Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup...
Transcript of Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup...
![Page 1: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/1.jpg)
Motivations The DNC and Blup constructions Exact sequences
Blowup and deformation groupoids in relationwith index theory
D. & Skandalis - Blowup constructions for Lie groupoids and a Boutet deMonvel type calculus
Claire Debord
IMJ-PRG
Orleans 2019 : Jean’s feast
![Page 2: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/2.jpg)
Motivations The DNC and Blup constructions Exact sequences
Some historical Groupoid successes in index theory
Smooth compact manifold M :
M ×M ⇒M the pair groupoid.
The tangent groupoid of A. Connes :
GtM = TM × {0} ∪M ×M×]0, 1] ⇒M × [0, 1]
It defines : 0→ C∗(GtM |M×]0,1])
' K ⊗ C0(]0, 1])
→ C∗(GtM )e0→ C∗(GtM |M×{0})
= C∗(TM)
→ 0
[e0] ∈ KK(C∗(GtM ), C∗(TM)) is invertible.Let e1 : C∗(GtM )→ C∗(GtM |M×{1}) = C∗(M ×M) ' K.
The index element
IndM×M := [e0]−1 ⊗ [e1] ∈ KK(C∗(TM),K) ' K0(C∗(TM)) .
![Page 3: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/3.jpg)
Motivations The DNC and Blup constructions Exact sequences
Some historical Groupoid successes in index theory
Smooth compact manifold M : M ×M ⇒M the pair groupoid.
The tangent groupoid of A. Connes :
GtM = TM × {0} ∪M ×M×]0, 1] ⇒M × [0, 1]
It defines : 0→ C∗(GtM |M×]0,1])
' K ⊗ C0(]0, 1])
→ C∗(GtM )e0→ C∗(GtM |M×{0})
= C∗(TM)
→ 0
[e0] ∈ KK(C∗(GtM ), C∗(TM)) is invertible.Let e1 : C∗(GtM )→ C∗(GtM |M×{1}) = C∗(M ×M) ' K.
The index element
IndM×M := [e0]−1 ⊗ [e1] ∈ KK(C∗(TM),K) ' K0(C∗(TM)) .
![Page 4: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/4.jpg)
Motivations The DNC and Blup constructions Exact sequences
Some historical Groupoid successes in index theory
Smooth compact manifold M : M ×M ⇒M the pair groupoid.
The tangent groupoid of A. Connes :
GtM = TM × {0} ∪M ×M×]0, 1] ⇒M × [0, 1]
It defines : 0→ C∗(GtM |M×]0,1])
' K ⊗ C0(]0, 1])
→ C∗(GtM )e0→ C∗(GtM |M×{0})
= C∗(TM)
→ 0
[e0] ∈ KK(C∗(GtM ), C∗(TM)) is invertible.
Let e1 : C∗(GtM )→ C∗(GtM |M×{1}) = C∗(M ×M) ' K.
The index element
IndM×M := [e0]−1 ⊗ [e1] ∈ KK(C∗(TM),K) ' K0(C∗(TM)) .
![Page 5: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/5.jpg)
Motivations The DNC and Blup constructions Exact sequences
Some historical Groupoid successes in index theory
Smooth compact manifold M : M ×M ⇒M the pair groupoid.
The tangent groupoid of A. Connes :
GtM = TM × {0} ∪M ×M×]0, 1] ⇒M × [0, 1]
It defines : 0→ C∗(GtM |M×]0,1])
' K ⊗ C0(]0, 1])
→ C∗(GtM )e0→ C∗(GtM |M×{0})
= C∗(TM)
→ 0
[e0] ∈ KK(C∗(GtM ), C∗(TM)) is invertible.Let e1 : C∗(GtM )→ C∗(GtM |M×{1}) = C∗(M ×M) ' K.
The index element
IndM×M := [e0]−1 ⊗ [e1] ∈ KK(C∗(TM),K) ' K0(C∗(TM)) .
![Page 6: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/6.jpg)
Motivations The DNC and Blup constructions Exact sequences
The algebra Ψ∗(G) = Ψ∗(M ×M) identifies with the C∗-algebra oforder 0 pseudodifferential operators on M and
0 // C∗(M ×M)' K
// Ψ∗(M ×M) // C(S∗TM) // 0
which gives a connecting element IndM×M ∈ KK1(C(S∗TM),K).
Let i be the inclusion of S∗TM × R∗+ as the open subset T ∗M \M ofT ∗M then
IndM×M = IndM×M ⊗ [i]
Proposition [Connes]
The morphism · ⊗ IndM×M : K0(T ∗M) ' KK(C, C∗(TM)) −→ Z isthe analytic index map of A-S.
Foliation F on M : Replace in the picture the groupoid M ×M bythe holonomy groupoid Hol(M,F) (i.e. the “smallest” Lie groupoidover M whose orbits are the leaves of the foliation) [Connes].
![Page 7: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/7.jpg)
Motivations The DNC and Blup constructions Exact sequences
The algebra Ψ∗(G) = Ψ∗(M ×M) identifies with the C∗-algebra oforder 0 pseudodifferential operators on M and
0 // C∗(M ×M)' K
// Ψ∗(M ×M) // C(S∗TM) // 0
which gives a connecting element IndM×M ∈ KK1(C(S∗TM),K).Let i be the inclusion of S∗TM × R∗+ as the open subset T ∗M \M ofT ∗M then
IndM×M = IndM×M ⊗ [i]
Proposition [Connes]
The morphism · ⊗ IndM×M : K0(T ∗M) ' KK(C, C∗(TM)) −→ Z isthe analytic index map of A-S.
Foliation F on M : Replace in the picture the groupoid M ×M bythe holonomy groupoid Hol(M,F) (i.e. the “smallest” Lie groupoidover M whose orbits are the leaves of the foliation) [Connes].
![Page 8: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/8.jpg)
Motivations The DNC and Blup constructions Exact sequences
The algebra Ψ∗(G) = Ψ∗(M ×M) identifies with the C∗-algebra oforder 0 pseudodifferential operators on M and
0 // C∗(M ×M)' K
// Ψ∗(M ×M) // C(S∗TM) // 0
which gives a connecting element IndM×M ∈ KK1(C(S∗TM),K).Let i be the inclusion of S∗TM × R∗+ as the open subset T ∗M \M ofT ∗M then
IndM×M = IndM×M ⊗ [i]
Proposition [Connes]
The morphism · ⊗ IndM×M : K0(T ∗M) ' KK(C, C∗(TM)) −→ Z isthe analytic index map of A-S.
Foliation F on M : Replace in the picture the groupoid M ×M bythe holonomy groupoid Hol(M,F) (i.e. the “smallest” Lie groupoidover M whose orbits are the leaves of the foliation) [Connes].
![Page 9: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/9.jpg)
Motivations The DNC and Blup constructions Exact sequences
The algebra Ψ∗(G) = Ψ∗(M ×M) identifies with the C∗-algebra oforder 0 pseudodifferential operators on M and
0 // C∗(M ×M)' K
// Ψ∗(M ×M) // C(S∗TM) // 0
which gives a connecting element IndM×M ∈ KK1(C(S∗TM),K).Let i be the inclusion of S∗TM × R∗+ as the open subset T ∗M \M ofT ∗M then
IndM×M = IndM×M ⊗ [i]
Proposition [Connes]
The morphism · ⊗ IndM×M : K0(T ∗M) ' KK(C, C∗(TM)) −→ Z isthe analytic index map of A-S.
Foliation F on M : Replace in the picture the groupoid M ×M bythe holonomy groupoid Hol(M,F) (i.e. the “smallest” Lie groupoidover M whose orbits are the leaves of the foliation) [Connes].
![Page 10: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/10.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G)) with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 11: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/11.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G))
with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 12: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/12.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G)) with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 13: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/13.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G)) with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 14: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/14.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G)) with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 15: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/15.jpg)
Motivations The DNC and Blup constructions Exact sequences
General Lie groupoid G⇒M [Monthubert-Pierrot, Nistor-Weinstein-Xu]
The adiabatic groupoid : GtM = AG× {0} ∪G×]0, 1] ⇒M × [0, 1]gives IndG := [e0]−1 ⊗ [e1] ∈ KK(C∗(AG), C∗(G)).
Pseudodifferential exact sequence :
0 // C∗(G) // Ψ∗(G) // C(S∗AG) // 0
which defines IndG ∈ KK1(C(S∗AG), C∗(G)) with IndG = IndG ⊗ [i]where i is the inclusion of S∗AG× R∗+ as the open subset A∗G \M ofA∗G.
Manifold with boundary - V ⊂M a hypersurface [Melrose & co.]
• 0-calculus, (pseudodifferential) operators vanishing on V :replace M ×M by G0 ⇒M equal to the pair groupoid on M \ Voutside V and isomorphic to GtV oR∗+ around V .
• b-calculus, (pseudodifferential) operators vanishing on the normaldirection of V : replace M ×M by Gb ⇒M equal toM \ V ×M \ V outside V and isomorphic to V × V × Ro R∗+around V .
![Page 16: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/16.jpg)
Motivations The DNC and Blup constructions Exact sequences
What about more general situations ...
Can we mix situation analogous to foliation and hypersurface ?
There is no reason to restrict to :
• V being a hypersurface,
• Operators which are usual operators outside V .
Framework : G⇒M a Lie groupoid, V ⊂M a submanifold,Γ ⇒ V a sub-groupoid of G and operators that “slow down” near Vin the normal direction and “propagate” along Γ inside V .
Today, in this talk :• Present general groupoid constructions involved in such
situations.
• Compute, compare the corresponding index elements andconnecting maps arising.
![Page 17: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/17.jpg)
Motivations The DNC and Blup constructions Exact sequences
What about more general situations ...
Can we mix situation analogous to foliation and hypersurface ?There is no reason to restrict to :
• V being a hypersurface,
• Operators which are usual operators outside V .
Framework : G⇒M a Lie groupoid, V ⊂M a submanifold,Γ ⇒ V a sub-groupoid of G and operators that “slow down” near Vin the normal direction and “propagate” along Γ inside V .
Today, in this talk :• Present general groupoid constructions involved in such
situations.
• Compute, compare the corresponding index elements andconnecting maps arising.
![Page 18: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/18.jpg)
Motivations The DNC and Blup constructions Exact sequences
What about more general situations ...
Can we mix situation analogous to foliation and hypersurface ?There is no reason to restrict to :
• V being a hypersurface,
• Operators which are usual operators outside V .
Framework : G⇒M a Lie groupoid, V ⊂M a submanifold,Γ ⇒ V a sub-groupoid of G and operators that “slow down” near Vin the normal direction and “propagate” along Γ inside V .
Today, in this talk :• Present general groupoid constructions involved in such
situations.
• Compute, compare the corresponding index elements andconnecting maps arising.
![Page 19: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/19.jpg)
Motivations The DNC and Blup constructions Exact sequences
What about more general situations ...
Can we mix situation analogous to foliation and hypersurface ?There is no reason to restrict to :
• V being a hypersurface,
• Operators which are usual operators outside V .
Framework : G⇒M a Lie groupoid, V ⊂M a submanifold,Γ ⇒ V a sub-groupoid of G and operators that “slow down” near Vin the normal direction and “propagate” along Γ inside V .
Today, in this talk :• Present general groupoid constructions involved in such
situations.
• Compute, compare the corresponding index elements andconnecting maps arising.
![Page 20: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/20.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Deformation to the Normal Cone construction
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V . The deformation to the normal cone is
DNC(M,V ) = M × R∗ ∪NMV × {0}
It is endowed with a smooth structure thanks to the choice of anexponential map θ : U ′ ⊂ NM
V → U ⊂M by asking the map
Θ : (x,X, t) 7→{
(θ(x, tX), t) for t 6= 0(x,X, 0) for t = 0
to be a diffeomorphism from the open neighborhoodW ′ = {(x,X, t) ∈ NM
V ×R | (x, tX) ∈ U ′} of NMV ×{0} in NM
V ×R onits image.
We define similarly
DNC+(M,V ) = M × R∗+ ∪NMV × {0}
![Page 21: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/21.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Deformation to the Normal Cone construction
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V . The deformation to the normal cone is
DNC(M,V ) = M × R∗ ∪NMV × {0}
It is endowed with a smooth structure thanks to the choice of anexponential map θ : U ′ ⊂ NM
V → U ⊂M by asking the map
Θ : (x,X, t) 7→{
(θ(x, tX), t) for t 6= 0(x,X, 0) for t = 0
to be a diffeomorphism from the open neighborhoodW ′ = {(x,X, t) ∈ NM
V ×R | (x, tX) ∈ U ′} of NMV ×{0} in NM
V ×R onits image.
We define similarly
DNC+(M,V ) = M × R∗+ ∪NMV × {0}
![Page 22: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/22.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Deformation to the Normal Cone construction
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V . The deformation to the normal cone is
DNC(M,V ) = M × R∗ ∪NMV × {0}
It is endowed with a smooth structure thanks to the choice of anexponential map θ : U ′ ⊂ NM
V → U ⊂M by asking the map
Θ : (x,X, t) 7→{
(θ(x, tX), t) for t 6= 0(x,X, 0) for t = 0
to be a diffeomorphism from the open neighborhoodW ′ = {(x,X, t) ∈ NM
V ×R | (x, tX) ∈ U ′} of NMV ×{0} in NM
V ×R onits image.
We define similarly
DNC+(M,V ) = M × R∗+ ∪NMV × {0}
![Page 23: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/23.jpg)
Motivations The DNC and Blup constructions Exact sequences
Alternative description of DNC
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V :
DNC(M,V ) = M × R∗ ∪NMV × {0}
The smooth structure is generated by asking the following maps to besmooth
For any x ∈M , λ ∈ R∗, y ∈ V , ξ ∈ TyM/TyV :
• p : DNC(M,V )→M ×R : p(x, λ) = (x, λ) , p(y, ξ, 0) = (y, 0);
• given f : M → R, smooth with f|V = 0,
f : DNC(M,V )→ R, f(x, λ) =f(x)
λ, f(y, ξ, 0) = (df)y(ξ)
![Page 24: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/24.jpg)
Motivations The DNC and Blup constructions Exact sequences
Alternative description of DNC
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V :
DNC(M,V ) = M × R∗ ∪NMV × {0}
The smooth structure is generated by asking the following maps to besmooth
For any x ∈M , λ ∈ R∗, y ∈ V , ξ ∈ TyM/TyV :
• p : DNC(M,V )→M ×R : p(x, λ) = (x, λ) , p(y, ξ, 0) = (y, 0);
• given f : M → R, smooth with f|V = 0,
f : DNC(M,V )→ R, f(x, λ) =f(x)
λ, f(y, ξ, 0) = (df)y(ξ)
![Page 25: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/25.jpg)
Motivations The DNC and Blup constructions Exact sequences
Alternative description of DNC
Let V be a closed submanifold of a smooth manifold M with normalbundle NM
V :
DNC(M,V ) = M × R∗ ∪NMV × {0}
The smooth structure is generated by asking the following maps to besmooth
For any x ∈M , λ ∈ R∗, y ∈ V , ξ ∈ TyM/TyV :
• p : DNC(M,V )→M ×R : p(x, λ) = (x, λ) , p(y, ξ, 0) = (y, 0);
• given f : M → R, smooth with f|V = 0,
f : DNC(M,V )→ R, f(x, λ) =f(x)
λ, f(y, ξ, 0) = (df)y(ξ)
![Page 26: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/26.jpg)
Motivations The DNC and Blup constructions Exact sequences
Functoriality of DNC
Consider a commutative diagram of smooth maps
V� � //
fV��
M
fM��
V ′ �� // M ′
Where the horizontal arrows are inclusions of submanifolds. Let{DNC(f)(x, λ) = (fM (x), λ) for x ∈M, λ ∈ R∗DNC(f)(x, ξ, 0) = (fV (x), (dfM )x(ξ), 0) for x ∈ V, ξ ∈ TxM/TxV
We get a smooth map DNC(f) : DNC(M,V )→ DNC(M ′, V ′).
![Page 27: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/27.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).
Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 28: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/28.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid;
its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 29: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/29.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 30: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/30.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 31: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/31.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 32: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/32.jpg)
Motivations The DNC and Blup constructions Exact sequences
Deformation groupoid
Let Γ be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒ G(0).Functoriality implies :
DNC(G,Γ) ⇒ DNC(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are DNC(s)and DNC(t); DNC(G,Γ)(2) identifies with DNC(G(2),Γ(2)) and its
product with DNC(m) where m : G(2)i → Gi is the product.
Remarks
• No transversality asumption !
• NGΓ is a VB-groupoid over NG(0)
Γ(0) denoted NGΓ ⇒ NG(0)
Γ(0) .
DNC(G,Γ) = G× R∗ ∪NGΓ × {0}⇒ G(0) × R∗ ∪NG(0)
Γ(0) × {0}
![Page 33: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/33.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
1. The adiabatic groupoid is the restriction of DNC(G,G(0)) overG(0) × [0, 1].
2. If V is a saturated submanifold of G(0) for G, DNC(G,GVV ) isthe normal groupoid of the immersion GVV ↪→ G which gives theshriek map [M. Hilsum, G. Skandalis].
3. 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].
![Page 34: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/34.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
1. The adiabatic groupoid is the restriction of DNC(G,G(0)) overG(0) × [0, 1].
2. If V is a saturated submanifold of G(0) for G, DNC(G,GVV ) isthe normal groupoid of the immersion GVV ↪→ G which gives theshriek map [M. Hilsum, G. Skandalis].
3. 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].
![Page 35: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/35.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
1. The adiabatic groupoid is the restriction of DNC(G,G(0)) overG(0) × [0, 1].
2. If V is a saturated submanifold of G(0) for G, DNC(G,GVV ) isthe normal groupoid of the immersion GVV ↪→ G which gives theshriek map [M. Hilsum, G. Skandalis].
3. 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].
![Page 36: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/36.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
1. The adiabatic groupoid is the restriction of DNC(G,G(0)) overG(0) × [0, 1].
2. If V is a saturated submanifold of G(0) for G, DNC(G,GVV ) isthe normal groupoid of the immersion GVV ↪→ G which gives theshriek map [M. Hilsum, G. Skandalis].
3. 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].
![Page 37: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/37.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
3. 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].
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GtM
)ππ
)|E×[0,1] E × E
T hom T � GtE
TE
Thom−1
==
TE × [0, 1] TE
IndEa
dd
![Page 38: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/38.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples
3. 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].
E ×M
TM ×M
E
IndMa
,,DNC(E × E,E ×
ME
'(GtM
)ππ
)|E×[0,1] E × E
T hom T � GtE
TE
Thom−1
==
TE × [0, 1] TE
IndEa
dd
Gives IndMa = IndMt [D.-Lescure-Nistor].
![Page 39: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/39.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Blowup construction
The scaling action of R∗ on M × R∗ extends to the gauge action onDNC(M,V ) = M × R∗ ∪NM
V × {0} :
DNC(M,V )× R∗ −→ DNC(M,V )(z, t, λ) 7→ (z, λt) for t 6= 0
(x,X, 0, λ) 7→ (x, 1λX, 0) for t = 0
The manifold V × R embeds in DNC(M,V ) : V� � //
��
V
��V �� // M
The gauge action is free and proper on the open subsetDNC(M,V ) \ V × R of DNC(M,V ). We let :
Blup(M,V ) =(DNC(M,V ) \ V × R
)/R∗ = M \ V ∪ P(NM
V ) and
SBlup(M,V ) =(DNC+(M,V ) \ V × R+
)/R∗+ = M \ V ∪ S(NM
V ) .
![Page 40: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/40.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Blowup construction
The scaling action of R∗ on M × R∗ extends to the gauge action onDNC(M,V ) = M × R∗ ∪NM
V × {0} :
DNC(M,V )× R∗ −→ DNC(M,V )(z, t, λ) 7→ (z, λt) for t 6= 0
(x,X, 0, λ) 7→ (x, 1λX, 0) for t = 0
The manifold V × R embeds in DNC(M,V ) : V� � //
��
V
��V �� // M
The gauge action is free and proper on the open subsetDNC(M,V ) \ V × R of DNC(M,V ). We let :
Blup(M,V ) =(DNC(M,V ) \ V × R
)/R∗ = M \ V ∪ P(NM
V ) and
SBlup(M,V ) =(DNC+(M,V ) \ V × R+
)/R∗+ = M \ V ∪ S(NM
V ) .
![Page 41: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/41.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Blowup construction
The scaling action of R∗ on M × R∗ extends to the gauge action onDNC(M,V ) = M × R∗ ∪NM
V × {0} :
DNC(M,V )× R∗ −→ DNC(M,V )(z, t, λ) 7→ (z, λt) for t 6= 0
(x,X, 0, λ) 7→ (x, 1λX, 0) for t = 0
The manifold V × R embeds in DNC(M,V ) : V� � //
��
V
��V �� // M
The gauge action is free and proper on the open subsetDNC(M,V ) \ V × R of DNC(M,V ).
We let :
Blup(M,V ) =(DNC(M,V ) \ V × R
)/R∗ = M \ V ∪ P(NM
V ) and
SBlup(M,V ) =(DNC+(M,V ) \ V × R+
)/R∗+ = M \ V ∪ S(NM
V ) .
![Page 42: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/42.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Blowup construction
The scaling action of R∗ on M × R∗ extends to the gauge action onDNC(M,V ) = M × R∗ ∪NM
V × {0} :
DNC(M,V )× R∗ −→ DNC(M,V )(z, t, λ) 7→ (z, λt) for t 6= 0
(x,X, 0, λ) 7→ (x, 1λX, 0) for t = 0
The manifold V × R embeds in DNC(M,V ) : V� � //
��
V
��V �� // M
The gauge action is free and proper on the open subsetDNC(M,V ) \ V × R of DNC(M,V ). We let :
Blup(M,V ) =(DNC(M,V ) \ V × R
)/R∗ = M \ V ∪ P(NM
V )
and
SBlup(M,V ) =(DNC+(M,V ) \ V × R+
)/R∗+ = M \ V ∪ S(NM
V ) .
![Page 43: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/43.jpg)
Motivations The DNC and Blup constructions Exact sequences
The Blowup construction
The scaling action of R∗ on M × R∗ extends to the gauge action onDNC(M,V ) = M × R∗ ∪NM
V × {0} :
DNC(M,V )× R∗ −→ DNC(M,V )(z, t, λ) 7→ (z, λt) for t 6= 0
(x,X, 0, λ) 7→ (x, 1λX, 0) for t = 0
The manifold V × R embeds in DNC(M,V ) : V� � //
��
V
��V �� // M
The gauge action is free and proper on the open subsetDNC(M,V ) \ V × R of DNC(M,V ). We let :
Blup(M,V ) =(DNC(M,V ) \ V × R
)/R∗ = M \ V ∪ P(NM
V ) and
SBlup(M,V ) =(DNC+(M,V ) \ V × R+
)/R∗+ = M \ V ∪ S(NM
V ) .
![Page 44: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/44.jpg)
Motivations The DNC and Blup constructions Exact sequences
Functoriality of Blup
V �� //
fV��
M
fM��
V ′� � // M ′
gives DNC(f) : DNC(M,V )→ DNC(M ′, V ′)
which is equivariant under the gauge action
: it passes to the quotientBlup as soon as it is defined.
Let Uf (M,V ) = DNC(M,V ) \DNC(f)−1(V ′ × R) and define
Blupf (M,V ) = Uf/R∗ ⊂ Blup(M,V )
Then DNC(f) passes to the quotient :
Blup(f) : Blupf (M,V )→ Blup(M ′, V ′)
Analogous constructions hold for SBlup.
![Page 45: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/45.jpg)
Motivations The DNC and Blup constructions Exact sequences
Functoriality of Blup
V �� //
fV��
M
fM��
V ′� � // M ′
gives DNC(f) : DNC(M,V )→ DNC(M ′, V ′)
which is equivariant under the gauge action : it passes to the quotientBlup as soon as it is defined.
Let Uf (M,V ) = DNC(M,V ) \DNC(f)−1(V ′ × R) and define
Blupf (M,V ) = Uf/R∗ ⊂ Blup(M,V )
Then DNC(f) passes to the quotient :
Blup(f) : Blupf (M,V )→ Blup(M ′, V ′)
Analogous constructions hold for SBlup.
![Page 46: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/46.jpg)
Motivations The DNC and Blup constructions Exact sequences
Functoriality of Blup
V �� //
fV��
M
fM��
V ′� � // M ′
gives DNC(f) : DNC(M,V )→ DNC(M ′, V ′)
which is equivariant under the gauge action : it passes to the quotientBlup as soon as it is defined.
Let Uf (M,V ) = DNC(M,V ) \DNC(f)−1(V ′ × R) and define
Blupf (M,V ) = Uf/R∗ ⊂ Blup(M,V )
Then DNC(f) passes to the quotient :
Blup(f) : Blupf (M,V )→ Blup(M ′, V ′)
Analogous constructions hold for SBlup.
![Page 47: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/47.jpg)
Motivations The DNC and Blup constructions Exact sequences
Functoriality of Blup
V �� //
fV��
M
fM��
V ′� � // M ′
gives DNC(f) : DNC(M,V )→ DNC(M ′, V ′)
which is equivariant under the gauge action : it passes to the quotientBlup as soon as it is defined.
Let Uf (M,V ) = DNC(M,V ) \DNC(f)−1(V ′ × R) and define
Blupf (M,V ) = Uf/R∗ ⊂ Blup(M,V )
Then DNC(f) passes to the quotient :
Blup(f) : Blupf (M,V )→ Blup(M ′, V ′)
Analogous constructions hold for SBlup.
![Page 48: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/48.jpg)
Motivations The DNC and Blup constructions Exact sequences
Blowup groupoidLet Γ be a closed Lie subgroupoid of a Lie groupoid G
t,s
⇒ G(0). Define
˜DNC(G,Γ) = Ut(G,Γ) ∩ Us(G,Γ)
elements whose image by DNC(s) and DNC(t) are not in Γ(0) × R.
Functoriality implies :
Blupt,s(G,Γ) = ˜DNC(G,Γ)/R∗ ⇒ Blup(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are Blup(s)and Blup(t) and its product is Blup(m).
Analogous constructions hold for SBlup.
Remark
Let NGΓ be the restriction of NG
Γ ⇒ NG(0)
Γ(0) to NG(0)
Γ(0) \ Γ(0) and G the
restriction of G to G(0) \ Γ(0).
NGΓ /R∗ inherits a structure of Lie groupoid : PNG
Γ ⇒ PNG(0)
Γ(0) .
Blupr,s(G,Γ) = G ∪ PNGΓ ⇒ G(0) \ Γ(0) ∪ PNG(0)
Γ(0) .
![Page 49: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/49.jpg)
Motivations The DNC and Blup constructions Exact sequences
Blowup groupoidLet Γ be a closed Lie subgroupoid of a Lie groupoid G
t,s
⇒ G(0). Define
˜DNC(G,Γ) = Ut(G,Γ) ∩ Us(G,Γ)
elements whose image by DNC(s) and DNC(t) are not in Γ(0) × R.Functoriality implies :
Blupt,s(G,Γ) = ˜DNC(G,Γ)/R∗ ⇒ Blup(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are Blup(s)and Blup(t) and its product is Blup(m).
Analogous constructions hold for SBlup.
Remark
Let NGΓ be the restriction of NG
Γ ⇒ NG(0)
Γ(0) to NG(0)
Γ(0) \ Γ(0) and G the
restriction of G to G(0) \ Γ(0).
NGΓ /R∗ inherits a structure of Lie groupoid : PNG
Γ ⇒ PNG(0)
Γ(0) .
Blupr,s(G,Γ) = G ∪ PNGΓ ⇒ G(0) \ Γ(0) ∪ PNG(0)
Γ(0) .
![Page 50: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/50.jpg)
Motivations The DNC and Blup constructions Exact sequences
Blowup groupoidLet Γ be a closed Lie subgroupoid of a Lie groupoid G
t,s
⇒ G(0). Define
˜DNC(G,Γ) = Ut(G,Γ) ∩ Us(G,Γ)
elements whose image by DNC(s) and DNC(t) are not in Γ(0) × R.Functoriality implies :
Blupt,s(G,Γ) = ˜DNC(G,Γ)/R∗ ⇒ Blup(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are Blup(s)and Blup(t) and its product is Blup(m).
Analogous constructions hold for SBlup.
Remark
Let NGΓ be the restriction of NG
Γ ⇒ NG(0)
Γ(0) to NG(0)
Γ(0) \ Γ(0) and G the
restriction of G to G(0) \ Γ(0).
NGΓ /R∗ inherits a structure of Lie groupoid : PNG
Γ ⇒ PNG(0)
Γ(0) .
Blupr,s(G,Γ) = G ∪ PNGΓ ⇒ G(0) \ Γ(0) ∪ PNG(0)
Γ(0) .
![Page 51: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/51.jpg)
Motivations The DNC and Blup constructions Exact sequences
Blowup groupoidLet Γ be a closed Lie subgroupoid of a Lie groupoid G
t,s
⇒ G(0). Define
˜DNC(G,Γ) = Ut(G,Γ) ∩ Us(G,Γ)
elements whose image by DNC(s) and DNC(t) are not in Γ(0) × R.Functoriality implies :
Blupt,s(G,Γ) = ˜DNC(G,Γ)/R∗ ⇒ Blup(G(0),Γ(0))
is naturally a Lie groupoid; its source and range maps are Blup(s)and Blup(t) and its product is Blup(m).
Analogous constructions hold for SBlup.
Remark
Let NGΓ be the restriction of NG
Γ ⇒ NG(0)
Γ(0) to NG(0)
Γ(0) \ Γ(0) and G the
restriction of G to G(0) \ Γ(0).
NGΓ /R∗ inherits a structure of Lie groupoid : PNG
Γ ⇒ PNG(0)
Γ(0) .
Blupr,s(G,Γ) = G ∪ PNGΓ ⇒ G(0) \ Γ(0) ∪ PNG(0)
Γ(0) .
![Page 52: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/52.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .
Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 53: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/53.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 54: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/54.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]
2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 55: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/55.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 56: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/56.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 57: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/57.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners.
Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 58: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/58.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V .
Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 59: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/59.jpg)
Motivations The DNC and Blup constructions Exact sequences
Examples of blowup groupoids
1. Take G⇒ G(0) a Lie groupoid ans G = G× R× R ⇒ G(0) × R .Recall that DNC(G,G(0)) = G× R∗ ∪ AG× {0}⇒ G(0) × R.
Blupr,s(G,G(0) × {(0, 0)}) = DNC(G,G(0)) oR∗ ⇒ G(0) × R
Gauge adiabatic groupoid [D.-Skandalis]2. Let V ⊂M be a hypersurface.
Gb = SBlupr,s(M ×M,V × V )︸ ︷︷ ︸The b-calculus groupoid
⊂ SBlup(M ×M,V × V )︸ ︷︷ ︸Melrose’s b-space
G0 = SBlupr,s(M ×M,∆(V ))︸ ︷︷ ︸The 0-calculus groupoid
⊂ SBlup(M ×M,∆(V ))︸ ︷︷ ︸Mazzeo-Melrose’s 0-space
Iterate these constructions to go to the study of manifolds withcorners. Or consider a foliation with no holonomy on V . Define theholonomy groupoid of a manifold with iterated fibred corners.
![Page 60: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/60.jpg)
Motivations The DNC and Blup constructions Exact sequences
Exact sequences coming from deformations and blowups
Let Γ ⇒ V be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒M ,suppose that Γ is amenable and let M = M \ V . Let NG
Γ be therestriction of the groupoid NG
Γ ⇒ NMV to NM
V \ V .
![Page 61: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/61.jpg)
Motivations The DNC and Blup constructions Exact sequences
Exact sequences coming from deformations and blowups
Let Γ ⇒ V be a closed Lie subgroupoid of a Lie groupoid Gt,s
⇒M ,suppose that Γ is amenable and let M = M \ V . Let NG
Γ be therestriction of the groupoid NG
Γ ⇒ NMV to NM
V \ V .
DNC+(G,Γ) = G× R∗+ ∪NGΓ × {0}⇒M × R∗+ ∪NM
V
˜DNC+(G,Γ) = GMM× R∗+ ∪ NG
Γ × {0}⇒ M × R∗+ ∪ NMV
SBlupt,s(G,Γ) = ˜DNC+(G,Γ)/R∗+ = GMM∪ SNG
Γ ⇒ M ∪ S(NMV )
![Page 62: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/62.jpg)
Motivations The DNC and Blup constructions Exact sequences
Exact sequences coming from deformations and blowups
DNC+(G,Γ) = G× R∗+ ∪NGΓ × {0}⇒M × R∗+ ∪NM
V
˜DNC+(G,Γ) = GMM× R∗+ ∪ NG
Γ × {0}⇒ M × R∗+ ∪ NMV
SBlupt,s(G,Γ) = ˜DNC+(G,Γ)/R∗+ = GMM∪ SNG
Γ ⇒ M ∪ S(NMV )
0 // C∗(G× R∗+) // C∗(DNC+(G,Γ)) // C∗(NGΓ ) // 0
0 // C∗(GMM× R∗+) // C∗( ˜DNC+(G,Γ)) // C∗(NGΓ ) // 0
0 // C∗(GMM
) // C∗(SBlupt,s(G,Γ)) // C∗(SNGΓ ) // 0
![Page 63: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/63.jpg)
Motivations The DNC and Blup constructions Exact sequences
Connecting elements
0 // C∗(G× R∗+) // C∗(DNC+(G,Γ)) // C∗(NGΓ ) // 0 ∂DNC+
0 // C∗(GMM× R∗+) // C∗( ˜DNC+(G,Γ)) // C∗(NGΓ ) // 0 ∂
DNC+
0 // C∗(GMM
) // C∗(SBlupt,s(G,Γ)) // C∗(SNGΓ ) // 0 ∂SBlup
Connecting elements : ∂DNC+∈ KK1(C∗(NG
Γ ), C∗(G× R∗+)),
∂DNC+
∈ KK1(C∗(NGΓ ), C∗(GM
M× R∗+)) and
∂SBlup ∈ KK1(C∗(SNGΓ ), C∗(GM
M)).
![Page 64: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/64.jpg)
Motivations The DNC and Blup constructions Exact sequences
Connecting elements
0 // C∗(G× R∗+) // C∗(DNC+(G,Γ)) // C∗(NGΓ ) // 0 ∂DNC+
0 // C∗(GMM× R∗+) //
β
C∗( ˜DNC+(G,Γ)) //
β
C∗(NGΓ )
β∂
// 0 ∂DNC+
0 // C∗(GMM
) // C∗(SBlupt,s(G,Γ)) // C∗(SNGΓ ) // 0 ∂SBlup
The β’s being KK-equivalences given by Connes-Thom elements.
![Page 65: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/65.jpg)
Motivations The DNC and Blup constructions Exact sequences
Connecting elements
0 // C∗(G× R∗+) // C∗(DNC+(G,Γ)) // C∗(NGΓ ) // 0 ∂DNC+
0 // C∗(GMM× R∗+) //
β
?�
j
OO
C∗( ˜DNC+(G,Γ)) //
β
?�
j
OO
C∗(NGΓ )
β∂
//?�
j∂
OO
0 ∂DNC+
0 // C∗(GMM
) // C∗(SBlupt,s(G,Γ)) // C∗(SNGΓ ) // 0 ∂SBlup
The j’s coming from inclusion.
![Page 66: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/66.jpg)
Motivations The DNC and Blup constructions Exact sequences
Connecting elements
0 // C∗(G× R∗+) // C∗(DNC+(G,Γ)) // C∗(NGΓ ) // 0 ∂DNC+
0 // C∗(GMM× R∗+) //
β
?�
j
OO
C∗( ˜DNC+(G,Γ)) //
β
?�
j
OO
C∗(NGΓ )
β∂
//?�
j∂
OO
0 ∂DNC+
0 // C∗(GMM
) // C∗(SBlupt,s(G,Γ)) // C∗(SNGΓ ) // 0 ∂SBlup
Proposition
∂SBlup ⊗ β ⊗ [j] = β∂ ⊗ [j∂ ]⊗ ∂DNC+∈ KK1(C∗(SNG
Γ ), C∗(G)).
![Page 67: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/67.jpg)
Motivations The DNC and Blup constructions Exact sequences
Index type connecting elements
0 // C∗(G× R∗+) // Ψ∗(DNC+(G,Γ)) // ΣDNC+// 0 IndDNC+
0 // C∗(GMM× R∗+) //
β
?�
j
OO
Ψ∗( ˜DNC+(G,Γ)) //
β
?�
j
OO
ΣDNC+
β∂
//?�
j∂
OO
0 IndDNC+
0 // C∗(GMM
) // Ψ∗(SBlupr,s(G,Γ)) // ΣSBlup // 0 IndSBlup
Proposition
IndSBlup⊗ β⊗ [j] = β∂⊗ [j∂ ]⊗ IndDNC+∈ KK1(C∗(ΣSBlup), C
∗(G)).
![Page 68: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/68.jpg)
Motivations The DNC and Blup constructions Exact sequences
Index type connecting elements
0 // C∗(G× R∗+) // Ψ∗(DNC+(G,Γ)) // ΣDNC+// 0 IndDNC+
0 // C∗(GMM× R∗+) //
β
?�
j
OO
Ψ∗( ˜DNC+(G,Γ)) //
β
?�
j
OO
ΣDNC+
β∂
//?�
j∂
OO
0 IndDNC+
0 // C∗(GMM
) // Ψ∗(SBlupr,s(G,Γ)) // ΣSBlup // 0 IndSBlup
Proposition
IndSBlup⊗ β⊗ [j] = β∂⊗ [j∂ ]⊗ IndDNC+∈ KK1(C∗(ΣSBlup), C
∗(G)).
![Page 69: Blowup and deformation groupoids in relation with index theory · Motivations The DNC and Blup constructions Exact sequences Blowup and deformation groupoids in relation with index](https://reader035.fdocuments.us/reader035/viewer/2022071215/6046914d8036832f9a5129b5/html5/thumbnails/69.jpg)
Motivations The DNC and Blup constructions Exact sequences
Thank you for your attention !