Analyse fonctionnelle/Functional analysis

C. R. Acad. Sci. Paris, t. ???, Série I, p. 1–??, 2002
Analyse fonctionnelle/Functional analysis
(/)
- PXMA????.TEX -
K-duality for pseudomanifolds with an isolated
singularity
Claire Debord a , Jean-Marie Lescure b ,
a
Laboratoire de Mathématiques et Applications, Université de Haute-Alsace, 4 rue des Frères Lumière, 68093
Mulhouse cedex, France
E-mail: [email protected]
b
Laboratoire de Mathématiques pures, Université Blaise Pascal, Complexe universitaire des Cézeaux, 124 Av.
des Landais, 63177 Aubière cedex, France
E-mail: [email protected]
(Reçu le jour mois année, accepté après révision le jour mois année)
Abstract.
We associate to a pseudomanifold X with an isolated singularity a differentiable groupoid G
which plays the role of the tangent space of X. We construct a Dirac element D and a Dual
Dirac element λ which induce a Poincaré duality in K-theory between the C ∗ -algebras
c 2002 Académie des sciences/Éditions scientifiques et médicales
C(X) and C ∗ (G). Elsevier SAS
K-dualité pour les pseudo-variétés à singularité isolée.
Résumé.
1.
Etant donnée une pseudo-variété X ayant une singularité isolée, nous lui associons un
groupoı̈de différentiable G qui joue le rôle d’espace tangent à X. Nous construisons un
élément Dirac D ainsi qu’un élément dual-Dirac λ qui induisent une dualité de Poincaré
c 2002 Académie des sciences/Édien K-théorie entre les C ∗ -algèbres C(X) et C ∗ (G). tions scientifiques et médicales Elsevier SAS
The tangent bundle of a singular manifold
Let X be a pseudomanifold with an isolated singularity c, that is X = cL ∪ X1 , where X1 is a smooth
compact manifold with boundary L glued along its boundary with the cone over L : cL = L×[0, 1]/L×{0}.
The singularity c is then the image of L × {0} in cL. We denote by M = L×] − 1, 1] ∪ X1 the manifold
obtained by gluing X1 with L×] − 1, 1] along the boundary.
If y is a point of M or of X \ {c} which is in L×] − 1, 1[ we note yL ∈ L its tangential componant and
ky ∈] − 1, 1[ its radial coordinate; the function ky is smoothly extended to X1 in such a way that ky > 1 on
X1 . We set M + = {y ∈ M | ky > 0}, M − = {y ∈ M | ky < 0} and M + = {y ∈ M | ky > 0}.
We suppose that the manifold M is equipped with a Riemannian metric whose injectivity radius is bigger
than 1 and which is a product metric on L×] − 1, 1[.
Note présentée par First name NAME
S0764-4442(00)0????-?/FLA
c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
1
C. Debord et J.M. Lescure
We define the groupoid G, with source map s , range map r and space of units G(0) = M :
s
G = M− × M− ∪ TM+ ⇒ M .
r
It is the union of the bundle T M + = {(z, v) ∈ T M | z ∈ M + } equipped with its standard groupoid
structure over M + and the pair groupoid M − × M − ⇒ M − .
In order to equip G with a smooth structure, we take the usual structure of manifold on M − × M − and on
T M + . We take a smooth positive function τ : ] − 1, +∞[→ R which satisfies τ −1 ({0}) = [0, +∞[. A
local chart around boundary points of T M + is provided by the following map :
(y, V ) 7→ (y, V ) if y ∈ M +
EG : V(T M ) −→ V(G) ,
(y, V ) 7→ (y, expy (−τ (ky )V )) if y ∈ M −
where exp is the exponential map of the Riemannian manifold M , V(T M ) = {(y, V ) ∈ T M ; −τ (ky )V ∈
dom(expy )} and V(G) is a neighborhood of G(0) in G. The groupoid G is called the tangent bundle of X.
Following the construction of A. Connes for smooth manifolds [3], we define the tangent groupoid of X
in the following way :
G = M × M ×]0, 1] ∪ G × {0}⇒M × [0, 1] .
The groupoid G is the union of the groupoid G×{0} ⇒ M ×{0} and the pair groupoid over M parametrized
by ]0, 1]. We equip G with a structure of smooth groupoid similarly as we did for G.
The groupoid G is amenable [1] so that its reduced C ∗ -algebra coincides with the maximal one and it is
nuclear. The same occurs for G. We denote respectively C ∗ (G) and C ∗ (G) these C ∗ -algebras. Moreover,
up to isomorphisms, these C ∗ -algebras do not depend on the map τ used to define the smooth structure of
G.
2.
The Dirac element
The partition M × {0} ∪ M ×]0, 1] of G (0) into a saturated open subset and a saturated closed subset
induces the following exact sequence of C ∗ -algebras [5] :
e
0 −−−−→ C ∗ (G|M ×]0,1] ) −−−−→ C ∗ (G) −−−0−→ C ∗ (G|M ×{0} ) = C ∗ (G) −−−−→ 0
where the first map is the inclusion and e0 is the evaluation map at 0. The C ∗ -algebra C ∗ (G|M ×]0,1] )
is isomorphic to K ⊗ C0 (]0, 1]) which is contractible. So, since C ∗ (G) is nuclear, the element [e0 ] of
KK(C ∗ (G), C ∗ (G)) corresponding to e0 , is invertible. We denote [e0 ]−1 ∈ KK(C ∗ (G), C ∗ (G)) its
inverse. Let e1 : C ∗ (G) → C ∗ (G|M ×{1} ) = K be the evaluation map at 1. We let b be the generator of
KK(K, C). We set :
∂ = [e0 ]−1 ⊗ [e1 ] ⊗ b ∈ KK(C ∗ (G), C) .
C ∗ (G)
K
The algebra C(X) maps to the center of the multiplier algebra of C ∗ (G). Let Ψ : C ∗ (G) ⊗ C(X) →
C ∗ (G) be the morphism induced by multiplication and [Ψ] be the corresponding element in KK(C ∗ (G) ⊗
C(X), C ∗ (G)).
The Dirac element is then defined as
D = [Ψ] ⊗ ∂ ∈ KK(C ∗ (G) ⊗ C(X), C) .
C ∗ (G)
2
K-duality for pseudomanifolds with an isolated singularity
3. The dual Dirac element
We are looking for an element λ in KK(C, C ∗ (G) ⊗ C(X)), that is a continuous family (λy )y∈X of
elements of KK(C, C ∗ (G)). In order to be dual to the Dirac element D, λ is constructed so that :
(i) λ is in the image of (iO )∗ : KK(C, C ∗ (G × X|O )) → KK(C, C ∗ (G × X)) where O is an open
subset of U = {(x, y) ∈ M × X | kx < 1, ky < 1} ∪ {(x, y) ∈ M × X | d(x, y) < 1}.
(ii) The equality λ ⊗ ∂ = 1 ∈ K 0 (X) holds.
C ∗ (G)
We first assign to each point y of X an open subset Oy of M which is a ball centered on y when ky > 1,
which is contained in M − when ky 6 1/2 and equal to M − when ky 6 ε (where 0 < ε < 1/2).
Furthermore the set O = ∪y∈X Oy × {y} is an open subset of M × X contained in U.
Notice that K(C ∗ (G|Oy )) ' Z for each y ∈ X.
We pull back the vector bundle of differential forms on M to G and G using their range maps and then
to G × X and G × X using the first projection. We denote all these bundles by the same letter Λ. The
following step is the construction of a continuous family (βy )y∈X , where βy = (Fy , C ∗ (G|Oy , Λ)) is an
element of E(C, C ∗ (G|Oy )) which class is a generator of K(C ∗ (G|Oy )) ' Z.
When Oy is a subset of M + it is natural to state Fy = ay where ay ∈ Cb∞ (T ∗ Oy , EndΛ) is the symbol of
a pseudo-differential operator Gy which satisfies :
• Gy belongs to Ψ0c (Oy , Λ) + Cb∞ (Oy , EndΛ) and G2y − 1 belongs to Ψ−1
c (Oy , Λ) ,
−
0 Gy
• Gy is of the form Gy =
with respect to the usual decomposition of Λ = Λev ⊕ Λodd ,
G+
0
y
+
∞
• G+
y is surjective and Ker(Gy ) = C · y where y belongs to Cb (Oy , Λ),
S
• the family (Gy ) y∈X defines a continuous section of B(H) where H is the bundle y∈X L2 (Oy , Λ).
ky >ε
ky >ε
The existence of such an operator is a consequence of theorem (19.2.12) of [6].
When y comes closer to the singularity c, we gradually pass from a situation where G|Oy = T Oy to a
situation where G|Oy = Oy × Oy . The pseudo-differential calculus on groupoids, first introduced by A.
Connes in [2] (see also [4]), enables us to construct a continuous family (Fy )y∈X,ky >ε such that up to a
compact operator Fy = ay when G|Oy = T Oy and Fy = Gy when G|Oy = Oy × Oy . Thus we “replace”
the symbol ay by the operator Gy .
The last case is when Oy becomes equal to M − . We first choose an appropriate extension of Fy which
belongs to B(L2 (Oy , Λ)) into an element of B(L2 (M − , Λ)). Afterwards, thanks to the properties of the
operator Gy listed above, we can replace Fy by a constant operator Fc .
We construct in this way a continuous family βy of elements of E(C, C ∗ (G|Oy )). This family induces
an element β of KK(C, C ∗ (G × X|O )). We set λ = (iO )∗ (β). The dual-Dirac element λ satisfies the
properties (i) and (ii) mentionned above.
4. The Poincaré duality
T HEOREM 4.1. – The Dirac element D and the dual-Dirac element λ induce a Poincaré duality between
the C ∗ -algebras C ∗ (G) and C(X), that is :
λ ⊗ D = 1C(X) ∈ KK(C(X), C(X)) and λ ⊗ D = 1C ∗ (G) ∈ KK(C ∗ (G), C ∗ (G)).
C ∗ (G)
C(X)
Idea of the proof : Let us consider the morphisms Ψ0 , ∆0 : C ∗ (G × X) ⊗ C(X) → C ∗ (G) ⊗ C(X) given
by Ψ0 (f ⊗ g ⊗ h) = Ψ(f ⊗ h) ⊗ g and ∆0 (f ⊗ g ⊗ h) = f ⊗ ∆(h ⊗ g) where ∆ : C(X) ⊗ C(X) → C(X)
is the multiplication map. Their restrictions C ∗ (G × X|O ) ⊗ C(X) → C ∗ (G) ⊗ C(X) are homotopic,
hence, since λ = (iO )∗ (β), the following equality holds :
λ ⊗ [Ψ] = λ ⊗ [∆].
C ∗ (G)
C(X)
3
C. Debord et J.M. Lescure
This ensures that
λ ⊗ D = (λ ⊗ ∂) ⊗ [∆] = 1C(X) .
C ∗ (G)
C ∗ (G)
C(X)
In order to show the second equality, we study the invariance of the element λ⊗C(X) [Ψ] under the flip
automorphism f of C ∗ (G × G), that is the automorphism induced by (γ, η) ∈ G × G 7→ (η, γ). The
motivation comes from the equality
((λ ⊗ [Ψ])
⊗
C(X)
C ∗ (G×G)
[f ])) ⊗
C ∗ (G)
∂ = (λ ⊗ ∂) ⊗ [Ψ] = 1C ∗ (G) , which implies
C ∗ (G)
λ ⊗ D − 1C ∗ (G) = ((λ ⊗ [Ψ])
C(X)
C(X)
C(X)
⊗
[id −f ]) ⊗
C ∗ (G×G)
∂.
C ∗ (G)
If B is a symmetric geodesically convex neighborhood of the diagonal of M + × M + contained in the range
of exp, then the flip automorphism of C ∗ (T B) is homotopic to identity. Let C = L×] − 1, 1[⊂ M and
F = M × M \ C × C. We denote by r∗ : KK(C ∗ (G), C ∗ (G × G)) → KK(C ∗ (G), C ∗ (G × G|F )) the
morphism induced by the restriction r.
Using again that λ is in the image of (iO )∗ and that O ∩ M + × M + is a small enough neigborhood of the
diagonal, we show that r∗ (λ⊗C(X) [Ψ]) is invariant under the flip automorphism of C ∗ (G × G|F ). Then,
the long exact sequence in KK-theory associated to :
iC×C
r
0 −−−−→ C ∗ (G × G|C×C ) −−−−→ C ∗ (G × G) −−−−→ C ∗ (G × G|F ) −−−−→ 0
ensures that (λ⊗C(X) [Ψ])⊗C ∗ (G×G) [id −f ] belongs to the image of (iC×C )∗ : KK(C ∗ (G), C ∗ (G ×
G|C×C )) → KK(C ∗ (G), C ∗ (G×G)). This enables us to show that λ⊗C(X) D −1C ∗ (G) is in the image of
(iC )∗ : KK(C ∗ (G), C ∗ (G|C )) → KK(C ∗ (G), C ∗ (G)) where iC is the inclusion morphism of C ∗ (G|C )
into C ∗ (G).
On the other hand the equality λ⊗C ∗ (G) D = 1C(X) ensures that λ⊗C(X) D − 1C ∗ (G) is in the kernel of
the map (·⊗C ∗ (G) D) : KK(C ∗ (G), C ∗ (G)) → KK(C ∗ (G) ⊗ C(X), C). To finish the proof we show
that this map is injective in restriction to the image of (iC )∗ . This last point comes from the fact that the
inclusion iK of C ∗ (G|M − ) ' K into C ∗ (G|C ) induces a KK-equivalence between K and C ∗ (G|C ); and
(·⊗C ∗ (G) D) ◦ (iC )∗ ◦ (iK )∗ = e∗c where e∗c : KK(C ∗ (G), C) → KK(C ∗ (G) ⊗ C(X), C) comes from
the evaluation map at c, ec : C(X) → C.
2
A consequence of the preceding theorem is a Poincaré duality between C0 (T ∗ M + ) and C0 (M + ). Thus we
get a second Poincaré duality for manifolds with boundary, the first one has been stated by G. Kasparov in
[7] for the algebras C0 (T ∗ M + ) and C(M + ).
References
[1] C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de l’Enseignement
Mathématique, 36, L’Enseignement Mathématique, Genève (2000).
[2] A. Connes, Sur la théorie non commutative de l’intégration, Lecture Notes in Math. 725, Springer (1979), 19143.
[3] A. Connes, Noncommutative Geometry, Academic Press (1994).
[4] A. Connes and G. Skandalis, The longitudinal index theorem for foliations, Publ. R.I.M.S. Kyoto Univ., vol 20
(1984), 1139-1183.
[5] M. Hilsum and G. Skandalis, Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov , Ann. Sci. Ecole Norm. Sup., vol. 20:4 (1987), 325-390.
[6] L. Hörmander, The analysis of linear partial differential operators, III, A series of comprehensive studies in
mathematics, Springer-Verlag, Berlin (1985).
[7] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91:1 (1988), 147-201.
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