Singular structures, groupoids and metrics of positive scalar - - PowerPoint PPT Presentation

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Singular structures, groupoids and metrics of positive scalar curvature.

Paolo Piazza. Sapienza Universit` a di Roma (based on joint work with Vito Felice Zenobi)

• Copenaghen. June 11th 2018.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Outline

1

Introduction: primary and secondary invariants

2

Stratified spaces and metrics

3

Groupoids

4

K-Theory invariants in the singular case

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

K-Theory invariants in the closed case

Let (X, g) be a spin compact manifold without boundary and let / Dg be the associated Dirac operator We can define the fundamental class [ / Dg] ∈ K∗(X). We can define the index class Ind( / DΓ

g) ∈ K∗(C ∗(XΓ)Γ).

g of PSC implies that Ind( / DΓ

g) = 0.

If g is of PSC we can define a rho class ρ( / DΓ

g) ∈ K∗+1(D∗(XΓ)Γ);

this is a secondary invariant and can distinguish metrics of PSC.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

The Higson-Roe surgery sequence

All this is encoded in the Higson-Roe analytic surgery sequence · · ·

K∗(D∗(XΓ)Γ) K∗(D∗(XΓ)Γ/C ∗(XΓ)Γ) ∂ K∗+1(C ∗(XΓ)Γ)

• which is associated to the short exact sequence of C ∗-algebras

C ∗(XΓ)Γ D∗(XΓ)Γ D∗(XΓ)Γ/C ∗(XΓ)Γ

Important facts: (i) K∗(D∗(XΓ)Γ/C ∗(XΓ)Γ) = K Γ

∗+1(XΓ) = K∗+1(X)

(ii) K∗(C ∗(XΓ)Γ) = K∗(C ∗

r Γ)

(iii) Ind( / DΓ

g) = ∂[ /

Dg], so ρ( / DΓ

g) ∈ K∗(D∗(XΓ)Γ) is a lift

• f [ /

Dg].

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Some developments.

P-Schick (2014): using these invariants and an APS index class one can map the Stolz sequence to the Higson-Roe sequence. Crucial in this work is the delocalized APS index theorem, relating the APS index class of a m.w.b. with the rho class of its boundary. In particular: ρ defines a map from concordance classes of PSC metrics, π0(R+(X)), to K∗(D∗(XΓ)Γ) Alternative approach using localization algebra of Yu. Interesting applications by Xie-Yu to the cardinality of

• π0(R+(X)).

geometric approach ` a la Baum-Douglas by Deeley and Goffeng.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

The approach by groupoids

Zenobi in his Ph.D. thesis (Rome 1 + Paris 7) proposed yet a different approach to the 3 classes [ / Dg] ∈ K∗(X), Ind( / DΓ

g) ∈ K∗(C ∗(XΓ)Γ) , ρ( /

DΓ

g) ∈ K∗+1(D∗(XΓ)Γ)

Our object of interest is the following Lie groupoid with units X: G(X) = XΓ ×Γ XΓ ⇒ X Alain Connes introduced the adiabatic deformation of G(X) : G(X)ad := TX × {0} ⊔ XΓ ×Γ XΓ × (0, 1] ⇒ X × [0, 1]; One can equip this set with a topology and a smooth structure. Thus we have obtained a new Lie groupoid, the adiabatic deformation of G(X).

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

We can associate a C ∗-algebra to a Lie groupoid: if we denote the restriction of G(X)ad to X × [0, 1) by G(X)0

ad, then using the

evaluation at 0 we have

C ∗(XΓ ×Γ XΓ × (0, 1)) C ∗(G(X)0

ad) ev0 C ∗(TX)

Zenobi shows that the long exact seq. in K-theory associated to this short exact seq. is isomorphic to the Higson-Roe sequence and that the 3 classes correspond. This will be explained in this workshop by Zenobi! Conclusion: if g is of PSC we can also define ρ( / Dg) ∈ K∗(C ∗(G(X)0

ad).

Very important: this works for any groupoid G ⇒ M with algebroid A, with Gad := A × {0} ⊔ G × (0, 1] ⇒ M × [0, 1] Also very important: there is a general delocalized APS index th.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Stratified spaces

Let SX be a locally compact metrizable space such that it is the union of two smooth manifolds X reg and S, there is an open neighbourhood N of S in SX, with a continuous retraction π: N → S and a continuous map ρ: N → [0, +∞[ such that ρ−1(0) = S. N is a fiber bundle over S with fiber C(Z), the cone over a compact manifold (Z is called the link)

SX is a Thom-Mather staratified space of depth 1. We can

associate to SX its resolution: let X be the manifold

SX \ ρ−1([0, 1)) with boundary H = ρ−1(1). This boundary is the

total space of a fibration π: H → S and we will denote the typical fiber by Z. From now on we denote by x the boundary defining function for H.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Metrics

We can consider in the interior of X, which is X reg, the following metrics: incomplete edge: gie = dx2 + π∗gS + x2gZ where gS is a metric on S and gZ is a vertical metric on the fibers of π : H → S complete edge: ge= dx2

x2 + π∗gS x2

+ gZ fibered cusp: gfc = dx2

x2 + π∗gS + x2gZ

fibered boundary: gfb = dx2

x4 + π∗gS x2

+ gZ Remark 1: gie = x2ge and gfc = x2gfb Remark 2: in this talk we are mainly interested in the complete metric gfb.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Tangent bundles

Let us consider the space of vector fields Vfb(X) given by {ξ ∈ Γ(TX) | ξ|H is tangent to the fibers of π and ξx ∈ x2·C ∞(X)}. It is a finitely generated projective C ∞(X)-module which is closed under Lie bracket. By Serre-Swan there exists a vector bundle fbTX → X and a natural map ιfb : fbTX → TX such that Vfb(X) = ιfbΓ(fbTX).

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

If p is a point in H, then we can consider the following system of coordinates (x, y1, . . . , yk, z1, . . . , zh) where the functions yi are coordinates on S and zj are coordinates

• n the fibers. The Lie algebra Vfb(X) is locally spanned by

x2 ∂ ∂x , x ∂ ∂y1 , . . . , x ∂ ∂yk , ∂ ∂z1 , . . . , ∂ ∂zk . These are the vector fields dual to the fibered boundary metric gfb = dx2 x4 + π∗gS x2 + gZ Thus a fibered boundary metric gfb extends to a smooth metric

• n fbTX .

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Differential Operators

We can define Diff ∗

fb(X) in terms of Vfb(X)

we can also define Diff ∗

fb(X; E, F)

ellipticity is defined in terms of the ”new” cotangent bundle

fbT ∗X

there is a pseudodifferential calculus Ψ∗

fb(X)

(Mazzeo-Melrose+ Debord-Lescure-Rochon) Make a spin assumption on X reg and consider / Dfb, the Dirac operator associated to gfb then we’ve that / Dfb∈ Diff 1

bf(X; /

S)

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

The groupoid associated to a stratified space

The vector bundle fbTX → X is a Lie algebroid with anchor map ιfb : fbTX → TX. Recall that the boundary of X is H and that H fibers over S through π: following Debord-Lescure-Rochon we change notation and use ιπ : πTX → TX instead of ιfb : fbTX → TX The anchor map ιπ : πTX → TX is injective on X reg := X \ H, a dense subset of X.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

A theorem of Claire Debord says that πTX integrates to a Lie groupoid Gπ ⇒ X. One can prove that Gπ is given by X reg × X reg

• ver

X reg and H ×

S TS × S H × R

• ver

H. In the equivariant case: G Γ

π := X reg Γ

×

Γ X reg Γ

∪ H ×

S TS × S H × R

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

More singular structures I

We consider manifolds with a foliated boundary; thus X is a m.w.b. and ∃ a foliation F on ∂X; we consider VF(X) {ξ ∈ Vb(X) ξ|∂X ∈ Γ(∂X, TF) and ξx ∈ x2C ∞(X)} this is a Lie algebra. we obtain the F-tangent bundle, FTX → X; this is a Lie algebroid on X. a F-metric gF is a metric on ˚ X that extends as a smooth metric on FTX → X.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

More singular structures II

we can also consider (X, H) with H transverse to the boundary ∂X; we assume that ∂X is foliated by a second foliation F such that F ⊂ H|∂X. we consider VF(X, H) {ξ ∈ Γ(X, TH)∩Vb(X), ξ|∂X ∈ Γ(∂X, TF) and ξx ∈ x2C ∞(X)} there exists a bundle FTH whose sections are precisely given by VF(X, H). an admissible metric gH

F in this situation is a foliated metric

• n (˚

X, G|˚

X) that extends to a smooth metric on FTH.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

More groupoids

the groupoid of a stratified space, Gπ, can be described by a blow-up technique in the groupoid setting due to Debord and Skandalis in fact, in their notation, Blup+

r,s(Blup+ r,s(X × X, ∂X × ∂X), ∂X ×S ∂X)

the construction is flexible enough to produce groupoids integrating the Lie algebroids FTX and FTH.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Summary

we have 3 singular structures described by 3 Lie algebras of vector fields Vfb(X) , VF(X) , VF(X, H) these produce 3 Lie algebroids and 3 algebras of differential

• perators

smooth metrics on the Lie algebroids are admissible metrics in this singular context geometric operators for these metrics are elements of the respective algebras of differential operators the Lie algebroids can be integrated to 3 groupoids NEXT: we want to use the adiabatic deformation in order to produce our favourite K-theoretic classes (fundamental class, index class, rho class).

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Poincar´ e duality

Recall that if M is a closed manifold, then M is K-dual to its cotangent bundle, that is K∗(M) ∼ = K∗(C0(T ∗M)) Let us consider the adiabatic deformation (Gπ)ad ⇒ X × [0, 1] and let us denote by T NCX its restriction to X reg × {0} ∪ H × [0, 1). Notice that it is open at 1. It is explicitly given by the disjoint union

πTX ∪ (H × S TS × S H × R) × (0, 1)

A Theorem of Debord and Lescure states that C(SX), the algebra

• f continuous functions on the stratified manifold, is K-dual to the

C*-algebra C ∗

r (T NCX).

In other words K∗(SX) ∼ = K∗(C ∗

r (T NCX))

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

How to produce K-theory classes

We consider a pseudodifferential Gπ-operator P; that is, a Gπ-equivariant family of pseudodifferential operators on the s-fibers of Gπ, parametrized by X. This boils down to: if q ∈ X reg, then s−1(q) = X reg and on these fibers we have a pseudodifferential operator P in the pseudodifferential calculus

• f Mazzeo-Melrose;

if q ∈ H, then s−1(q) ∼ = Z × Tπ(q)S × R with Z the fiber of H over π(q). Thus s−1(q) is isomorphic to Z × Rdim S × R. So Pq is a pseudodifferential operator on Z × Rdim S × R and, by equivariance, Pq is translation invariant on the euclidean part and such that Pq = Pr for all q, r ∈ H such that π(q) = π(r). This is precisely the normal family N(P) of Mazzeo-Melrose.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Fully elliptic operators

So Ψ∗

c(Gπ) injects into Ψfb(X) and Diff∗(Gπ) is in bijection with

Diff∗

fb(X).

We want to consider elliptic operators. But simple ellipticity will not suffice. From Mazzeo-Melrose we know that in order to produce a parametrix with compact error we need fully elliptic operators.

Definition

P is fully elliptic if it is elliptic and the normal family N(P) is invertible.

Proposition

If P ∈ Ψ0

c(Gπ) is fully elliptic, then it defines a class [P] ∈ K∗(SX)

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

The spin Dirac operator

Let us consider again the Dirac operator associated to gfb denoted simply by /

• D. This defines in a natural way a Gπ-operator.

Recall that the metric gπ near the boundary is of the form dx2 x4 + π∗gS x2 + gZ Assume that the scalar curvature of gZ is positive and that the singular stratum is spin.

Proposition

If gZ has positive scalar curvature and S is spin, then the normal family of / D is invertible (i.e. / D is fully elliptic). The bounded transform of / D defines a class [ / D] ∈ K∗(SX)

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

The fundamental class in the adiabatic picture

We want to produce from / D a class [σnc( / D)] in K∗(C ∗

r (T NCX))

which corresponds to [ / D] ∈ K∗(SX) in Poincar´ e duality. We assume that the links Z have PSC and that S is spin. Recall that T NCX is equal to

πTX ∪ (H × S TS × S H × R) × (0, 1)

Consider

πTX ∪ (H × S TS × S H × R) × (0, 1]

We want to produce a KK-class for the C ∗-algebra of this groupoid which is degenerate at 1.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Let ψ: R → R be t →

t √ 1+t2 and let ψs(t) := ψ( t 1−s ) be, for

s ∈ [0, 1], the path of functions from ψ to χ: t →

t |t|.

Then the class [σnc( / D)] is constructed in the following way: consider the Dirac operator of the Lie groupoid

πTX ∪ (H × S TS × S H × R) × (0, 1]

(it is equal to N( / D) on H ×

S TS × S H × R × {1});

take the functional calculus of this operator by ψ; concatenate ψs(N( / D)) to ψ(N( / D)) and rescale to have [0, 1] again; this operator is degenerate at 1 and gives the desired class.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

K-theory invariants in the adiabatic picture

Let us denote by (G Γ

π)0 ad the restriction of the adiabatic

deformation of G Γ

π to X × [0, 1).

The C*-algebra of this groupoid fits into the following exact sequence

C ∗

r (X reg Γ

×

Γ X reg Γ

× (0, 1))

i C ∗ r ((G Γ π)0 ad) r C ∗ r (T NCX)

0 .

Let / D be associated to a fiber-boundary metric gfb. Assume PSC along the links and that S is spin. We have defined [σnc( / D)] in K∗(C ∗

r (T NCX)).

It is Poincar´ e dual to [ / D] ∈ K∗(SX) (Debord-Lescure-Rochon). Take the K-theory sequence associated to the above short exact

• sequence. The adiabatic index class of /

D is the image of [σnc( / D)] through the boundary homomorphism δab followed by Bott periodicity. It is an element in K∗(C ∗

r (X reg Γ

×

Γ X reg Γ

)) ∼ = K∗(C ∗

r Γ)

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Theorem

(P.-Zenobi) The adiabatic index class corresponds, through the above isomorphism, to the classic index class defined through a Γ-equivariant parametrix in the Mazzeo-Melrose calculus. Assume that gπ has positive scalar curvature everywhere on X, then the adiabatic index class is zero and we can define a class ρ(gπ) ∈ K∗(C ∗

r ((G Γ π)0 ad))

as a (natural) lift of [σnc( / D)]. This is our rho-class.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Let πfb

0 (X) be the set of concordance classes of fibered boundary

metrics on X: two metrics g0 and g1 are concordant if there exists a fibered boundary metric G with psc on X × [0, 1] such that the restriction of G to X × {i} is equal to gi (for i = 0, 1).

Theorem

The application ρ: πfb

0 (X) → K∗(C ∗ r ((G Γ π)0 ad)) given by

g → ρ(g) is well defined.

Proof.

Since G has psc, the index class associated to G vanishes; the result is then an application of the delocalized APS index Theorem for Lie groupoids, proved by Zenobi.

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Final remarks

there are corresponding results for incomplete metrics gw = dx2 + x2gF + π∗gS (Albin-Gell Redman/Albin-P.) most probably the two sets of invariants are compatible as in Zenobi’s thesis but this needs a proof and could be difficult (complete versus incomplete) advantage of the groupoid method: everything can be generalized to the singular foliations we have considered. future: use these invariants to investigate, for example,

• πfb

0 (X) (project with Botvinnik and Rosenberg).

Introduction: primary and secondary invariants Stratified spaces and metrics Groupoids K-Theory invariants in the singular case

Thank you!