Introduction Groupoids PDO Index Theory Bonus
Index theory through Lie groupoids
Joint works with J.-M. Lescure and G. Skandalis.
Inspired by ideas of A. Connes.
Claire Debord
Universite Paris-Diderot Paris 7Institut de Mathematiques de Jussieu - Paris Rive Gauche
Potsdam, March 2019
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý They are already there ....We have all encountered several convolution formulas.
• On a group : f ∗ g(x) =
∫G
f(y) g(y−1x) dy.
• Kernels : f ∗ g(x, y) =
∫M
f(x, z) g(z, y) dz.
These are particular cases of convolution on Lie groupoids.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý They are already there ....
We have all encountered several convolution formulas.
• On a group : f ∗ g(x) =
∫G
f(y) g(y−1x) dy.
• Kernels : f ∗ g(x, y) =
∫M
f(x, z) g(z, y) dz.
These are particular cases of convolution on Lie groupoids.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý They are already there ....We have all encountered several convolution formulas.
• On a group : f ∗ g(x) =
∫G
f(y) g(y−1x) dy.
• Kernels : f ∗ g(x, y) =
∫M
f(x, z) g(z, y) dz.
These are particular cases of convolution on Lie groupoids.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý They are already there ....We have all encountered several convolution formulas.
• On a group : f ∗ g(x) =
∫G
f(y) g(y−1x) dy.
• Kernels : f ∗ g(x, y) =
∫M
f(x, z) g(z, y) dz.
These are particular cases of convolution on Lie groupoids.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý They are already there ....We have all encountered several convolution formulas.
• On a group : f ∗ g(x) =
∫G
f(y) g(y−1x) dy.
• Kernels : f ∗ g(x, y) =
∫M
f(x, z) g(z, y) dz.
These are particular cases of convolution on Lie groupoids.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Gelfand’s theorem
X ←→ C0(X)Locally compact space Commutative C∗-algebra
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Gelfand’s theorem
X ←→ C0(X)Locally compact space Commutative C∗-algebra
Noncommutative geometry propose to replace the study of a singularspace by the study of a convenient C∗-algebra :
Z� Singular � space
//
))
C∗(Z) = C∗(G)Noncommutative C∗-algebra
G⇒ G(0), G(0)/G ' ZGroupoid
OO
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Z� Singular � space
//
))
C∗(Z) = C∗(G)Noncommutative C∗-algebra
G⇒ G(0), G(0)/G ' ZGroupoid
OO
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Z� Singular � space
//
))
C∗(Z) = C∗(G)Noncommutative C∗-algebra
G⇒ G(0), G(0)/G ' ZGroupoid
OO
Singular geometrical spaces : space of leaves of a foliation, manifoldwith corners, stratified pseudo-manifold...
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Z� Singular � space
//
))
C∗(Z) = C∗(G)Noncommutative C∗-algebra
G⇒ G(0), G(0)/G ' ZGroupoid
OO
Singular geometrical spaces : space of leaves of a foliation, manifoldwith corners, stratified pseudo-manifold...
Ingredients : algebra of � continuous functions � on the space ofparameter, pseudodifferential calculus, � tangent space �...
Introduction Groupoids PDO Index Theory Bonus
Introduction : Why put groupoids into the picture ?
ý Convolution
ý Following Gelfand’s theorem, Noncommutative geometry proposeto replace the study of a singular space by the study of a convenientC∗-algebra.
Method approach - initiated by A. Connes in ’79 : get to geometrythanks to (Lie) groupoids. The C∗-algebra of a groupoid is from J.Renault ’80.
Introduction Groupoids PDO Index Theory Bonus
Today in this talk...
1. Groupoids and a few words on the C*-algebra of a groupoid
2. Pseudodifferential operators and analytic index
3. Constructions of Lie groupoids in connexion with index theory
Introduction Groupoids PDO Index Theory Bonus
Today in this talk...
1. Groupoids and a few words on the C*-algebra of a groupoid
2. Pseudodifferential operators and analytic index
3. Constructions of Lie groupoids in connexion with index theory
Introduction Groupoids PDO Index Theory Bonus
Today in this talk...
1. Groupoids and a few words on the C*-algebra of a groupoid
2. Pseudodifferential operators and analytic index
3. Constructions of Lie groupoids in connexion with index theory
Introduction Groupoids PDO Index Theory Bonus
1. Groupoidsand a few words on the C*-algebra of a groupoid
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;
• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and
x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;
• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and
x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0)
s(x)
x##r(x)
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;
• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and
x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0)
s(x)
x##r(x)
• x, y composable if s(x) = r(y),we obtain x · y (or xy) withsource s(y) and range r(x). s(y)
y %%
x · y
##r(y)=s(x)
x %%r(x)
• Range and source maps r, s : G→ G(0).• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• x, y, z with s(x) = r(y) and s(y) = r(z), then (x · y) · z = x · (y · z) ;• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and
x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;
• u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) · x = x and
x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;
• Units : u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) ·x = x
and x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G• Range and source maps r, s : G→ G(0).• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;
• Units : u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) ·x = x
and x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G withr(x−1) = s(x), s(x−1) = r(x), x · x−1 = er(x)
and x−1 · x = es(x).
s(x)
x##r(x)
x−1
dd
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
We denote : G⇒ G(0)
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;
• Units : u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) ·x = x
and x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
We denote : G⇒ G(0)
! G acts on G(0) : the orbit of x ∈ G(0) is r(s−1(x)).
Introduction Groupoids PDO Index Theory Bonus
Groupoids
Definition
A groupoid is a small category such that every arrow is invertible.
• Set of objects G(0), set of arrows G(1) = G
• Range and source maps r, s : G→ G(0).
• x, y composable if s(x) = r(y), we obtain x · y (or xy) with sources(y) and range r(x).
• Associativity : x, y, z with s(x) = r(y) and s(y) = r(z), then(x · y) · z = x · (y · z) ;
• Units : u ∈ G(0), unit eu ∈ G with r(eu) = s(eu) = u ; er(x) ·x = x
and x · es(x) = x for all x ∈ G ; identification G(0) ⊂ G ;
• Inverse : ∀x ∈ G, ∃x−1 ∈ G with r(x−1) = s(x), s(x−1) = r(x),x · x−1 = er(x) and x−1 · x = es(x).
We denote : G⇒ G(0)
! G acts on G(0) : the orbit of x ∈ G(0) is r(s−1(x)).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• inclusion u 7→ eu of G(0) to G, inverse are smooth
• s, r : G→ G(0) are smooth submersionsG(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;
• composition G(2) → G is smooth.
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• inclusion u 7→ eu of G(0) to G, inverse are smooth
• s, r : G→ G(0) are smooth submersions
G(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;
• composition G(2) → G is smooth.
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• inclusion u 7→ eu of G(0) to G, inverse are smooth
• s, r : G→ G(0) are smooth submersionsG(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;
• composition G(2) → G is smooth.
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• inclusion u 7→ eu of G(0) to G, inverse are smooth
• s, r : G→ G(0) are smooth submersionsG(2) = {(x, y); s(x) = r(y)} is a submanifold of G×G ;
• composition G(2) → G is smooth.
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
Examples
1.
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
Examples
1. A manifold M is a Lie groupoid. All maps r, s, composition...idM .
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
Examples
1. A manifold is a Lie groupoid M ⇒M .
2. A Lie group is a Lie groupoid with just one unit.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
Examples
1. A manifold is a Lie groupoid M ⇒M .
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
Lie Groupoids
Definition
A Lie groupoid is a groupoid G⇒ G(0) such that
• G and G(0) are manifolds ;
• All maps smooth.
Examples
1. A manifold is a Lie groupoid M ⇒M .
2. A Lie group is a Lie groupoid.
3. A smooth vector bundle is a Lie groupoid.
4. Pair groupoid M ×Mr,s
⇒M ; s(x, y) = y, r(x, y) = x,u(x) = (x, x), (x, y) · (y, z) = (x, z) et (x, y)−1 = (y, x).
Introduction Groupoids PDO Index Theory Bonus
More examples
5. Group actionSmooth action of a Lie group H on a manifold M : H nM ⇒M .
s(h, x) = x, r(h, x) = h · x, u(x) = (e, x)(k, h · x) · (h, x) = (kh, x) et (h, x)−1 = (h−1, h · x)
6. Poincare Groupoidγ a path on M , [γ] homotopy class with fixed end points of γ,s[γ] = γ(0), r[γ] = γ(1), concatenation product...
Π(M) = {[γ] | γ path on M}⇒M
For x ∈M , π1(M,x) = s−1(x)∩ r−1(x) is the fondamental group,it acts (on the right) on the universal cover s−1(x).
Introduction Groupoids PDO Index Theory Bonus
More examples
5. Group actionSmooth action of a Lie group H on a manifold M : H nM ⇒M .
s(h, x) = x, r(h, x) = h · x, u(x) = (e, x)(k, h · x) · (h, x) = (kh, x) et (h, x)−1 = (h−1, h · x)
6. Poincare Groupoidγ a path on M , [γ] homotopy class with fixed end points of γ,s[γ] = γ(0), r[γ] = γ(1), concatenation product...
Π(M) = {[γ] | γ path on M}⇒M
For x ∈M , π1(M,x) = s−1(x)∩ r−1(x) is the fondamental group,it acts (on the right) on the universal cover s−1(x).
Introduction Groupoids PDO Index Theory Bonus
More examples
5. Group actionSmooth action of a Lie group H on a manifold M : H nM ⇒M .
s(h, x) = x, r(h, x) = h · x, u(x) = (e, x)(k, h · x) · (h, x) = (kh, x) et (h, x)−1 = (h−1, h · x)
6. Poincare Groupoidγ a path on M , [γ] homotopy class with fixed end points of γ,s[γ] = γ(0), r[γ] = γ(1), concatenation product...
Π(M) = {[γ] | γ path on M}⇒M
For x ∈M , π1(M,x) = s−1(x)∩ r−1(x) is the fondamental group,it acts (on the right) on the universal cover s−1(x).
Introduction Groupoids PDO Index Theory Bonus
7. Graph of an equivalence relationThe pair groupoid M ×M ⇒M ; (x, y) · (y, z) = (x, z).
I Sub-groupoids (with M as units space) of the pair groupoidover M are exactly graphs of equivalence relations.
I Let R be an equivalence relation on M . Its graphGR = {(x, y) ∈M ×M | xRy}⇒M is a Lie groupoid when R isthe relation � being on the same leaf of a regular foliation withno holonomy. �
8. Holonomy groupoid of a regular foliation on M
Winkelnkemper - Pradines ’80
Construction of the holonomy groupoid : the � smallest � Liegroupoid with units M and the leaves of F as orbits.
Introduction Groupoids PDO Index Theory Bonus
7. Graph of an equivalence relationThe pair groupoid M ×M ⇒M ; (x, y) · (y, z) = (x, z).
I Sub-groupoids (with M as units space) of the pair groupoidover M are exactly graphs of equivalence relations.
I Let R be an equivalence relation on M . Its graphGR = {(x, y) ∈M ×M | xRy}⇒M is a Lie groupoid when R isthe relation � being on the same leaf of a regular foliation withno holonomy. �
8. Holonomy groupoid of a regular foliation on M
Winkelnkemper - Pradines ’80
Construction of the holonomy groupoid : the � smallest � Liegroupoid with units M and the leaves of F as orbits.
Introduction Groupoids PDO Index Theory Bonus
7. Graph of an equivalence relationThe pair groupoid M ×M ⇒M ; (x, y) · (y, z) = (x, z).
I Sub-groupoids (with M as units space) of the pair groupoidover M are exactly graphs of equivalence relations.
I Let R be an equivalence relation on M . Its graphGR = {(x, y) ∈M ×M | xRy}⇒M is a Lie groupoid when R isthe relation � being on the same leaf of a regular foliation withno holonomy. �
8. Holonomy groupoid of a regular foliation on M
Winkelnkemper - Pradines ’80
Construction of the holonomy groupoid : the � smallest � Liegroupoid with units M and the leaves of F as orbits.
Introduction Groupoids PDO Index Theory Bonus
7. Graph of an equivalence relationThe pair groupoid M ×M ⇒M ; (x, y) · (y, z) = (x, z).
I Sub-groupoids (with M as units space) of the pair groupoidover M are exactly graphs of equivalence relations.
I Let R be an equivalence relation on M . Its graphGR = {(x, y) ∈M ×M | xRy}⇒M is a Lie groupoid when R isthe relation � being on the same leaf of a regular foliation withno holonomy. �
8. Holonomy groupoid of a regular foliation on M
Winkelnkemper - Pradines ’80
Construction of the holonomy groupoid : the � smallest � Liegroupoid with units M and the leaves of F as orbits.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])
• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).
3. Lie algebroid of the holonomy groupoid of a regular foliation F :(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
A few words about Lie theory for groupoids
The Lie algebroid of the Lie groupoid G⇒M is : (AG, ], [·, ·])• AG = KerTs|G(0) → G(0) - smooth vector bundle.
• ] = Tr : AG→ TG(0) - smooth bundle map.
• [·, ·] Lie bracket on smooth sections of AG - constructed from leftinvariant vector fields.
It satisfies for any X, Y ∈ Γ(AG), f ∈ C∞(G(0)) :
][X,Y ] = [](X), ](Y )] and [X, fY ] = f [X,Y ] + ](X)(f).Y
Examples
1. Lie algebroid of a Lie group : Lie algebra.
2. Lie algebroid of the pair groupoid M ×M ⇒M : (TM, Id, [·, ·]).3. Lie algebroid of the holonomy groupoid of a regular foliation F :
(TF , Id, [·, ·]).
! Lie third theorem fails : Lie algebroid may not be the Lie algebroidof a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.
• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Convolution on a Lie groupoid G
Algebra C∞c (G) : f1 ∗ f2(x) =
∫(x1,x2)∈G; x1x2=x
f1(x1)f2(x2) dν
• The set {(x1, x2) ∈ G×G; x1x2 = x} : smooth manifoldx1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1
1 x.
• dν is a smooth “Haar system”• i.e. smooth choice of a Lebesgue measure νu on every Gu.• Left invariance : for every x ∈ G, the measure νs(x) ↔ νr(x)
through diffeomorphism y 7→ x · y from Gs(x) with Gr(x).
Convolution formula f1 ∗ f2(x) =
∫Gr(x)
f1(y)f2(y−1x) dνr(x)(y).
Convolution associative by invariance of the Haar system (andFubini).
Adjoint of f ∈ C∞c (G) : function f∗ : x 7→ f(x−1).
We choose an operator nom : C∗(G) = completion of C∞c (G).
In the � good cases � : C0(G(0)/G)! C∗(G).
Introduction Groupoids PDO Index Theory Bonus
Examples of groupoids and their C∗-algebras
1. G = M ×M .
Then C∗(G) = K : algebra of compact operators.Equivalent to C.
2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).
3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).
4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :
0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.
Introduction Groupoids PDO Index Theory Bonus
Examples of groupoids and their C∗-algebras
1. G = M ×M . Then C∗(G) = K : algebra of compact operators.Equivalent to C.
2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).
3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).
4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :
0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.
Introduction Groupoids PDO Index Theory Bonus
Examples of groupoids and their C∗-algebras
1. G = M ×M . Then C∗(G) = K : algebra of compact operators.Equivalent to C.
2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).
3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).
4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :
0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.
Introduction Groupoids PDO Index Theory Bonus
Examples of groupoids and their C∗-algebras
1. G = M ×M . Then C∗(G) = K : algebra of compact operators.Equivalent to C.
2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).
3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).
4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :
0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.
Introduction Groupoids PDO Index Theory Bonus
Examples of groupoids and their C∗-algebras
1. G = M ×M . Then C∗(G) = K : algebra of compact operators.Equivalent to C.
2. G = M ×B M where M → B is a submersion.C∗(G) = C(B)⊗K. Equivalent to C(B).
3. If G is a vector bundle E →M , then C∗(G) = C0(E∗) : whereE∗ is the dual bundle and C0(E∗) : functions that vanish atinfinity on (the total space of) E∗ (using Fourier transform).
4. If U ⊂ G(0) is open and saturated (i.e. s(x) ∈ U ⇔ r(x) ∈ U) andF = G(0) \ U , exact sequence :
0→ C∗(GU )→ C∗(G)→ C∗(GF )→ 0.
Introduction Groupoids PDO Index Theory Bonus
2. Pseudodifferential operators andanalytic index
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.
• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential operators on a Lie groupoid G
Differential operators : (enveloping) algebra generated by sections ofthe algebroid AG = ∪x∈G(0)TGx ' the normal bundle of the inclusion
G(0) ⊂ G.
Pm(G) - pseudodifferential operators of order m ∈ Z are distributionswith singular support G(0) ⊂ G, conormal, i.e. of the form P + kwhere k ∈ C∞c (G) and
P (x) = (2π)−d∫
(A∗G)s(x)
ei〈θ(x)|ξ〉χ(x)a(s(x), ξ) dξ
• d is the dimension of the algebroid AG ;• θ : U → AG is a diffeomorphism, inverse of an exp map, from a
tubular neighbourhood of G(0) in G to the normal bundle AG.• χ is a smooth bump function (1 on G(0), and 0 outside U) ;
• a is a classical polyhomogeneous symbol a(u, ξ) ∼∑j
am−j(u, ξ)
where a` is homogeneous of order `.
• oscilatory integral
∫(A∗G)s(x)
= limR→∞
∫‖ξ‖≤R
(as a distribution).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Pseudodifferential calculus
These pseudodifferential operators form a convolution ∗-algebra :
• P of < 0 order - with compact support - is in C∗(G).
• P of order ≤ 0 - with compact support - bounded.
Ψ∗(G) = completion of P0(G).
We obtain an exact sequence of C∗ algebras.
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
• SA∗G is the sphere bundle of the dual bundle to the algebroidAG.
• σ : principal symbol map (a ∼∑
a−j 7→ a0).
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ;
C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
Analytic index
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0
Connecting map : ∂G : K∗+1(C(SA∗G))→ K∗(C∗(G)).
Consider the inclusion i : SA∗G× R∗+ ' A∗G \G(0) → A∗G.
∂G = ∂G ◦ [i]
Example
G = M ×M pair groupoid ; AG = TM ; C∗(G) = K(L2(M)).
∂G : K0(C0(T ∗M))→ K0(K) = Z is the Atiyah-Singer analytic index.
Thus ∂G : K∗(C0(A∗G))→ K∗(C∗(G)) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M .
Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M .
Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F).
Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
In the examples...
1. G = M ×M . Atiyah-Singer index - with values in Z.
2. G = M ×B M . Atiyah-Singer index for families - with values inK∗(C0(B)).
3. G holonomy groupoid of a foliation (M,F). Connes’ index withvalues in K∗(C
∗(M,F)).
4. Manifolds with boundary, with corners... Corresponding indexproblems.
Introduction Groupoids PDO Index Theory Bonus
3. Constructions of Lie groupoidsin connection with index theory
or PDOs through geometry
Introduction Groupoids PDO Index Theory Bonus
3. Constructions of Lie groupoidsin connection with index theory
or PDOs through geometry
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle.
DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid
Deformation to the normal cone : V ⊂M a submanifold, NMV the
normal bundle. DNC(M,V ) = (M × R∗) t (NMV × {0}).
Natural smooth structure. Generated by :
• the natural map ϕ : DNC(M,V )→M × R is smooth(given by (x, t) 7→ (x, t) and (x, ξ, 0) 7→ (x, 0)).
• if f : M → R smooth and vanishes on V , the function
fdnc : DNC(M,V )→ R given by (x, t) 7→ f(x)
tand
(x, ξ, 0) 7→ df(ξ) is smooth.
Remarks (D.-Skandalis)
1. This construction is functorial
2. If G is a Lie groupoid and H is a subgroupoid,DNC(G,H)⇒ DNC(G(0), H(0)) is a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) (M ⊂M ×M diagonally)
= (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) (M ⊂M ×M diagonally)= (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.
More preciselyGT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) // 0
K
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Connes’ tangent groupoid : Analytic index without PDO’s
Definition (Connes’ tangent groupoid)
DNC(M ×M,M) = (M ×M × R∗) t (TM × {0})⇒M × R.
GT = DNC[0,1](M ×M,M) = (M ×M × (0, 1]) t (TM × {0}).
Exact sequence 0→ K⊗ C0((0, 1]) −→ C∗(GT )ev0−→ C∗(TM)→ 0.
K ⊗ C0((0, 1]) is contractible : ev0 isomorphism in K-theory.
0 // K ⊗ C0((0, 1]) // C∗(GT )ev0 //
ev1
��
C∗(TM) = C0(T ∗M) //
indavv
0
KTheorem (Connes) : The Atiyah-Singer analytic index is equal to∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M))→ K0(K) = Z
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
Let G⇒M be a Lie groupoid. Start with the adiabatic groupoid
Gad = DNCR+(G,M) = (G× R∗+) t (AG× {0})⇒M × R+
ä Natural � zooming action � of R∗+ :
λ(x, t) = (x, λt) for t 6= 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.
Gauge adiabatic groupoid :Gag = Gad oR∗+ = (G× R∗+ × R∗+) t (AG oR∗+)⇒M × R
ä Look at the evaluation map : ev0 : C∗(Gad)→ C∗(AG) ' C0(A∗G)and consider the ideal J(G) = ev−1
0 (C0(A∗G \M))
0→ C∗(G)⊗ C0(R∗+) −→ J(G)ev0−→ C0(A∗G \G(0))→ 0
Which is equivariant under the action of R∗+ and leads to
0→(C∗(G)⊗ C0(R∗+)
)oR∗+
' C∗(G)⊗K→ J(G) oR∗+⊂ C∗(Gga)
→ C0(A∗G \G(0)) oR∗+' C(S∗AG)⊗K
→ 0
(GAG)
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
We have an exact sequence
0→ C∗(G)⊗K −→ J(G) oR∗+ −→ C(SA∗G)⊗K → 0 (GAG)
Compare with...
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
Theorem (D.-Skandalis)
There is a natural isomorphism J(G) oR∗+ ' Ψ∗(G)⊗K.
In other words, pseudodifferential operators can be expressed asconvolution kernels on a groupoid.
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
We have an exact sequence
0→ C∗(G)⊗K −→ J(G) oR∗+ −→ C(SA∗G)⊗K → 0 (GAG)
Compare with...
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
Theorem (D.-Skandalis)
There is a natural isomorphism J(G) oR∗+ ' Ψ∗(G)⊗K.
In other words, pseudodifferential operators can be expressed asconvolution kernels on a groupoid.
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
We have an exact sequence
0→ C∗(G)⊗K −→ J(G) oR∗+ −→ C(SA∗G)⊗K → 0 (GAG)
Compare with...
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
Theorem (D.-Skandalis)
There is a natural isomorphism J(G) oR∗+ ' Ψ∗(G)⊗K.
In other words, pseudodifferential operators can be expressed asconvolution kernels on a groupoid.
Introduction Groupoids PDO Index Theory Bonus
Gauge adiabatic groupoid and PDOs (D.-Skandalis)
We have an exact sequence
0→ C∗(G)⊗K −→ J(G) oR∗+ −→ C(SA∗G)⊗K → 0 (GAG)
Compare with...
0→ C∗(G) −→ Ψ∗(G)σ−→ C(SA∗G)→ 0 (PDO)
Theorem (D.-Skandalis)
There is a natural isomorphism J(G) oR∗+ ' Ψ∗(G)⊗K.
In other words, pseudodifferential operators can be expressed asconvolution kernels on a groupoid.
Introduction Groupoids PDO Index Theory Bonus
Theorem (D. & Skandalis)
There is an ideal J (G) ⊂ C∞c (Gad) such that forf = (ft)t∈R+
∈ J (G) and m ∈ N let
P =
∫ +∞
0
tmftdt
tand σ : (x, ξ) ∈ A∗G 7→
∫ +∞
0
tmf(x, tξ, 0)dt
t
Then P belongs to P−m(G) and its principal symbol is σ.
What does it mean : There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :
P ∗ g =
∫ +∞
0
tmft ∗ gdt
tand g ∗ P =
∫ +∞
0
tmg ∗ ftdt
t
Remark : Moreover any P ∈ P−m(G) is a Pf =
∫ +∞
0
tmftdt
tfor some
f = (ft)t∈R+ ∈ J (G).
Introduction Groupoids PDO Index Theory Bonus
Theorem (D. & Skandalis)
There is an ideal J (G) ⊂ C∞c (Gad) such that forf = (ft)t∈R+
∈ J (G) and m ∈ N let
P =
∫ +∞
0
tmftdt
tand σ : (x, ξ) ∈ A∗G 7→
∫ +∞
0
tmf(x, tξ, 0)dt
t
Then P belongs to P−m(G) and its principal symbol is σ.
What does it mean :
There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :
P ∗ g =
∫ +∞
0
tmft ∗ gdt
tand g ∗ P =
∫ +∞
0
tmg ∗ ftdt
t
Remark : Moreover any P ∈ P−m(G) is a Pf =
∫ +∞
0
tmftdt
tfor some
f = (ft)t∈R+ ∈ J (G).
Introduction Groupoids PDO Index Theory Bonus
Theorem (D. & Skandalis)
There is an ideal J (G) ⊂ C∞c (Gad) such that forf = (ft)t∈R+
∈ J (G) and m ∈ N let
P =
∫ +∞
0
tmftdt
tand σ : (x, ξ) ∈ A∗G 7→
∫ +∞
0
tmf(x, tξ, 0)dt
t
Then P belongs to P−m(G) and its principal symbol is σ.
What does it mean : There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :
P ∗ g =
∫ +∞
0
tmft ∗ gdt
tand g ∗ P =
∫ +∞
0
tmg ∗ ftdt
t
Remark : Moreover any P ∈ P−m(G) is a Pf =
∫ +∞
0
tmftdt
tfor some
f = (ft)t∈R+ ∈ J (G).
Introduction Groupoids PDO Index Theory Bonus
Theorem (D. & Skandalis)
There is an ideal J (G) ⊂ C∞c (Gad) such that forf = (ft)t∈R+
∈ J (G) and m ∈ N let
P =
∫ +∞
0
tmftdt
tand σ : (x, ξ) ∈ A∗G 7→
∫ +∞
0
tmf(x, tξ, 0)dt
t
Then P belongs to P−m(G) and its principal symbol is σ.
What does it mean : There exists a pseudodifferential operatorP ∈ P−m(G) with principal symbol σ such that if g ∈ C∞c (G) :
P ∗ g =
∫ +∞
0
tmft ∗ gdt
tand g ∗ P =
∫ +∞
0
tmg ∗ ftdt
t
Remark : Moreover any P ∈ P−m(G) is a Pf =
∫ +∞
0
tmftdt
tfor some
f = (ft)t∈R+ ∈ J (G).
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators
in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
In conclusion...
Lie groupoids allow us
1. to generalise the analytic index ;
2. to express the analytic index in a (pseudo)differential operatorfree way ;
3. to give proofs of index theorems ;
4. to express the order 0 pseudodifferential operators in a(pseudo)differential operator free way.
Introduction Groupoids PDO Index Theory Bonus
Thank you for your attention !
Introduction Groupoids PDO Index Theory Bonus
Analytic index for groupoids
Let G⇒M be a Lie groupoid.
Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram
0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //
ev1
��
C0(A∗G) //
indayy
0
C∗(G)
Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)
Analytic index = [ev1] ◦ [ev0]−1.
Introduction Groupoids PDO Index Theory Bonus
Analytic index for groupoids
Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).
Diagram
0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //
ev1
��
C0(A∗G) //
indayy
0
C∗(G)
Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)
Analytic index = [ev1] ◦ [ev0]−1.
Introduction Groupoids PDO Index Theory Bonus
Analytic index for groupoids
Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram
0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //
ev1
��
C0(A∗G) //
indayy
0
C∗(G)
Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)
Analytic index = [ev1] ◦ [ev0]−1.
Introduction Groupoids PDO Index Theory Bonus
Analytic index for groupoids
Let G⇒M be a Lie groupoid.Define the adiabatic groupoid Gad = DNC[0,1](G,M).Diagram
0 // C∗(G)⊗ C0((0, 1]) // C∗(Gad)ev0 //
ev1
��
C0(A∗G) //
indayy
0
C∗(G)
Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)
Analytic index = [ev1] ◦ [ev0]−1.
Introduction Groupoids PDO Index Theory Bonus
Digression : Pullback groupoid and Morita equivalence
Suppose G⇒M is a smooth groupoid and let f : N →M be asurjective submersion. The pullback groupoid of G by f is the smoothgroupoid
Gff := {(x, γ, y) ∈ N ×G×N | r(γ) = f(x), s(γ) = f(y)}⇒ N
Two smooth groupoids G⇒M and H ⇒ N are Morita equivalent ifone can find a manifold Z with two surjective submersions p : Z →Mand q : Z → N such that the pullbacks Gpp ⇒ Z and Hq
q ⇒ Z areisomorphic.
Theorem (Muhly, Renault, Williams)
The C∗-algebras of two Morita equivalent groupoids are Moritaequivalent.
Introduction Groupoids PDO Index Theory Bonus
Digression : Pullback groupoid and Morita equivalence
Suppose G⇒M is a smooth groupoid and let f : N →M be asurjective submersion. The pullback groupoid of G by f is the smoothgroupoid
Gff := {(x, γ, y) ∈ N ×G×N | r(γ) = f(x), s(γ) = f(y)}⇒ N
Two smooth groupoids G⇒M and H ⇒ N are Morita equivalent ifone can find a manifold Z with two surjective submersions p : Z →Mand q : Z → N such that the pullbacks Gpp ⇒ Z and Hq
q ⇒ Z areisomorphic.
Theorem (Muhly, Renault, Williams)
The C∗-algebras of two Morita equivalent groupoids are Moritaequivalent.
Introduction Groupoids PDO Index Theory Bonus
Digression : Pullback groupoid and Morita equivalence
Suppose G⇒M is a smooth groupoid and let f : N →M be asurjective submersion. The pullback groupoid of G by f is the smoothgroupoid
Gff := {(x, γ, y) ∈ N ×G×N | r(γ) = f(x), s(γ) = f(y)}⇒ N
Two smooth groupoids G⇒M and H ⇒ N are Morita equivalent ifone can find a manifold Z with two surjective submersions p : Z →Mand q : Z → N such that the pullbacks Gpp ⇒ Z and Hq
q ⇒ Z areisomorphic.
Theorem (Muhly, Renault, Williams)
The C∗-algebras of two Morita equivalent groupoids are Moritaequivalent.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :
T = DNC(DNC(E × E,E ×
ME),∆E × {0}
)⇒ E × R× R
Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].
Proposition (D.-Lescure-Nistor)
For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the
inverse of the Thom isomorphism.
Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity
τRn : K0(C∗(TRn))→ Z.
Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with
(deformation) groupoids.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :
T = DNC(DNC(E × E,E ×
ME),∆E × {0}
)⇒ E × R× R
Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].
The Thom groupoid T hom = TE × {0} t TMππ×]0, 1]⇒ E × [0, 1]
and the Morita equivalence TMππ ∼ TM provides :
τE : K∗(C∗(TE)) = K∗(C0(T ∗E))→ K∗(C
∗(TM)) = K∗(C0(T ∗M))
Proposition (D.-Lescure-Nistor)
For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the
inverse of the Thom isomorphism.
Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity
τRn : K0(C∗(TRn))→ Z.
Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with
(deformation) groupoids.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :
T = DNC(DNC(E × E,E ×
ME),∆E × {0}
)⇒ E × R× R
Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].
The Thom groupoid T hom = TE × {0} t TMππ×]0, 1]⇒ E × [0, 1]
and the Morita equivalence TMππ ∼ TM provides :
τE : K∗(C∗(TE)) = K∗(C0(T ∗E))→ K∗(C
∗(TM)) = K∗(C0(T ∗M))
Proposition (D.-Lescure-Nistor)
For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the
inverse of the Thom isomorphism.
Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity
τRn : K0(C∗(TRn))→ Z.
Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with
(deformation) groupoids.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :
T = DNC(DNC(E × E,E ×
ME),∆E × {0}
)⇒ E × R× R
Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].
Proposition (D.-Lescure-Nistor)
For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the
inverse of the Thom isomorphism.
Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity
τRn : K0(C∗(TRn))→ Z.
Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with
(deformation) groupoids.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
π : E →M a vector bundle ; consider ∆E ⊂ E ×ME ⊂ E × E :
T = DNC(DNC(E × E,E ×
ME),∆E × {0}
)⇒ E × R× R
Let T � = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].
Proposition (D.-Lescure-Nistor)
For j : M ↪→ Rn and E = NRnM the normal of the inclusion, τE is the
inverse of the Thom isomorphism.
Particular case The normal bundle of · ↪→ Rn is just Rn → · andleads to the Bott periodicity
τRn : K0(C∗(TRn))→ Z.
Conclusion Indt = τRn ◦ [j] ◦ τ−1E is entirely described with
(deformation) groupoids.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
E = NRnM , T � = DNC
(DNC(E × E,E ×
ME),∆E × {0}
)|E×[0,1]×[0,1]
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GT (M))ππ
)|E×[0,1] E × E
T hom T � GT (E)
TE
Thom−1
<<
TE × [0, 1] TE
IndEa
dd
Gives IndMa = IndMt [D.-Lescure-Nistor].
Can be extended to M with isolated conical singularities... and to Ma general pseudomanifold.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
E = NRnM , T � = DNC
(DNC(E × E,E ×
ME),∆E × {0}
)|E×[0,1]×[0,1]
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GT (M))ππ
)|E×[0,1] E × E
T hom T � GT (E)
TE
Thom−1
<<
TE × [0, 1] TE
IndEa
dd
Gives IndMa = IndMt [D.-Lescure-Nistor].
Can be extended to M with isolated conical singularities... and to Ma general pseudomanifold.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
E = NRnM , T � = DNC
(DNC(E × E,E ×
ME),∆E × {0}
)|E×[0,1]×[0,1]
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GT (M))ππ
)|E×[0,1] E × E
T hom T � GT (E)
TE
Thom−1
<<
TE × [0, 1] TE
IndEa
dd
Gives IndMa = IndMt [D.-Lescure-Nistor].
Can be extended to M with isolated conical singularities...
and to Ma general pseudomanifold.
Introduction Groupoids PDO Index Theory Bonus
Atiyah-Singer index theorem
E = NRnM , T � = DNC
(DNC(E × E,E ×
ME),∆E × {0}
)|E×[0,1]×[0,1]
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GT (M))ππ
)|E×[0,1] E × E
T hom T � GT (E)
TE
Thom−1
<<
TE × [0, 1] TE
IndEa
dd
Gives IndMa = IndMt [D.-Lescure-Nistor].
Can be extended to M with isolated conical singularities... and to Ma general pseudomanifold.