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 Introduction: primary and secondary invariants 1 Stratified spaces and metrics 2 Groupoids 3 K-Theory invariants in the singular case 4

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 / D g be the associated Dirac operator We can define the fundamental class [ / D g ] ∈ K ∗ ( X ). D Γ We can define the index class Ind( / g ) ∈ K ∗ ( C ∗ ( X Γ ) Γ ). D Γ g of PSC implies that Ind( / g ) = 0. If g is of PSC we can define a rho class D Γ g ) ∈ K ∗ +1 ( D ∗ ( X Γ ) Γ ); ρ ( / 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 ∗ ( X Γ ) Γ ) � K ∗ ( D ∗ ( X Γ ) Γ / C ∗ ( X Γ ) Γ ) ∂ � K ∗ +1 ( C ∗ ( X Γ ) Γ ) · · · which is associated to the short exact sequence of C ∗ -algebras � C ∗ ( X Γ ) Γ � D ∗ ( X Γ ) Γ � D ∗ ( X Γ ) Γ / C ∗ ( X Γ ) Γ � 0 0 Important facts: (i) K ∗ ( D ∗ ( X Γ ) Γ / C ∗ ( X Γ ) Γ ) = K Γ ∗ +1 ( X Γ ) = K ∗ +1 ( X ) (ii) K ∗ ( C ∗ ( X Γ ) Γ ) = K ∗ ( C ∗ r Γ) D Γ D Γ g ) ∈ K ∗ ( D ∗ ( X Γ ) Γ ) is a lift (iii) Ind( / g ) = ∂ [ / D g ], so ρ ( / of [ / D g ].

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 π 0 ( R + ( X )), to K ∗ ( D ∗ ( X Γ ) Γ ) PSC metrics, � Alternative approach using localization algebra of Yu. Interesting applications by Xie-Yu to the cardinality of π 0 ( 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 D Γ D Γ g ) ∈ K ∗ ( C ∗ ( X Γ ) Γ ) , ρ ( / g ) ∈ K ∗ +1 ( D ∗ ( X Γ ) Γ ) [ / D g ] ∈ K ∗ ( X ) , Ind( / Our object of interest is the following Lie groupoid with units X : G ( X ) = X Γ × Γ X Γ ⇒ X Alain Connes introduced the adiabatic deformation of G ( X ) : G ( X ) ad := TX × { 0 } ⊔ X Γ × Γ X Γ × (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 ev 0 � C ∗ ( TX ) � C ∗ ( X Γ × Γ X Γ × (0 , 1)) � C ∗ ( G ( X ) 0 � 0 0 ad ) 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 ( / D g ) ∈ K ∗ ( C ∗ ( G ( X ) 0 ad ). Very important: this works for any groupoid G ⇒ M with algebroid A , with G ad := 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 S X 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 S X , 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) S X is a Thom-Mather staratified space of depth 1 . We can associate to S X its resolution : let X be the manifold S X \ 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: g ie = dx 2 + π ∗ g S + x 2 g Z where g S is a metric on S and g Z is a vertical metric on the fibers of π : H → S complete edge: g e = dx 2 x 2 + π ∗ g S + g Z x 2 fibered cusp: g fc = dx 2 x 2 + π ∗ g S + x 2 g Z fibered boundary: g fb = dx 2 x 4 + π ∗ g S + g Z x 2 Remark 1: g ie = x 2 g e and g fc = x 2 g fb Remark 2: in this talk we are mainly interested in the complete metric g fb .

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 V fb ( X ) given by { ξ ∈ Γ( TX ) | ξ | H is tangent to the fibers of π and ξ x ∈ x 2 · 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 fb TX → X and a natural map ι fb : fb TX → TX such that V fb ( X ) = ι fb Γ( fb TX ) .

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 , y 1 , . . . , y k , z 1 , . . . , z h ) where the functions y i are coordinates on S and z j are coordinates on the fibers. The Lie algebra V fb ( X ) is locally spanned by x 2 ∂ ∂ x , x ∂ , . . . , x ∂ , ∂ , . . . , ∂ . ∂ y 1 ∂ y k ∂ z 1 ∂ z k These are the vector fields dual to the fibered boundary metric g fb = dx 2 x 4 + π ∗ g S + g Z x 2 Thus a fibered boundary metric g fb extends to a smooth metric on fb TX .

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 V fb ( X ) we can also define Diff ∗ fb ( X ; E , F ) ellipticity is defined in terms of the ”new” cotangent bundle fb T ∗ X there is a pseudodifferential calculus Ψ ∗ fb ( X ) (Mazzeo-Melrose+ Debord-Lescure-Rochon) Make a spin assumption on X reg and consider / D fb , the Dirac operator associated to g fb D fb ∈ Diff 1 then we’ve that / 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 fb TX → X is a Lie algebroid with anchor map ι fb : fb TX → 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 ι π : π TX → TX ι fb : fb TX → TX instead of The anchor map ι π : π TX → 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 π TX integrates to a Lie groupoid G π ⇒ X . One can prove that G π is given by X reg × X reg X reg over and H × S TS × S H × R over H . In the equivariant case: G Γ π := X reg Γ X reg × ∪ H × S TS × S H × R Γ Γ

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 V F ( X ) and ξ x ∈ x 2 C ∞ ( X ) } { ξ ∈ V b ( X ) ξ | ∂ X ∈ Γ( ∂ X , T F ) this is a Lie algebra. we obtain the F -tangent bundle, F TX → X ; this is a Lie algebroid on X . a F -metric g F is a metric on ˚ X that extends as a smooth metric on F TX → X .