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 e u ∈ G with r ( e u ) = s ( e u ) = u ; e r ( x ) · x = x and x · e s ( 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 = e r ( x ) and x − 1 · x = e s ( 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.
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 �→ e u of G (0) to G , inverse are smooth
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 �→ e u of G (0) to G , inverse are smooth • s, r : G → G (0) are smooth submersions
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 �→ e u 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 ;
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 �→ e u 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.
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
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... id M .
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.
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.
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. r,s 4. Pair groupoid M × M ⇒ 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 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 )
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...
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 , π 1 ( 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 relation The pair groupoid M × M ⇒ M ; ( x, y ) · ( y, z ) = ( x, z ).
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 over M are exactly graphs of equivalence relations.
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 over M are exactly graphs of equivalence relations. ◮ Let R be an equivalence relation on M . Its graph G R = { ( x, y ) ∈ M × M | x R y } ⇒ M is a Lie groupoid when R is the relation ≪ being on the same leaf of a regular foliation with no holonomy. ≫
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 over M are exactly graphs of equivalence relations. ◮ Let R be an equivalence relation on M . Its graph G R = { ( x, y ) ∈ M × M | x R y } ⇒ 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.
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ])
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle.
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map.
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields.
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields. It satisfies for any X, Y ∈ Γ( A G ), f ∈ C ∞ ( G (0) ) : ♯ [ X, Y ] = [ ♯ ( X ) , ♯ ( Y )] and [ X, fY ] = f [ X, Y ] + ♯ ( X )( f ) .Y
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields. It satisfies for any X, Y ∈ Γ( A G ), 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.
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields. It satisfies for any X, Y ∈ Γ( A G ), 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, [ · , · ]).
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields. It satisfies for any X, Y ∈ Γ( A G ), 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 : ( T F , Id, [ · , · ]).
Introduction Groupoids PDO Index Theory Bonus A few words about Lie theory for groupoids The Lie algebroid of the Lie groupoid G ⇒ M is : ( A G, ♯, [ · , · ]) • A G = KerTs | G (0) → G (0) - smooth vector bundle. • ♯ = Tr : A G → TG (0) - smooth bundle map. • [ · , · ] Lie bracket on smooth sections of A G - constructed from left invariant vector fields. It satisfies for any X, Y ∈ Γ( A G ), 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 : ( T F , Id, [ · , · ]). ! Lie third theorem fails : Lie algebroid may not be the Lie algebroid of a Lie groupoid.
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x .
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u .
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) .
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) . � G r ( x ) f 1 ( y ) f 2 ( y − 1 x ) dν r ( x ) ( y ) . Convolution formula f 1 ∗ f 2 ( x ) =
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) . � G r ( x ) f 1 ( y ) f 2 ( y − 1 x ) dν r ( x ) ( y ) . Convolution formula f 1 ∗ f 2 ( x ) = Convolution associative by invariance of the Haar system (and Fubini).
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) . � G r ( x ) f 1 ( y ) f 2 ( y − 1 x ) dν r ( x ) ( y ) . Convolution formula f 1 ∗ f 2 ( x ) = Convolution associative by invariance of the Haar system (and Fubini). c ( G ) : function f ∗ : x �→ f ( x − 1 ). Adjoint of f ∈ C ∞
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) . � G r ( x ) f 1 ( y ) f 2 ( y − 1 x ) dν r ( x ) ( y ) . Convolution formula f 1 ∗ f 2 ( x ) = Convolution associative by invariance of the Haar system (and Fubini). c ( G ) : function f ∗ : x �→ f ( x − 1 ). Adjoint of f ∈ C ∞ We choose an operator nom : C ∗ ( G ) = completion of C ∞ c ( G ).
Introduction Groupoids PDO Index Theory Bonus Convolution on a Lie groupoid G � Algebra C ∞ c ( G ) : f 1 ∗ f 2 ( x ) = f 1 ( x 1 ) f 2 ( x 2 ) dν ( x 1 ,x 2 ) ∈ G ; x 1 x 2 = x • The set { ( x 1 , x 2 ) ∈ G × G ; x 1 x 2 = x } : smooth manifold x 1 ∈ G r ( x ) = { y ∈ G ; r ( y ) = r ( x ) } and x 2 = x − 1 1 x . • dν is a smooth “Haar system” • i.e. smooth choice of a Lebesgue measure ν u on every G u . • Left invariance : for every x ∈ G , the measure ν s ( x ) ↔ ν r ( x ) through diffeomorphism y �→ x · y from G s ( x ) with G r ( x ) . � G r ( x ) f 1 ( y ) f 2 ( y − 1 x ) dν r ( x ) ( y ) . Convolution formula f 1 ∗ f 2 ( x ) = Convolution associative by invariance of the Haar system (and Fubini). c ( G ) : function f ∗ : x �→ f ( x − 1 ). Adjoint of f ∈ C ∞ We choose an operator nom : C ∗ ( G ) = completion of C ∞ c ( G ). In the ≪ good cases ≫ : C 0 ( G (0) /G ) � C ∗ ( G ).
Introduction Groupoids PDO Index Theory Bonus Examples of groupoids and their C ∗ -algebras 1. G = M × M .
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 .
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 ).
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 ) = C 0 ( E ∗ ) : where E ∗ is the dual bundle and C 0 ( E ∗ ) : functions that vanish at infinity on (the total space of) E ∗ (using Fourier transform).
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 ) = C 0 ( E ∗ ) : where E ∗ is the dual bundle and C 0 ( 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 ∗ ( G U ) → C ∗ ( G ) → C ∗ ( G F ) → 0 .
Introduction Groupoids PDO Index Theory Bonus 2. Pseudodifferential operators and analytic index
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G .
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x )
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x ) • d is the dimension of the algebroid A G ; • θ : U → A G is a diffeomorphism, inverse of an exp map, from a tubular neighbourhood of G (0) in G to the normal bundle A G .
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x ) • d is the dimension of the algebroid A G ; • θ : U → A G is a diffeomorphism, inverse of an exp map, from a tubular neighbourhood of G (0) in G to the normal bundle A G . • χ is a smooth bump function (1 on G (0) , and 0 outside U ) ;
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x ) • d is the dimension of the algebroid A G ; • θ : U → A G is a diffeomorphism, inverse of an exp map, from a tubular neighbourhood of G (0) in G to the normal bundle A G . • χ is a smooth bump function (1 on G (0) , and 0 outside U ) ; � • a is a classical polyhomogeneous symbol a ( u, ξ ) ∼ a m − j ( u, ξ ) j where a ℓ is homogeneous of order ℓ .
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x ) • d is the dimension of the algebroid A G ; • θ : U → A G is a diffeomorphism, inverse of an exp map, from a tubular neighbourhood of G (0) in G to the normal bundle A G . • χ is a smooth bump function (1 on G (0) , and 0 outside U ) ; � • a is a classical polyhomogeneous symbol a ( u, ξ ) ∼ a m − j ( u, ξ ) j where a ℓ is homogeneous of order ℓ .
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential operators on a Lie groupoid G Differential operators : (enveloping) algebra generated by sections of the algebroid A G = ∪ x ∈ G (0) TG x ≃ the normal bundle of the inclusion G (0) ⊂ G . P m ( 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 ) = (2 π ) − d e i � θ ( x ) | ξ � χ ( x ) a ( s ( x ) , ξ ) dξ ( A ∗ G ) s ( x ) • d is the dimension of the algebroid A G ; • θ : U → A G is a diffeomorphism, inverse of an exp map, from a tubular neighbourhood of G (0) in G to the normal bundle A G . • χ is a smooth bump function (1 on G (0) , and 0 outside U ) ; � • a is a classical polyhomogeneous symbol a ( u, ξ ) ∼ a m − j ( u, ξ ) j where a ℓ is homogeneous of order ℓ . � � • oscilatory integral = lim (as a distribution). R →∞ ( A ∗ G ) s ( x ) � ξ �≤ R
Introduction Groupoids PDO Index Theory Bonus Pseudodifferential calculus These pseudodifferential operators form a convolution ∗ -algebra :
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 ).
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.
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 P 0 ( G ).
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 P 0 ( G ). We obtain an exact sequence of C ∗ algebras. σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − ( PDO )
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 P 0 ( G ). We obtain an exact sequence of C ∗ algebras. σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − ( PDO )
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 P 0 ( G ). We obtain an exact sequence of C ∗ algebras. σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − ( PDO ) • S A ∗ G is the sphere bundle of the dual bundle to the algebroid A G .
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 P 0 ( G ). We obtain an exact sequence of C ∗ algebras. σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − ( PDO ) • S A ∗ G is the sphere bundle of the dual bundle to the algebroid A G . � • σ : principal symbol map ( a ∼ a − j �→ a 0 ).
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 −
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )).
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )). + ≃ A ∗ G \ G (0) → A ∗ G . Consider the inclusion i : S A ∗ G × R ∗ ∂ G = ˜ ∂ G ◦ [ i ]
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )). + ≃ A ∗ G \ G (0) → A ∗ G . Consider the inclusion i : S A ∗ G × R ∗ ∂ G = ˜ ∂ G ◦ [ i ] Example G = M × M pair groupoid ; A G = TM ;
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )). + ≃ A ∗ G \ G (0) → A ∗ G . Consider the inclusion i : S A ∗ G × R ∗ ∂ G = ˜ ∂ G ◦ [ i ] Example G = M × M pair groupoid ; A G = TM ; C ∗ ( G ) = K ( L 2 ( M )).
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )). + ≃ A ∗ G \ G (0) → A ∗ G . Consider the inclusion i : S A ∗ G × R ∗ ∂ G = ˜ ∂ G ◦ [ i ] Example G = M × M pair groupoid ; A G = TM ; C ∗ ( G ) = K ( L 2 ( M )). ∂ G : K 0 ( C 0 ( T ∗ M )) → K 0 ( K ) = Z is the Atiyah-Singer analytic index. ˜
Introduction Groupoids PDO Index Theory Bonus Analytic index σ 0 → C ∗ ( G ) − → Ψ ∗ ( G ) → C ( S A ∗ G ) → 0 − Connecting map : ∂ G : K ∗ +1 ( C ( S A ∗ G )) → K ∗ ( C ∗ ( G )). + ≃ A ∗ G \ G (0) → A ∗ G . Consider the inclusion i : S A ∗ G × R ∗ ∂ G = ˜ ∂ G ◦ [ i ] Example G = M × M pair groupoid ; A G = TM ; C ∗ ( G ) = K ( L 2 ( M )). ∂ G : K 0 ( C 0 ( T ∗ M )) → K 0 ( K ) = Z is the Atiyah-Singer analytic index. ˜ Thus ˜ ∂ G : K ∗ ( C 0 ( A ∗ G )) → K ∗ ( C ∗ ( G )) is a generalised analytic index.
Introduction Groupoids PDO Index Theory Bonus In the examples... 1. G = M × M .
Introduction Groupoids PDO Index Theory Bonus In the examples... 1. G = M × M . Atiyah-Singer index - with values in Z .
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 .
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 ∗ ( C 0 ( B )).
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 ∗ ( C 0 ( B )). 3. G holonomy groupoid of a foliation ( M, F ).
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 ∗ ( C 0 ( B )). 3. G holonomy groupoid of a foliation ( M, F ). Connes’ index with values in K ∗ ( C ∗ ( M, F )).
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 ∗ ( C 0 ( 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.
Introduction Groupoids PDO Index Theory Bonus 3. Constructions of Lie groupoids in connection with index theory
Introduction Groupoids PDO Index Theory Bonus 3. Constructions of Lie groupoids in 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, N M the V normal bundle.
Introduction Groupoids PDO Index Theory Bonus Connes’ tangent groupoid Deformation to the normal cone : V ⊂ M a submanifold, N M the V normal bundle. DNC ( M, V ) = ( M × R ∗ ) ⊔ ( N M V × { 0 } ).
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