Examples of deformation groupoids and blowup groupoids

We examine some particular cases of inclusions of groupoidsG₁ ⊂G₂. The various constructions of deformation to the normal cone and blow-up allow us to recover many well known groupoids.

As already noted in the introduction, our constructions immediately extend to the case where we restrict to a closed saturated subset of a smooth groupoid, in particular for manifolds with corners.

4.3.1 Inclusion F ⊂E of vector spaces

LetE be a real vector space - considered as a group - and F a vector subspace ofE. The inclusion of groupsF →E gives rise to a groupoid DN C(E, F). Using any supplementary subspace of F in E, we may identify the groupoidDN C(E, F) withE×R⇒R. ItsC^∗-algebra identifies then with C₀(E^∗×R).

More generally, ifF is a vector-subbundle of a vector bundle E over a manifold M (considered as a family of groups indexed byM), then the groupoidDN C(E, F)⇒M ×Ridentifies with E×R and itsC^∗-algebra is C₀(E^∗×R).

Let p_E : E → M be a vector bundle over a manifold M and let V be a submanifold of M. Let pF :F →V be a subbundle of the restriction ofE toV. We use a tubular construction and find an open subsetU ofM which is a vector bundleπ:Q→V. Usingπ, we may extendF to a subbundle F_U of the restriction to F on U. Using that, we may identify DN C(E, F) with the open subset E×R^∗∪p⁻¹_E (U)×Rof E×R. ItsC^∗-algebra identifies then withC0(E^∗×R^∗∪p⁻¹_E∗(U)×R).

4.3.2 Inclusion G⁽⁰⁾ ⊂G: adiabatic groupoid

The deformation to the normal cone DN C(G, G⁽⁰⁾) is the adiabatic groupoid G_ad ([31, 34]), it is obtained by using the deformation to the normal cone construction for the inclusion ofG⁽⁰⁾ as a Lie subgroupoid ofG. The normal bundleN_G^G(0) is the total space of the Lie algebroidA(G) ofG. Note that its groupoid structure coincides with its vector bundle structure. Thus,

DN C(G, G⁽⁰⁾) =G×R^∗∪A(G)× {0}⇒G⁽⁰⁾×R.

We often denote DN C(G, G⁽⁰⁾) by G_ad and G⁺_ad, G^[0,1]_ad ,G^[0,1)_ad its restriction respectively toR+, to [0,1] and to [0,1).

Note thatBlup(G⁽⁰⁾, G⁽⁰⁾) =∅=Blup_r,s(G, G⁽⁰⁾).

The particular case whereGis the pair groupoidM×M is the original construction of the “tangent groupoid” of Alain Connes ([8]).

4.3.3 Gauge adiabatic groupoid Start with a Lie groupoidG⇒V. LetG×(R×R)

˜ r,˜s

⇒V ×Rbe the product groupoid ofG with the pair groupoid over R.

First notice that sinceV × {0} is a codimension 1 submanifold in V ×R, SBlup(V ×R, V × {0}) is canonically isomorphic toV ×(R−tR+). Then SBlup_˜r,˜s(G×(R×R), V × {(0,0)})^V_V^×_×^R+

R+ is the semi-direct product groupoid G_ad(V ×R+)o R^∗+:

SBlup_˜r,˜s(G×(R×R), V × {(0,0)})^V_V^×_×^R+

R+ =G⁺_ado R^∗ ⇒V ×R+ . In other words, SBlup˜r,˜s(G×(R×R), V × {(0,0)})^V_V^×_×^R+

R+ is the gauge adiabatic groupoid used in [11]; we often denote it byGga .

Indeed, asG×(R×R) is a vector bundle overG,DN C(G×(R×R), V × {(0,0)})'DN C(G, V)× R² (remark 3.4.e). Under this identification, the gauge action of R^∗ is given by λ.(w, t, t⁰) = (λ.w, λ⁻¹t, λ⁻¹t⁰). The maps DN C(˜s) and DN C(˜r) are respectively (w, t, t⁰) 7→ (DN C(s)(w), t⁰) and (w, t, t⁰)7→(DN C(r)(w), t). It follows thatSBlup_r,˜˜s(G×(R×R), V × {(0,0)}) is the quotient by the diagonal action ofR^∗₊ of the open subsetDN C(G, V)×(R^∗)² of DN C+(G, V)×R². According to the description of the groupoid of a group action on a groupoid given in [13, section 2.3] it is isomorphic to DN C(G, V)₊o R^∗+× {−1,+1}² where{−1,+1}² is the pair groupoid over {−1,+1}.

4.3.4 Inclusion of a transverse submanifold of the unit space

LetG be a Lie groupoid with set of objectsM =G⁽⁰⁾ and let V be a submanifold of M. We now study the special case of normal and blowup groupoids DN C(G, V) and Blupr,s(G, V) (as well as SBlup_r,s(G, V)) associated to the groupoid morphismV →G.

Put ˚M = M \V. Let N = N_V^G and N⁰ = N_V^M be the normal bundles. We identify N⁰ with a subbundle of N by means of the inclusion M ⊂ G. The submersions r, s : G → M give rise to bundle morphisms r^N, s^N :N → N⁰ that are sections of the inclusion N⁰ → N. By construction, using remark 4.2.a), the groupoid DN C(G, V) is the union of G×R^∗ with the family of linear groupoidsN_rN,s^N(N). It follows thatBlup_r,s(G, V) is the union ofG^M˚_˚

M with the family (PN, r^N, s^N) of projective groupoids.

If V is transverse to G, the bundle map r^N −s^N :N =N_V^G → N⁰ = N_V^M is surjective; it follows that

• L_rN,s^N(N) identifies with the pull-back groupoid A(G^V_V)q

qwhereq :N⁰→V is the projection,

• (PN, r^N, s^N) with the pull-back groupoid A(G^V_V)o R^∗ρ

ρ whereρ:P(N⁰)→V is the projec-tion,

• (SN, r^N, s^N) with the pull-back groupoid A(G^V_V)o R^∗+

p

p wherep:S(N⁰)→V is the projec-tion.

Let us give a local description of these groupoids in the neighborhood of the transverse submanifold V.

Put ˚G=G^M_M^\V_\V. Upon arguing locally, we can assume that V is compact.

By Remark 2.2, V admits a tubular neighborhoodW 'N_V^M such that G^W_W is the pull back of G^V_V by the retractionq :W →V.

The normal groupoidDN C(G^W_W, V) identifies with the pull back groupoid (DN C(G^V_V, V))^q_q of the adiabatic deformationDN C(G^V_V, V) = (G^V_V)ad by the mapq:N_V^M →V.

The (spherical) blowup groupoidSBlup_r,s(G^W_W, V) identifies with the pull back groupoid (DN C₊(G^V_V, V)o R^∗₊)^p_p of the gauge adiabatic deformationDN C+(G^V_V, V)o R^∗₊= (G^V_V)ga by the mapp:SN_V^M →V.

In order to getSBlupr,s(G, V), we then may glue (DN C+(G^V_V, V)o R^∗+)^p_p with ˚Gin their common open subset (G^V_V)^q_qW\V

W\V 'G^W\V_W\V.

4.3.5 Inclusion G^V_V ⊂G for a transverse hypersurface V of G: b-groupoid

IfV is a hypersurface ofM, the blowupBlup(M×M, V×V) is just the construction of Melrose of the b-space. Its open subspaceBlupr,s(M×M, V×V) is the associated groupoid of Monthubert [29, 30].

Moreover, ifG is a groupoid on M and V is transverse to G we can form the restriction groupoid G^V_V ⊂ G which is a submanifold of G. The corresponding blow up construction Blupr,s(G, G^V_V) identifies with the fibered productBlupr,s(M ×M, V ×V)×_M×M G (cf. remark 3.4.f).

Iterating (at least locally) this construction, we obtain theb-groupoid of Monthubert for manifolds with corners -cf. [29, 30].

Remark 4.7. The groupoid Blupr,s(G, V) corresponds to inflating all the distances when getting close toV.

The groupoid Blupr,s(G, G^V_V) is a kind of cylindric deformation groupoid which is obtained by pushing the boundary V at infinity but keeping the distances along V constant.

Remark 4.8. Intermediate examples between these two are given by a subgroupoid Γ⇒V ofG^V_V. In the case where G = M ×M, such a groupoid Γ is nothing else than the holonomy groupoid Hol(V,F) of a regular foliationF ofV (with trivial holonomy groups). The groupoidSBlup_r,s(M× M, Hol(V,F)) is a holonomy groupoid of a singular foliation of M: the sections of its algebroid. Its leaves areM\V and the leaves of (V,F). The corresponding calculus, whenM is a manifold with a boundaryV is Rochon’s generalization ([36]) of theφ calculus of Mazzeo and Melrose ([24]).

Iterating (at least locally) this construction, we obtain the holonomy groupoid associated to a stratified space in [10].

4.3.6 Inclusion G^V_V ⊂Gfor a saturated submanifold V of G: shriek map for immersion Suppose now thatV is saturated thusG^V_V =GV =G^V.

In such a situation the groupoidG^V_V acts on the normal bundleN_G^GV

V =r^∗(N_V^G(0)) andDN C(G, G^V_V)⇒ DN C(G⁽⁰⁾, V) coincides with the normal groupoid of the immersionϕ:G^V_V →G. This construction was defined in the case of foliation groupoids in [19, section 3] and was used in order to defineϕ!as its associatedKK-element.

4.3.7 Inclusion G₁ ⊂G₂ with G⁽⁰⁾₁ =G⁽⁰⁾₂

This is the case for the tangent and adiabatic groupoid discussed above. Let us mention two other kinds of this situation² that can be encountered in the literature:

a) The case of a subfoliationF₁ of a foliation F₂ on a manifold M: shriek map for submersion.

As pointed out in remark 3.19 of [19] the corresponding deformation groupoidDN C(G₂, G₁) gives an alternative construction of the element ϕ! whereϕ:M/F₁ →M/F₂ is a submersion of leaf spaces.

b) The case of a subgroup of a Lie group.

• If K is a maximal compact subgroup of a reductive Lie group G, the connecting map associated to the exact sequence of DN C(G, K) is the Dirac extension mapping the twisted K-theory ofK to theK-theory ofC_r^∗(G) (see [16]).

• In the case where Γ is a dense (non amenable) countable subgroup of a compact Lie group K, the groupoidDN C(K,Γ) was used in [17] in order to produce a Hausdorff groupoid for which the Baum-Connes map is not injective.

2Note that in this caseBlup(G⁽⁰⁾₂ , G⁽⁰⁾₁ ) =∅, whenceBlupr,s(G2, G1) =∅.

4.3.8 Wrong way functoriality

Letf :G1 →G2 be a morphism of Lie groupoids. If f is an (injective) immersion the construction ofDN C+(G2, G1) gives rise to a short exact sequence

0−→C^∗(G2×R^∗+)−→C^∗(DN C+(G2, G1))−→C^∗(N_G^G2

1)−→0.

and consequently to a connecting map from the K-theory of the C^∗-algebra of the groupoid N_G^G2

1, which is aVB groupoid overG1, to theK-theory ofC^∗(G2). This wrong way functoriality map will be discussed in the next section.

More generally letG=G⁽⁰⁾₁ ×G₂×G⁽⁰⁾₁ be the product ofG₂ by the pair groupoid ofG⁽⁰⁾₁ . Assume that the mapx7→(r(x), f(x), s(x)) is an immersion fromG₁→ G.

The above construction gives a map from K∗(C^∗(N_G^G

1)) to K∗(C^∗(G)) which is isomorphic to K∗(C^∗(G2)) since the groupoids G2 and G are canonically Morita equivalent.

5 The C

^∗

-algebra of a deformation and of a blowup groupoid, full symbol and index map

Let G⇒ M be a Lie groupoid and Γ⇒ V a Lie subgroupoid of G. The groupoids DN C+(G,Γ) and SBlup_r,s(G,Γ) that we constructed admit the closed saturated subsets N_V^M × {0} and SN_V^M respectively. We apply results of [13] in order to compute variousKK-elements involved in index theory for such situation.

In order to shorten the notation we put ˚M =M \V. Thefull symbol algebras are the quotientC^∗-algebras:

• Σ_{DN C+}(G,Γ) = Ψ^∗(DN C+(G,Γ))/C^∗(G×R+^∗);

• Σ_SBlup(G,Γ) = Ψ^∗(SBlup_r,s(G,Γ))/C^∗(G^M_M^˚_˚).

They give rise to the exact sequences

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}^∂ ₊ of groupoidC^∗-algebras as well as index type ones

0−→C^∗(G^M_M^˚_˚)−→Ψ^∗(SBlupr,s(G,Γ))−→Σ_SBlup(G,Γ)−→0 E_SBlup^indf and

0−→C^∗(G×R^∗+)−→Ψ^∗(DN C₊(G,Γ))−→Σ_{DN C+}(G,Γ)−→0 E_{DN C}^indf ₊ We will compare the exact sequences given byDN C and bySBlup.

IfV isAG-small (see notation 5.5 below), we will show that, in a sense,DN C and SBlupgive rise to equivalent exact sequences - both for the “connecting” ones and for the “index” ones.

We will then compare these elements with a coboundary construction.

We will compute these exact sequences when Γ =V ⊂ M. Finally, we will study a refinement of these constructions using relativeK-theory.

5.1 “DNC” versus “Blup”

Let Γ⇒V be a submanifold and a subgroupoid of a Lie-groupoidG⇒M. We will further assume that the groupoid Γ is amenable. We still put ˚M = M \V and let ˚N_Γ^G be the restriction of the groupoidN_Γ^G to the open subset ˚N_V^M =N_V^M \V of its unit space N_V^M.

5.1.1 The connecting element

As the groupoid Γ is amenable we have exact sequences both for the reduced and for the maximal C^∗-algebras:

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}^∂ ₊ By amenability, these exact sequences admit completely positive cross sections and therefore define elements ∂_SBlup^G,Γ ∈KK¹(C^∗( ˚N_Γ^G/R^∗+), C^∗(G^M_M^˚_˚)) and∂_{DN C}^G,Γ

+ ∈KK¹(C^∗(N_Γ^G), C^∗(G×R^∗+)).

With the notation of section 4.1, write DN C+ for DN C restricted to R+ and DN C^+ for DN C^ restricted to R+.

By [13, section 5.3], we have a diagram where the vertical arrows are KK¹-equivalences and the squares commute inKK-theory.

We also have a commutative diagram where the vertical maps are inclusions:

0 ^//C^∗(G^M˚_˚

Put Σ

DN C^+(G,Γ) = Ψ^∗(DN C^+(G,Γ))/C^∗(G^M_M^˚_˚×R^∗₊). By [13, prop. 5.4], we have a diagram where the vertical arrows areKK¹-equivalences and the squares commute in KK-theory.

0 ^//C^∗(G^M˚_˚

We also have a commutative diagram where the vertical maps are inclusions:

0 ^//C^∗(G^M˚_˚

Also, if ˚M meets all the orbits of G, the inclusioniis a Morita equivalence. In these cases ∂_{DN C}^G,Γ

+

determines ∂_SBlup^G,Γ .

Definition 5.5. We will say that V is AG-small if for every x ∈ V, the composition AG_x −→^\x TxM −→(N_V^M)x is not the zero map

If V is AG-small, then the orbits of the groupoid N_Γ^G are never contained in the 0 section, i.e.

they meet the open subset ˚N_V^M, and in fact the set V × {0}is small in every orbit of the groupoid DN C(G,Γ). It follows that the mapjis an isomorphism - as well of course as j⁰ andj⁰⁰of diagram (5.1). In that case,∂_{DN C}^G,Γ

+ and ∂_SBlup^G,Γ correspond to each other under these isomorphisms.

Proposition 5.6. (cf. [18, 12]) Let G ⇒Y be a Lie groupoid and let X ⊂Y be a (locally closed)

Proposition 5.7. We assume that Γ is amenable and thatV isAG-small.

Then, the inclusionsjΣ : Σ

DN C^+(G,Γ)→ΣDN C+(G,Γ),jΨ: Ψ^∗(DN C^+(G,Γ))→Ψ^∗(DN C+(G,Γ)) andj_symb :C₀(SA^∗(DN C^₊(G,Γ)))→C₀(SA^∗(DN C₊(G,Γ))) are KK-equivalences.

Proof. We have a diagram

0

0 ^//C^∗(DN C^₊(G,Γ)) ^//

j

Ψ^∗(DN C^₊(G,Γ)) ^//

jΨ

C₀(SA^∗(DN C^₊(G,Γ)) ^//

jsymb

0

0 ^//C^∗(DN C₊(G,Γ)) ^//Ψ^∗(DN C₊(G,Γ)) ^//C₀(SA^∗(DN C₊(G,Γ)) ^//

0

C0(SA^∗G|V ×R+)

0

Asj is an equality, we find an exact sequence

0−→Ψ^∗(DN C^+(G,Γ))−→^jΨ Ψ^∗(DN C+(G,Γ))−→C0(SA^∗G_|V ×R+)−→0.

As j⁰ : C^∗(G^M˚_˚

M ×R^∗₊) → C^∗(G×R^∗₊) is also an equality, we find (using diagram (5.2)) an exact sequence

0−→Σ

DN C^+(G,Γ))−→^jΣ Σ_{DN C+}(G, V)−→C₀(SA^∗G_|V ×R+)−→0.

As the algebra C₀(SA^∗G_|V ×R+) is contractible, we deduce that j_symb and then j_Ψ and j_Σ are KK-equivalences.

As a summary of these considerations, we find:

Theorem 5.8. Let G⇒M be a Lie groupoid and Γ⇒V a Lie subgroupoid of G. Assume that Γ is amenable and put M˚ = M\V. Let i:C^∗(G^M_M^˚_˚) → C^∗(G) be the inclusion. Put βˆ⁰⁰ =j_∗⁰⁰(β⁰⁰) ∈ KK¹(C^∗(SN_Γ^G), C^∗(N_Γ^G))and βˆΣ = (jΣ)∗(βΣ)∈KK¹(ΣSBlup(G,Γ),ΣDN C+(G, V)).

a) We have equalities

• ∂_SBlup^G,Γ ⊗[i] = ˆβ⁰⁰⊗∂_{DN C}^G,Γ

+ ∈KK¹(C^∗(SN_Γ^G), C^∗(G))and

• indf^G,Γ_SBlup⊗[i] = ˆβΣ⊗indf^G,Γ_{DN C+} ∈KK¹(C^∗(Σ_SBlup(G,Γ), C^∗(G))

b) If V isAG-small, then iis an isomorphism and the elements βˆ⁰⁰ andβˆ_Σ are invertible.

5.2 The KK-element associated with DNC

Dans le document Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus (Page 17-23)