverse toG. This is essentially the gluing construction described in 4.3.4 below.

4.2 Description of the normal groupoids

In this section, we study the normal groupoidN_Γ^G i.e. the restriction ofDN C(G,Γ) to its singular part N_V^M, as well as the projective normal groupoid PN_Γ^G the restriction of Blup_r,s,(G,Γ) to its singular part PN_V^M. The groupoid N_Γ^G is a VB groupoid in the sense of Pradines [35, 22]. In the particular case where Γ = V is just a space, the groupoids N_Γ^G and PN_Γ^G are bundles of linear and projective groupoids over the baseV in a sense defined bellow. In that case, a Thom-Connes isomorphism computes theKK-theory ofC^∗(N_Γ^G) (prop. 4.5).

4.2.1 Linear groupoids

LetE be a vector space over a fieldK and let F be a vector sub-space. Let r, s:E→ F be linear retractions of the inclusionF →E.

The linear groupoid. The space E is endowed with a groupoid structure E with base F. The range and source maps are r and s and the product is (x, y) 7→ (x·y) = x+y−s(x) for (x, y) composable,i.e. such thats(x) =r(y). One can easily check:

• Sincer ands are linear retractions: r(x·y) =r(x) and s(x·y) =s(y).

• If (x, y, z) are composable, then (x·y)·z=x+y+z−(r+s)(y) =x·(y·z).

• The inverse ofx is (r+s)(x)−x.

Remarks 4.2. a) Note that, given E and linear retractions r and s on F, E ⇒ F is the only possible linear groupoid structure(¹) onE . Indeed, for anyx∈E one must havex·s(x) =x

1A linear groupoid is a groupoidGsuch that G⁽⁰⁾ and Gare vector spaces and all structure maps (unit, range, source, product) are linear.

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and r(x)·x = x. By linearity, it follows that for every composable pair (x, y) = (x, s(x)) + (0, y−s(x)) we havex·y=x·s(x) + 0·(y−s(x)) =x+y−s(x).

b) The morphism r −s : E/F → F gives an action of E/F on F by addition. The groupoid associated to this action is in factE.

c) Given a linear groupoid structure on a vector spaceE, we obtain the “dual” linear groupoid structure E^∗ on the dual space E^∗ given by the subspace F^⊥ ={ξ ∈ E^∗; ξ|_F = 0} and the two retractions r^∗, s^∗ :E^∗ → F^⊥ with kernels (kerr)^⊥ and (kers)^⊥: for ξ ∈E^∗ and x ∈ E, r^∗(ξ)(x) =ξ(x−r(x)) and similarlys^∗(ξ)(x) =ξ(x−s(x)).

The projective groupoid. The multiplicative group K^∗ acts on E by groupoid automorphisms.

This action is free on the restrictionEe=E \(kerr∪kers) of the groupoid E to the subset F\ {0}

ofE⁽⁰⁾ =F.

The projective groupoid is the quotient groupoidPE=Ee/K^∗. It is described as follows.

As a set PE =P(E)\(P(kerr)∪P(kers)) and P⁽⁰⁾ = P(F) ⊂P(E). The source and range maps r, s:PE →P(F) are those induced byr, s:E →F. The product ofx, y∈ PE withs(x) =r(y) is the linex·y={u+v−s(u); u∈x, v∈y; s(u) =r(v)}. The inverse of x∈ PE is (r+s−id)(x).

Remarks 4.3. a) WhenF is just a vector line,PE is a group. Let us describe it:

we have a canonical morphismh:PE→K^∗ defined byr(u) =h(x)s(u) foru∈x. The kernel of h is P(ker(r−s))\P(kerr). Note that F ⊂ ker(r −s) and therefore ker(r−s) 6⊂ kerr, whence kerr∩ker(r−s) is a hyperplane in ker(r−s). The group kerh is then easily seen to be isomorphic to ker(r)∩ker(s). Indeed, choose a non zero vectorw inF; then the map which assigns tou ∈ker(r)∩ker(s) the line with direction w+u gives such an isomorphism onto kerh.

Then:

• Ifr=s,PE is isomorphic to the abelian group ker(r) = ker(s).

• Ifr 6=s, choosex such that r and sdo not coincide on x and let P be the plane F⊕x.

The subgroupP(P)\ {kerr∩P,kers∩P}ofPEis isomorphic throughhwithK^∗. It thus defines a section ofh. In that casePE is the group of dilations (ker(r)∩ker(s))o K^∗. b) In the general case, letd∈P(F). PutE_d^d=r⁻¹(d)∩s⁻¹(d).

• The stabilizer (PE)^d_d is the groupPE_d^d=P(E_d^d)\(P(kerr)∪P(kers)) described above.

• The orbit of a line d is the set of r(x) for x ∈ PE such that s(x) = d. It is therefore P(d+r(kers)).

c) the following are equivalent:

(i) (r, s) :E→F ×F is onto;

(ii) r(kers) =F;

(iii) (r−s) :E/F →F is onto;

(iv) the groupoidPE has just one orbit.

d) Whenr =s, the groupoid PE is the product of the abelian groupE/F by the space P(F).

Whenr 6=s, the groupoid Eeis Morita equivalent toE sinceF\ {0}meets all the orbits of E.

IfKis a locally compact field and r6=s, the smooth groupoidPE is Morita equivalent to the groupoid crossed-productEeo K^∗.

In all cases, whenKis a locally compact field, PE is amenable.

Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus