The case of a submanifold of the space of units

Let G be a Lie groupoid with objects M and let Γ = V ⊂ M be a closed submanifold of M. In this section, we push further the computations the connecting maps and indicesi.e. the connecting maps of the exact sequencesE_SBlup^∂ , E_{DN C}^∂ ₊, E^ind_SBlup^f and E_{DN C}^indf ₊.

5.3.1 Connecting map and index map

From [13, propositions 4.1, 4.6, 4.7] and fact 5.10, we find Proposition 5.11. a) The index elementind_NG

V ∈KK(C₀(A^∗N_V^G), C^∗(N_V^G))is invertible.

b) The inclusion j: Σ_N^M

V ×{0}(DN C+(G, V)),→ΣDN C+(G, V) is invertible in KK-theory.

c) The C^∗-algebra ΣDN C+(G, V) is naturally KK¹-equivalent with the mapping cone Cχ of the map χ:C₀(A^∗G×R^∗+)→C₀(DN C₊(M, V))defined by χ(f)(x) =

(f(x,0) ifx∈M×R^∗+

0 ifx∈N_V^M. d) The connecting element ∂_{DN C}^G,V ₊ ∈ KK¹(C^∗(N_V^G), C^∗(G×R^∗₊)) = KK(C^∗(N_V^G), C^∗(G)) is

δ_V^G= ind⁻¹_NG V

⊗[ϕ]⊗ind_G where ϕ:C₀(A^∗N_V^G)→ C₀(A^∗G) is the inclusion using the tubular neighborhood construction.

e) Under the KK¹ equivalence of c), the full index element

indf^G,V_{DN C+} ∈KK¹(Σ_{DN C+}(G, V), C^∗(G×R^∗+)) =KK¹(C_χ, C^∗(G)) isq^∗([Bott]⊗

C

ind_G) where q :Cχ →C0(A^∗G×R^∗+) is evaluation at 0.

The element [χ]∈KK(C₀(A^∗G×R^∗+), C₀(DN C₊(M, V))) is the Kasparov product of the “Euler ele-ment” of the bundleA^∗Gwhich is the class inKK(C₀(A^∗G), C₀(M)) =KK(C₀(A^∗G×R^∗+), C₀(M× R^∗₊)) of the map x 7→ (x,0) with the inclusionC0(M ×R^∗₊) →C0(DN C+(M, V)). It follows that [χ] is often the zero element of KK(C₀(A^∗G×R^∗+), C₀(DN C₊(M, V))). In particular, this is the case when the Euler class of the bundle A^∗G vanishes. In that case, the algebra Σ_{DN C+}(G, V) is KK-equivalent toC0(A^∗G)⊕C0(DN C+(M, V)).

IfV is AG small, then, by theorem 5.8,∂_SBlup^G,V and indf^G,V_SBlup are immediately deduced from propo-sition 5.11.

Remark 5.12. LetM_b be a manifold with boundary and V =∂M_b. Put ˚M =M_b\V. LetGbe a piece of Lie groupoid onMb in the sense of section 4.1.4. ThusGis the restriction of a Lie groupoid Ge⇒M, whereM is a neighborhood ofM_b. Recall that in this situation,SBlup(M, V) =M_btM−, where M = M_b ∪M− and M ∩M− = V, and we let SBlup_r,s(G, V) ⇒ M_b be the restriction of SBlup_r,s(G, Ve ) to M_b.

Let us denote by ˚N_V^G the open subset ofN_V^Ge made of (normal) tangent vectors whose image under the differential of the source and range maps of Ge are non vanishing elements of N_V^M pointing in the direction ofM_b. The groupoidSBlupr,s(G, V) is the union ˚N_V^G/R^∗+∪G^M_M^˚_˚.

We have exact sequences

0→C^∗(G^M˚_˚

M)→C^∗(SBlup_r,s(G, V))→C^∗( ˚N_V^G/R^∗₊)→0 0→C^∗(G^M_M^˚_˚)→Ψ^∗(SBlupr,s(G, V))→Σ_SBlup(G, V)→0.

AsV is of codimension 1, we find thatV isAG-small if and only if it is transverse toe G. In that case,e Proposition 5.11 computes the KK-theory of C^∗( ˚N_V^G/R^∗₊) and of ΣSBlup(G, V) and the KK-class of the connecting maps of these exact sequences.

In particular, we obtain a six term exact sequence

K₀(C(M_b)) ^//K₀(Σ_SBlup(G, V)) ^//K₁(C₀(A^∗G^M˚_˚

M))

χ

K₀(C₀(A^∗G^M˚_˚

M))

χ

OO

K₁(Σ_SBlup(G, V))

oo K₀(C(M_b))^oo

and the index map K∗(Σ_SBlup(G, V)) → K∗+1(G^M˚_˚

M) is the composition of K∗(Σ_SBlup(G, V)) → K∗+1(C₀(A^∗G^M_M^˚_˚)) with the index map of the groupoid G^M_M^˚_˚.

This holds, in particular, ifG=Mb×Mb since the boundaryV =∂Mb is transverse toGe=M×M.

Note that in that case, χ = 0 (in KK(C₀(T^∗M˚), C₀(M_b))) so that we obtain a (non canonically) split short exact sequence:

0 ^//K∗(C₀(M_b)) ^//K∗(Σ_SBlup(G, V)) ^//K∗+1(C₀(A^∗G^M˚_˚

M)) ^//0.

5.3.2 The index map via relative K-theory It follows now from [13, prop. 4.8]:

Proposition 5.13.Letψ_{DN C} :C₀(DN C₊(M, V))→Ψ^∗(DN C₊(G, V))be the inclusion map which associates to a (smooth) functionf the order0 (pseudo)differential operator multiplication byf and σ_{f ull} : Ψ^∗(DN C+(G, V))→ΣDN C+(G, V)the full symbol map. PutµDN C =σ_{f ull}◦ψ_{DN C}. Then the relative K-group K∗(µ_{DN C}) is naturally isomorphic to K∗+1(C₀(A^∗G)). Under this isomorphism, ind_rel :K∗(µ_{DN C})→K∗(C^∗(G×R^∗₊)) =K∗+1(C^∗(G))identifies with ind_G. Let us say also just a few words on the relative index map for SBlup_r,s(G, V), i.e. for the map µSBlup : C0(SBlup+(M, V)) → ΣSBlup(G, V) which is the composition of the inclusion ψSBlup : C0(SBlup(M, V) → Ψ^∗(SBlupr,s(G, V)) with the full index map σ_{f ull} : Ψ^∗(SBlupr,s(G, V)) → Σ_SBlup(G, V)), and the corresponding relative index map ind_rel : K∗(µ_SBlup) → K∗(C^∗(G^M˚_˚

M)).

Equivalently we wish to compute the relative index map ind_rel : K∗(µ

DN C^) → K∗+1(C^∗(G^M_M^˚_˚)), whereµ

DN C^ :C0(DN C^+(M, V))→Σ

DN C^+(G, V). We restrict to the case whenV is AG small.

We have a diagram

0 ^//C₀(DN C^₊(M, V)) ^//C₀(DN C₊(M, V)) ^//C₀(V ×R+) ^//0

and it follows that the inclusion C₀(DN C^₊(M, V))→C₀(DN C₊(M, V)) is aKK-equivalence.

Since the inclusions Ψ^∗(DN C^+(G, V)) → Ψ^∗(DN C+(G, V)) and Σ

DN C^+(G, V) → Σ_{DN C+}(G, V) are also KK-equivalences (prop. 5.7), it follows that the inclusion C_µ

^

DN C → C_{µDN C} of mapping cones is a KK-equivalence - and therefore the relative K-groups K∗(µ

DN C^) and K∗(µ_{DN C}) are naturally isomorphic. Using this, together with the Connes-Thom isomorphism, we deduce:

Corollary 5.14. We assume thatV is AGsmall a) The relative K-group K∗(µ

DN C^) is naturally isomorphic to K∗+1(C₀(A^∗G)). Under this iso-morphism, indrel:K∗(µ

DN C^)→K∗(C^∗(G×R^∗₊)) =K∗+1(C^∗(G)) identifies withindG. b) The relative K-group K∗(µ_SBlup) is naturally isomorphic to K∗(C₀(A^∗G)). Under this

iso-morphism, ind_rel:K∗(µ_SBlup)→K∗(C^∗(G))identifies with indG.

6 A Boutet de Monvel type calculus

In this section, we consider theSBlupconstruction in the special case of a transverse submanifold of the unit space of a groupoid. We use the bimodule that we constructed in [11] in order to obtain an algebra resembling the algebra of 2×2 matrices in the Boutet de Monvel pseudodifferential calculus (of order 0) on manifolds with boundary.

From now on, we suppose thatV is atransverse submanifold ofM with respect to the Lie groupoid G⇒M. In particularV isAG-small - of course, we assume that (in every connected component of V), the dimension ofV is strictly smaller than the dimension ofM.

6.1 The Poisson-trace bimodule

As V is transverse to G, the groupoid G^V_V is a Lie groupoid, so that we can construct its “gauge adiabatic groupoid” (G^V_V)ga (see section 4.3.3).

In [11], we constructed a bi-module relating the C^∗-algebra of the groupoid (G^V_V)_ga and the C^∗ -algebra of pseudodifferential operators ofG^V_V.

In this section,

• We first show that the groupoid (G^V_V)ga, is (sub-) Morita equivalent to SBlupr,s(G, V) (cf.

also section 4.3.4 for a local construction).

• Composing the resulting bimodules, we obtain the “Poisson-trace” bimodule that relates the C^∗-algebrasC^∗(SBlup_r,s(G, V)) and Ψ^∗(G^V_V).

6.1.1 The SBlup_r,s(G, V)−(G^V_V)_ga-bimodule E(G, V)

Define the mapj :Mt(V ×R)→M by letting j₀ :M →M be the identity and j₁ :V ×R→M the composition of the projectionV ×R→V with the inclusion. Let G=G^j_j. AsV is assumed to be transverse, the map j is also transverse, and therefore G is a Lie groupoid.

It is the union of four clopen subsets

• the groupoidsG^j_j⁰

0 =G=G_M^M and G^j_j¹

1 =G^V_V ×(R×R) =G_V^V_×^×R

R.

• thelinking spaces G^j_j⁰

1 =G_V^M_×

R=G_V ×R andG^j_j¹

0 =G_M^V×R=G^V ×R.

By functoriality, we obtain a sub-Morita equivalence ofSBlup_r,s(G^V_V×R×R, V) andSBlup_r,s(G, V) (see section 4.1.3).

Let us describe this sub-Morita equivalence in a slightly different way:

Let also Γ =V × {0,1}², sitting inG:

V × {(0,0)} ⊂G=G^j_j⁰

0 ; V × {(0,1)} ⊂GV × {0} ⊂G^j_j⁰

1 ; V × {(1,0)} ⊂G^V × {0} ⊂G^j_j¹

0 ; V × {(1,1)} ⊂G^V_V × {(0,0)} ⊂G^j_j¹

1 .

It is a subgroupoid of G. The blowup construction applied to Γ ⊂ G gives then a groupoid SBlupr,s(G,Γ) which is the union of:

SBlupr,s(G, V) ; SBlupr,s(GV ×R, V) ; SBlupr,s(G^V ×R, V) ; SBlupr,s(G^V_V ×R×R, V).

Recall thatSBlup(V×R, V×{0})'V×(R−tR+). ThusSBlup_r,s(G,Γ) is a groupoid with objects SBlup(M, V)tV ×R−tV ×R+.

The restriction ofSBlup_r,s(G,Γ) toV×R+coincides with the restriction ofSBlup_r,s(G^V_V×R×R, V) toV ×R+: it is the gauge adiabatic (G^V_V)_ga groupoid of G^V_V (cf. section 4.3.3).

PutSBlup_r,s(G_V ×R, V)₊=SBlup_r,s(G,Γ)^SBlup(M,V_V× ⁾

R+ . It is a linking space between the groupoids SBlup_r,s(G, V) and (G^V_V)_ga. Put also SBlup_r,s(G^V ×R, V)₊=SBlup_r,s(G,Γ)^V_SBlup(M,V^×R+ ₎.

With the notation used in fact 2.3, we define the C^∗(SBlupr,s(G, V)) −C^∗((G^V_V)ga)-bimodule E(G, V) to be C^∗(SBlup_r,s(G_V × R, V)₊). It is the closure of C_c(SBlup_r,s(G_V × R, V)₊) in C^∗(SBlup_r,s(G,Γ)). It is a full Hilbert-C^∗(SBlup_r,s(G, V))−C^∗((G^V_V)_ga)-module.

The Hilbert-C^∗((G^V_V)ga)-module E(G, V) is full and K(E(G, V)) is the ideal C^∗(SBlupr,s(G^Ω_Ω, V)) where Ω =r(G_V) is the union of orbits which meet V.

Notice that Ω = M \V tV ×R^∗ and F = SN_V^M tV tV gives a partition by respectively open and closed saturated subsets of the units ofSBlup_r,s(G,Γ). FurthermoreSBlup_r,s(G,Γ)^Ω_Ω =G_Ω^Ωand

C^∗(G_Ω^Ω) =C^∗(G) according to proposition 5.6. This decomposition gives rise to an exact sequence of C^∗-algebras.

0 ^//C^∗(G) ^//C^∗(SBlup_r,s(G,Γ)) ^//C^∗(SN_Γ^G) ^//0 This exact sequence gives rise to an exact sequence of bimodules:

0 ^//C^∗(G) ^// rise to an exact sequence of bimodule as above:

0 ^//C^∗(H×R^∗+×R^∗+) ^// justEP T. It leads to the exact sequence of bimodule:

0 ^//C^∗(G) ^//

We have an exact sequence (where ˚M tV 6=M denotes the topological disjoint union of ˚M with V):

0→C^∗(G^MtV_MtV^˚_˚ )→C_BM^∗ (G, V) ^r

C∗

−→V Σ^C_bound^∗ (G, V)→0,

where the quotient Σ^C_bound^∗ (G, V) is the algebra of the Boutet de Monvel type boundary symbols.

It is the algebra of matrices of the form

k p t q

where k ∈ C^∗(SN_V^G), q ∈ C(SA^∗G^V_V), p, t^∗ ∈