Index theory through Lie groupoids Joint works with J.-M. Lescure - - PowerPoint PPT Presentation

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Index theory through Lie groupoids Joint works with J.-M. Lescure - - PowerPoint PPT Presentation

Mar 30, 2023 31 likes •1.52k views

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 Universit e Paris-Diderot Paris 7 Institut de Math

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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

Universit´ e Paris-Diderot Paris 7 Institut de Math´ ematiques de Jussieu - Paris Rive Gauche

Potsdam, March 2019

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ They are already there ....

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ They are already there .... We have all encountered several convolution formulas.

  • On a group : f ∗ g(x) =
  • G

f(y) g(y−1x) dy.

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ 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.

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ 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.

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Gelfand’s theorem X ← → C0(X) Locally compact space Commutative C∗-algebra

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Gelfand’s theorem X ← → C0(X) Locally compact space Commutative C∗-algebra Noncommutative geometry propose to replace the study of a singular space by the study of a convenient C∗-algebra : Z

≪ Singular ≫ space

  • C∗(Z) = C∗(G)

Noncommutative C∗-algebra G ⇒ G(0), G(0)/G ≃ Z Groupoid

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Following Gelfand’s theorem, Noncommutative geometry propose to replace the study of a singular space by the study of a convenient C∗-algebra. Z

≪ Singular ≫ space

  • C∗(Z) = C∗(G)

Noncommutative C∗-algebra G ⇒ G(0), G(0)/G ≃ Z Groupoid

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Following Gelfand’s theorem, Noncommutative geometry propose to replace the study of a singular space by the study of a convenient C∗-algebra. Z

≪ Singular ≫ space

  • C∗(Z) = C∗(G)

Noncommutative C∗-algebra G ⇒ G(0), G(0)/G ≃ Z Groupoid

  • Singular geometrical spaces : space of leaves of a foliation, manifold

with corners, stratified pseudo-manifold...

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Following Gelfand’s theorem, Noncommutative geometry propose to replace the study of a singular space by the study of a convenient C∗-algebra. Z

≪ Singular ≫ space

  • C∗(Z) = C∗(G)

Noncommutative C∗-algebra G ⇒ G(0), G(0)/G ≃ Z Groupoid

  • Singular geometrical spaces : space of leaves of a foliation, manifold

with corners, stratified pseudo-manifold... Ingredients : algebra of ≪ continuous functions ≫ on the space of parameter, pseudodifferential calculus, ≪ tangent space ≫...

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Introduction Groupoids PDO Index Theory Bonus

Introduction : Why put groupoids into the picture ?

➽ Convolution ➽ Following Gelfand’s theorem, Noncommutative geometry propose to replace the study of a singular space by the study of a convenient C∗-algebra. Method approach - initiated by A. Connes in ’79 : get to geometry thanks to (Lie) groupoids. The C∗-algebra of a groupoid is from J. Renault ’80.

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Introduction Groupoids PDO Index Theory Bonus

Today in this talk...

  • 1. Groupoids and a few words on the C*-algebra of a groupoid
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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
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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
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Introduction Groupoids PDO Index Theory Bonus

  • 1. Groupoids

and a few words on the C*-algebra of a groupoid

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Introduction Groupoids PDO Index Theory Bonus

Groupoids Definition

A groupoid is a small category such that every arrow is invertible.

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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
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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)
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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) with source s(y) and range r(x).

s(y)

y

  • x · y
  • r(y)=s(x) x
  • r(x)
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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 source

s(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) ;

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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 source

s(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 ;

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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 source

s(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).

s(x)

x

  • r(x)

x−1

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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 source

s(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)

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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 source

s(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)).

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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.
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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 → eu of G(0) to G, inverse are smooth
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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 → eu of G(0) to G, inverse are smooth
  • s, r : G → G(0) are smooth submersions
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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 → 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 ;

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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 → 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.
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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

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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.

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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.
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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.
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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 × M

r,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).

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Introduction Groupoids PDO Index Theory Bonus

More examples

  • 5. Group action

Smooth action of a Lie group H on a manifold M : H ⋉ M ⇒ 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)

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Introduction Groupoids PDO Index Theory Bonus

More examples

  • 5. Group action

Smooth action of a Lie group H on a manifold M : H ⋉ M ⇒ 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. Poincar´

e Groupoid γ a path on M, [γ] homotopy class with fixed end points of γ, s[γ] = γ(0), r[γ] = γ(1), concatenation product...

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Introduction Groupoids PDO Index Theory Bonus

More examples

  • 5. Group action

Smooth action of a Lie group H on a manifold M : H ⋉ M ⇒ 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. Poincar´

e Groupoid γ 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).

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Introduction Groupoids PDO Index Theory Bonus

  • 7. Graph of an equivalence relation

The pair groupoid M × M ⇒ M ; (x, y) · (y, z) = (x, z).

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Introduction Groupoids PDO Index Theory Bonus

  • 7. Graph of an equivalence relation

The pair groupoid M × M ⇒ M ; (x, y) · (y, z) = (x, z).

◮ Sub-groupoids (with M as units space) of the pair groupoid

  • ver M are exactly graphs of equivalence relations.
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Introduction Groupoids PDO Index Theory Bonus

  • 7. Graph of an equivalence relation

The pair groupoid M × M ⇒ M ; (x, y) · (y, z) = (x, z).

◮ Sub-groupoids (with M as units space) of the pair groupoid

  • ver M are exactly graphs of equivalence relations.

◮ Let R be an equivalence relation on M. Its graph

GR = {(x, y) ∈ M × M | xRy} ⇒ M is a Lie groupoid when R is the relation ≪ being on the same leaf of a regular foliation with no holonomy. ≫

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Introduction Groupoids PDO Index Theory Bonus

  • 7. Graph of an equivalence relation

The pair groupoid M × M ⇒ M ; (x, y) · (y, z) = (x, z).

◮ Sub-groupoids (with M as units space) of the pair groupoid

  • ver M are exactly graphs of equivalence relations.

◮ Let R be an equivalence relation on M. Its graph

GR = {(x, y) ∈ M × M | xRy} ⇒ M is a Lie groupoid when R is the relation ≪ being on the same leaf of a regular foliation with no holonomy. ≫

  • 8. Holonomy groupoid of a regular foliation on M

Winkelnkemper - Pradines ’80 Construction of the holonomy groupoid : the ≪ smallest ≫ Lie groupoid with units M and the leaves of F as orbits.

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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, ♯, [·, ·])

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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.
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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.
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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 left

invariant vector fields.

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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 left

invariant 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

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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 left

invariant 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.
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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 left

invariant 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, [·, ·]).
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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 left

invariant 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, [·, ·]).

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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 left

invariant 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 algebroid

  • f a Lie groupoid.
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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

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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 manifold

x1 ∈ Gr(x) = {y ∈ G; r(y) = r(x)} and x2 = x−1

1 x.

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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 manifold

x1 ∈ 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.
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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 manifold

x1 ∈ 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 → x · y from Gs(x) with Gr(x).

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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 manifold

x1 ∈ 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 → x · y from Gs(x) with Gr(x).

Convolution formula f1 ∗ f2(x) =

  • Gr(x) f1(y)f2(y−1x) dνr(x)(y).
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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 manifold

x1 ∈ 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 → 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 (and Fubini).

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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 manifold

x1 ∈ 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 → 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 (and Fubini). Adjoint of f ∈ C∞

c (G) : function f ∗ : x → f(x−1).

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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 manifold

x1 ∈ 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 → 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 (and Fubini). Adjoint of f ∈ C∞

c (G) : function f ∗ : x → f(x−1).

We choose an operator nom : C∗(G) = completion of C∞

c (G).

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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 manifold

x1 ∈ 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 → 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 (and Fubini). Adjoint of f ∈ C∞

c (G) : function f ∗ : x → 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).

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Introduction Groupoids PDO Index Theory Bonus

Examples of groupoids and their C∗-algebras

  • 1. G = M × M.
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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.

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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).

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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∗) : where

E∗ is the dual bundle and C0(E∗) : functions that vanish at infinity on (the total space of) E∗ (using Fourier transform).

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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∗) : where

E∗ is the dual bundle and C0(E∗) : functions that vanish at infinity 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) and

F = G(0) \ U, exact sequence : 0 → C∗(GU) → C∗(G) → C∗(GF ) → 0.

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Introduction Groupoids PDO Index Theory Bonus

  • 2. Pseudodifferential operators and

analytic index

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G.

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(x)a(s(x), ξ) dξ

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(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.

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(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) ;
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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(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 aℓ is homogeneous of order ℓ.

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(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 aℓ is homogeneous of order ℓ.

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential operators on a Lie groupoid G

Differential operators : (enveloping) algebra generated by sections of the algebroid AG = ∪x∈G(0)TGx ≃ the normal bundle of the inclusion G(0) ⊂ G. Pm(G) - pseudodifferential operators of order m ∈ Z are distributions with singular support G(0) ⊂ G, conormal, i.e. of the form P + k where k ∈ C∞

c (G) and

P(x) = (2π)−d

  • (A∗G)s(x)

eiθ(x)|ξχ(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 aℓ is homogeneous of order ℓ.

  • oscilatory integral
  • (A∗G)s(x)

= lim

R→∞

  • ξ≤R

(as a distribution).

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Introduction Groupoids PDO Index Theory Bonus

Pseudodifferential calculus

These pseudodifferential operators form a convolution ∗-algebra :

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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).
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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.
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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).

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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)

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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)

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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 algebroid

AG.

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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 algebroid

AG.

  • σ : principal symbol map (a ∼
  • a−j → a0).
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Introduction Groupoids PDO Index Theory Bonus

Analytic index

0 → C∗(G) − → Ψ∗(G)

σ

− → C(SA∗G) → 0

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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)).

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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]

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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 ;

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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)).

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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.

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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.

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In the examples...

  • 1. G = M × M.
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Introduction Groupoids PDO Index Theory Bonus

In the examples...

  • 1. G = M × M. Atiyah-Singer index - with values in Z.
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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.
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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 in

K∗(C0(B)).

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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 in

K∗(C0(B)).

  • 3. G holonomy groupoid of a foliation (M, F).
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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 in

K∗(C0(B)).

  • 3. G holonomy groupoid of a foliation (M, F). Connes’ index with

values in K∗(C∗(M, F)).

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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 in

K∗(C0(B)).

  • 3. G holonomy groupoid of a foliation (M, F). Connes’ index with

values in K∗(C∗(M, F)).

  • 4. Manifolds with boundary, with corners... Corresponding index

problems.

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  • 3. Constructions of Lie groupoids

in connection with index theory

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Introduction Groupoids PDO Index Theory Bonus

  • 3. Constructions of Lie groupoids

in connection with index theory

  • r PDOs through geometry
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Introduction Groupoids PDO Index Theory Bonus

Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle.

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Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle. DNC(M, V ) = (M × R∗) ⊔ (N M

V × {0}).

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Introduction Groupoids PDO Index Theory Bonus

Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle. DNC(M, V ) = (M × R∗) ⊔ (N M

V × {0}).

Natural smooth structure. Generated by :

  • the natural map ϕ : DNC(M, V ) → M × R is smooth

(given by (x, t) → (x, t) and (x, ξ, 0) → (x, 0)).

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Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle. DNC(M, V ) = (M × R∗) ⊔ (N M

V × {0}).

Natural smooth structure. Generated by :

  • the natural map ϕ : DNC(M, V ) → M × R is smooth

(given by (x, t) → (x, t) and (x, ξ, 0) → (x, 0)).

  • if f : M → R smooth and vanishes on V , the function

f dnc : DNC(M, V ) → R given by (x, t) → f(x) t and (x, ξ, 0) → d f(ξ) is smooth.

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Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle. DNC(M, V ) = (M × R∗) ⊔ (N M

V × {0}).

Natural smooth structure. Generated by :

  • the natural map ϕ : DNC(M, V ) → M × R is smooth

(given by (x, t) → (x, t) and (x, ξ, 0) → (x, 0)).

  • if f : M → R smooth and vanishes on V , the function

f dnc : DNC(M, V ) → R given by (x, t) → f(x) t and (x, ξ, 0) → d f(ξ) is smooth.

Remarks (D.-Skandalis)

  • 1. This construction is functorial
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Connes’ tangent groupoid

Deformation to the normal cone : V ⊂ M a submanifold, N M

V

the normal bundle. DNC(M, V ) = (M × R∗) ⊔ (N M

V × {0}).

Natural smooth structure. Generated by :

  • the natural map ϕ : DNC(M, V ) → M × R is smooth

(given by (x, t) → (x, t) and (x, ξ, 0) → (x, 0)).

  • if f : M → R smooth and vanishes on V , the function

f dnc : DNC(M, V ) → R given by (x, t) → f(x) t and (x, ξ, 0) → d f(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.

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) (M ⊂ M × M diagonally)

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) (M ⊂ M × M diagonally) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R.

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R. More precisely GT = DNC[0,1](M × M, M) = (M × M × (0, 1]) ⊔ (TM × {0}).

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R. GT = DNC[0,1](M × M, M) = (M × M × (0, 1]) ⊔ (TM × {0}). Exact sequence 0 → K ⊗ C0((0, 1]) − → C∗(GT )

ev0

− → C∗(TM) → 0.

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R. GT = DNC[0,1](M × M, M) = (M × M × (0, 1]) ⊔ (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.

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R. GT = DNC[0,1](M × M, M) = (M × M × (0, 1]) ⊔ (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. K ⊗ C0((0, 1]) C∗(GT )

ev0 ev1

  • C∗(TM) = C0(T ∗M)

K

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Connes’ tangent groupoid : Analytic index without PDO’s Definition (Connes’ tangent groupoid)

DNC(M × M, M) = (M × M × R∗) ⊔ (TM × {0}) ⇒ M × R. GT = DNC[0,1](M × M, M) = (M × M × (0, 1]) ⊔ (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. K ⊗ C0((0, 1]) C∗(GT )

ev0 ev1

  • C∗(TM) = C0(T ∗M)
  • inda
  • K

Theorem (Connes) : The Atiyah-Singer analytic index is equal to ˜ ∂M×M = [ev1] ◦ [ev0]−1 : K0(C0(T ∗M)) → K0(K) = Z

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

➤ Natural ≪ zooming action ≫ of R∗

+ :

λ(x, t) = (x, λt) for t = 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere.

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

➤ Natural ≪ zooming action ≫ of R∗

+ :

λ(x, t) = (x, λt) for t = 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere. Gauge adiabatic groupoid : Gag = Gad ⋊ R∗

+ = (G × R∗ + × R∗ +) ⊔ (AG ⋊ R∗ +) ⇒ M × R

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

➤ Natural ≪ zooming action ≫ of R∗

+ :

λ(x, t) = (x, λt) for t = 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere. Gauge adiabatic groupoid : Gag = Gad ⋊ R∗

+ = (G × R∗ + × R∗ +) ⊔ (AG ⋊ R∗ +) ⇒ M × R

➤ 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))

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

➤ Natural ≪ zooming action ≫ of R∗

+ :

λ(x, t) = (x, λt) for t = 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere. Gauge adiabatic groupoid : Gag = Gad ⋊ R∗

+ = (G × R∗ + × R∗ +) ⊔ (AG ⋊ R∗ +) ⇒ M × R

➤ 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

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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∗

+) ⊔ (AG × {0}) ⇒ M × R+

➤ Natural ≪ zooming action ≫ of R∗

+ :

λ(x, t) = (x, λt) for t = 0 and λ(u, ξ, 0) = (u, λ−1ξ, 0) elsewhere. Gauge adiabatic groupoid : Gag = Gad ⋊ R∗

+ = (G × R∗ + × R∗ +) ⊔ (AG ⋊ R∗ +) ⇒ M × R

➤ 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∗

+)

  • ⋊ R∗

+

≃ C∗(G) ⊗ K → J(G) ⋊ R∗

+

⊂ C∗(Gga) → C0(A∗G \ G(0)) ⋊ R∗

+

≃ C(S∗AG) ⊗ K → 0 (GAG)

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Gauge adiabatic groupoid and PDOs (D.-Skandalis)

We have an exact sequence 0 → C∗(G) ⊗ K − → J(G) ⋊ R∗

+ −

→ C(SA∗G) ⊗ K → 0 (GAG)

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Gauge adiabatic groupoid and PDOs (D.-Skandalis)

We have an exact sequence 0 → C∗(G) ⊗ K − → J(G) ⋊ R∗

+ −

→ C(SA∗G) ⊗ K → 0 (GAG) Compare with... 0 → C∗(G) − → Ψ∗(G)

σ

− → C(SA∗G) → 0 (PDO)

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Gauge adiabatic groupoid and PDOs (D.-Skandalis)

We have an exact sequence 0 → C∗(G) ⊗ K − → J(G) ⋊ R∗

+ −

→ 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) ⋊ R∗

+ ≃ Ψ∗(G) ⊗ K.

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Gauge adiabatic groupoid and PDOs (D.-Skandalis)

We have an exact sequence 0 → C∗(G) ⊗ K − → J(G) ⋊ R∗

+ −

→ 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) ⋊ R∗

+ ≃ Ψ∗(G) ⊗ K.

In other words, pseudodifferential operators can be expressed as convolution kernels on a groupoid.

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Theorem (D. & Skandalis)

There is an ideal J (G) ⊂ C∞

c (Gad) such that for

f = (ft)t∈R+ ∈ J (G) and m ∈ N let P = +∞ tmft dt t and σ : (x, ξ) ∈ A∗G → +∞ tm f(x, tξ, 0)dt t Then P belongs to P−m(G) and its principal symbol is σ.

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Theorem (D. & Skandalis)

There is an ideal J (G) ⊂ C∞

c (Gad) such that for

f = (ft)t∈R+ ∈ J (G) and m ∈ N let P = +∞ tmft dt t and σ : (x, ξ) ∈ A∗G → +∞ tm f(x, tξ, 0)dt t Then P belongs to P−m(G) and its principal symbol is σ. What does it mean :

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Theorem (D. & Skandalis)

There is an ideal J (G) ⊂ C∞

c (Gad) such that for

f = (ft)t∈R+ ∈ J (G) and m ∈ N let P = +∞ tmft dt t and σ : (x, ξ) ∈ A∗G → +∞ tm f(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 operator P ∈ P−m(G) with principal symbol σ such that if g ∈ C∞

c (G) :

P ∗ g = +∞ tmft ∗ g dt t and g ∗ P = +∞ tmg ∗ ft dt t

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Theorem (D. & Skandalis)

There is an ideal J (G) ⊂ C∞

c (Gad) such that for

f = (ft)t∈R+ ∈ J (G) and m ∈ N let P = +∞ tmft dt t and σ : (x, ξ) ∈ A∗G → +∞ tm f(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 operator P ∈ P−m(G) with principal symbol σ such that if g ∈ C∞

c (G) :

P ∗ g = +∞ tmft ∗ g dt t and g ∗ P = +∞ tmg ∗ ft dt t Remark : Moreover any P ∈ P−m(G) is a Pf = +∞ tmft dt t for some f = (ft)t∈R+ ∈ J (G).

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In conclusion...

Lie groupoids allow us

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In conclusion...

Lie groupoids allow us

  • 1. to generalise the analytic index ;
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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 operator

free way ;

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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 operator

free way ;

  • 3. to give proofs of index theorems ;
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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 operator

free way ;

  • 3. to give proofs of index theorems ;
  • 4. to express the order 0 pseudodifferential operators
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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 operator

free way ;

  • 3. to give proofs of index theorems ;
  • 4. to express the order 0 pseudodifferential operators in a

(pseudo)differential operator free way.

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Thank you for your attention !

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Introduction Groupoids PDO Index Theory Bonus

Analytic index for groupoids

Let G ⇒ M be a Lie groupoid.

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Analytic index for groupoids

Let G ⇒ M be a Lie groupoid. Define the adiabatic groupoid Gad = DNC[0,1](G, M).

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Analytic index for groupoids

Let G ⇒ M be a Lie groupoid. Define the adiabatic groupoid Gad = DNC[0,1](G, M). Diagram C∗(G) ⊗ C0((0, 1]) C∗(Gad)

ev0 ev1

  • C0(A∗

G)

  • inda
  • C∗(G)
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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 C∗(G) ⊗ C0((0, 1]) C∗(Gad)

ev0 ev1

  • C0(A∗

G)

  • inda
  • C∗(G)

Theorem (Monthubert-Pierrot and Nistor-Weinstein-Xu)

Analytic index = [ev1] ◦ [ev0]−1.

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Digression : Pullback groupoid and Morita equivalence

Suppose G ⇒ M is a smooth groupoid and let f : N → M be a surjective submersion. The pullback groupoid of G by f is the smooth groupoid Gf

f := {(x, γ, y) ∈ N × G × N | r(γ) = f(x), s(γ) = f(y)} ⇒ N

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Digression : Pullback groupoid and Morita equivalence

Suppose G ⇒ M is a smooth groupoid and let f : N → M be a surjective submersion. The pullback groupoid of G by f is the smooth groupoid Gf

f := {(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 if

  • ne can find a manifold Z with two surjective submersions p : Z → M

and q : Z → N such that the pullbacks Gp

p ⇒ Z and Hq q ⇒ Z are

isomorphic.

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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 a surjective submersion. The pullback groupoid of G by f is the smooth groupoid Gf

f := {(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 if

  • ne can find a manifold Z with two surjective submersions p : Z → M

and q : Z → N such that the pullbacks Gp

p ⇒ Z and Hq q ⇒ Z are

isomorphic.

Theorem (Muhly, Renault, Williams)

The C∗-algebras of two Morita equivalent groupoids are Morita equivalent.

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Atiyah-Singer index theorem

π : E → M a vector bundle ; consider ∆E ⊂ E ×

M E ⊂ E × E :

T = DNC

  • DNC(E × E, E ×

M E), ∆E × {0}

  • ⇒ E × R × R

Let T = T |E×[0,1]×[0,1] and T hom = T |E×{0}×[0,1].

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Atiyah-Singer index theorem

π : E → M a vector bundle ; consider ∆E ⊂ E ×

M E ⊂ E × E :

T = DNC

  • DNC(E × E, E ×

M E), ∆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} ⊔ TM π

π ×]0, 1] ⇒ E × [0, 1]

and the Morita equivalence TM π

π ∼ TM provides :

τE : K∗(C∗(TE)) = K∗(C0(T ∗E)) → K∗(C∗(TM)) = K∗(C0(T ∗M))

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Atiyah-Singer index theorem

π : E → M a vector bundle ; consider ∆E ⊂ E ×

M E ⊂ E × E :

T = DNC

  • DNC(E × E, E ×

M E), ∆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} ⊔ TM π

π ×]0, 1] ⇒ E × [0, 1]

and the Morita equivalence TM π

π ∼ 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 = N Rn

M the normal of the inclusion, τE is the

inverse of the Thom isomorphism.

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Atiyah-Singer index theorem

π : E → M a vector bundle ; consider ∆E ⊂ E ×

M E ⊂ E × E :

T = DNC

  • DNC(E × E, E ×

M E), ∆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 = N Rn

M the normal of the inclusion, τE is the

inverse of the Thom isomorphism.

Particular case The normal bundle of · ֒

→ Rn is just Rn → · and leads to the Bott periodicity τRn : K0(C∗(TRn)) → Z.

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Atiyah-Singer index theorem

π : E → M a vector bundle ; consider ∆E ⊂ E ×

M E ⊂ E × E :

T = DNC

  • DNC(E × E, E ×

M E), ∆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 = N Rn

M the normal of the inclusion, τE is the

inverse of the Thom isomorphism.

Particular case The normal bundle of · ֒

→ Rn is just Rn → · and leads to the Bott periodicity τRn : K0(C∗(TRn)) → Z.

Conclusion Indt = τRn ◦ [j] ◦ τ −1

E

is entirely described with (deformation) groupoids.

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Atiyah-Singer index theorem

E = N Rn

M , T = DNC

  • DNC(E × E, E ×

M E), ∆E × {0}

  • |E×[0,1]×[0,1]

E ×

M TM × M E IndM

a

  • DNC(E × E, E ×

M E ≃(GT (M))π

π

)|E×[0,1] E × E T hom T GT (E) TE

T hom−1

  • TE × [0, 1]

TE

IndE

a

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Atiyah-Singer index theorem

E = N Rn

M , T = DNC

  • DNC(E × E, E ×

M E), ∆E × {0}

  • |E×[0,1]×[0,1]

E ×

M TM × M E IndM

a

  • DNC(E × E, E ×

M E ≃(GT (M))π

π

)|E×[0,1] E × E T hom T GT (E) TE

T hom−1

  • TE × [0, 1]

TE

IndE

a

  • Gives IndM

a = IndM t

[D.-Lescure-Nistor].

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Atiyah-Singer index theorem

E = N Rn

M , T = DNC

  • DNC(E × E, E ×

M E), ∆E × {0}

  • |E×[0,1]×[0,1]

E ×

M TM × M E IndM

a

  • DNC(E × E, E ×

M E ≃(GT (M))π

π

)|E×[0,1] E × E T hom T GT (E) TE

T hom−1

  • TE × [0, 1]

TE

IndE

a

  • Gives IndM

a = IndM t

[D.-Lescure-Nistor]. Can be extended to M with isolated conical singularities...

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Atiyah-Singer index theorem

E = N Rn

M , T = DNC

  • DNC(E × E, E ×

M E), ∆E × {0}

  • |E×[0,1]×[0,1]

E ×

M TM × M E IndM

a

  • DNC(E × E, E ×

M E ≃(GT (M))π

π

)|E×[0,1] E × E T hom T GT (E) TE

T hom−1

  • TE × [0, 1]

TE

IndE

a

  • Gives IndM

a = IndM t

[D.-Lescure-Nistor]. Can be extended to M with isolated conical singularities... and to M a general pseudomanifold.