Higher genera for proper actions of Lie groups
aa r X i v : . [ m a t h . K T ] M a y HIGHER GENERA FOR PROPER ACTIONS OF LIE GROUPS
PAOLO PIAZZA AND HESSEL B. POSTHUMA
Abstract.
Let G be a Lie group with finitely many connected components and let K be a maximal compactsubgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has a non-positive sectionalcurvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G -propermanifold with compact quotient M/G . Building on [6] and [27] we establish index formulae for the C ∗ -higherindices of a G -equivariant Dirac-type operator on M . We use these formulae to investigate geometric propertiesof suitably defined higher genera on M . In particular, we establish the G -homotopy invariance of the highersignatures of a G -proper manifold and the vanishing of the b A -genera of a G -spin G -proper manifold admittinga G -invariant metric of positive scalar curvature. Introduction
The aim of this paper is introduce certain geometric invariants associated to proper actions of Lie groups,generalizing the (higher) signatures and b A -genera. Let G be a Lie group satisfying the following assumptions: • G has finitely many components. • Because | π ( G ) | < ∞ , G has a maximal compact subgroup K , unique up to conjugation, and we assumethat the homogeneous space G/K has non-positive sectional curvature with respect the G-invariantmetric induced by a AdK-invariant inner product h , i on the Lie algebra g . • G satisfies the rapid decay (RD) property.We will explain these last two hypothesis in the course of the paper; it suffices for now to remark that naturalexamples of groups satisfying our assumptions are given by connected semisimple Lie groups. The homogeneousspace G/K is a smooth model for EG , the classifying space for proper actions of G , c.f. [2]: for any smoothproper action of G on a manifold M , there exists a smooth G -equivariant classifying map ψ M : M → G/K ,unique up to G -equivariant homotopy. Assuming in addition that the action is cocompact , i.e., that the quotient M/G is compact, we can fix a cut-off function χ M for M : this is a smooth function χ M ∈ C ∞ c ( M ) satisfying Z G χ M ( g − x ) dg = 1 , for all x ∈ M. For any proper action of G on M , we consider Ω • inv ( M ), the complex of G -invariant differential forms on M and its cohomology denoted by H • inv ( M ). In the universal case this cohomology can be identified with the K -relative Lie algebra cohomology of the Lie algebra g of G : H • inv ( G/K ) ∼ = H • CE ( g ; K ) where CE stands forChevalley-Eilenberg. For any α ∈ Ω • inv ( G/K ), consider its pull-back ψ ∗ M α ∈ Ω • inv ( M ). The higher signatureassociated to α is the real number(1.1) σ ( M, α ) := Z M χ M L ( M ) ∧ ψ ∗ M ( α ) , where L ( M ) is the invariant de Rham form representing the L -class of M . The insertion of the cut-off function χ M , which has compact support, ensures that the integral is well-defined, and it can be shown that it onlydepends on the class [ L ( M ) ∧ ψ ∗ M ( α )] ∈ H • inv ( M ). The collection(1.2) { σ ( M, α ) , [ α ] ∈ H • inv ( G/K ) } Mathematics Subject Classification.
Primary: 58J20. Secondary: 58J42, 19K56.
Key words and phrases.
Lie groups, proper actions, group cocycles, Van Est isomorphism, cyclic cohomology, K-theory, indexclasses, higher indices, higher index formulae, higher signatures, G-homotopy invariance, higher b A -genera, positive scalar curvature. are called the higher signatures of M . Similarly, the higher b A genus associated to M and to [ α ] ∈ H • inv ( G/K )is the real number(1.3) b A ( M, α ) := Z M χ M b A ( M ) ∧ ψ ∗ M ( α )with b A ( M ) the de Rham class associated to the b A -differential form for a G -invariant metric. The collection(1.4) { b A ( M, α ) , α ∈ H • inv ( G/K ) } are called the higher b A -genera of M .In this paper we establish the following result: Theorem 1.5.
Let G be a Lie group with finitely many connected components satisfying property RD, and suchthat G/K is of non-positive sectional curvature for a maximal compact subgroup K . Let M be an orientablemanifold with a proper, cocompact action of G . Then the following holds true: ( i ) each higher signatures σ ( M, α ) , α ∈ H • inv ( G/K ) , is a G -homotopy invariant of M . ( ii ) if M admits a G -invariant spin structure and a G -invariant metric of positive scalar curvature, theneach higher b A -genus b A ( M, α ) , α ∈ H • inv ( G/K ) , vanishes. We prove this result by adapting to the G -proper context the seminal paper of Connes and Moscovici on thecyclic cohomological appraoch to the Novikov conjecture for discrete Gromov hyperbolic groups.Crucial to this program is the proof of a higher index formula for higher indices associated to elements in H • diff ( G ) and to the index class Ind C ∗ r ( G ) ( D ) ∈ K ∗ ( C ∗ r ( G )) of a G -equivariant Dirac operator on M , M evendimensional, acting on the sections of a complex vector bundle E . Here are the main steps for establishing thisresult (for this introduction we expunge from the notation the vector bundle E ):(1) first, we remark that for any almost connected Lie group G there is a van Est isomorphism H • diff ( G ) ≃ H • inv ( G/K ) ≡ H • inv ( EG );(2) under the assumption of non-positive sectional curvature for G/K we prove that each α ∈ H • diff ( G ) hasa representative cocyle of polynomial growth;(3) if G is unimodular then for each α ∈ H evendiff ( G ) we define a cyclic cocycle τ Gα for the convolution algebra C ∞ c ( G ) and thus a homomorphism (cid:10) τ Gα , · (cid:11) : K ( C ∞ c ( G )) → C ;(4) for each α ∈ H evendiff ( G ) we also consider a cyclic cocycle τ Mα for the algebra of G -equivariant smoothkernels of G -compact support A cG ( M ); this defines a homomorphism (cid:10) τ Mα , · (cid:11) : K ( A cG ( M )) → C ;(5) we show that if in addition G satisfies the RD property, for example, if G is semisimple connected, then τ Gα extends to K ( C ∗ r ( G )) and τ Mα extends to K ( C ∗ ( M ) G ), with C ∗ ( M ) G denoting the Roe algebra of M ;(6) if D is a G -equivariant Dirac operator we consider its index class Ind C ∗ r ( G ) ( D ) ∈ K ( C ∗ r ( G )) and itsMorita equivalent index class Ind C ∗ ( M ) G ( D ) ∈ K ( C ∗ ( M ) G ) and show that (cid:10) τ Gα , Ind C ∗ r ( G ) ( D ) (cid:11) = (cid:10) τ Mα , Ind C ∗ ( M ) G ( D ) (cid:11) ;(7) we apply the index theorem of Pflaum-Posthuma-Tang [27] in order to compute (cid:10) τ Mα , Ind C ∗ ( M ) G ( D ) (cid:11) ,thus establishing our higher C ∗ -index formula in the even dimensional case.We remark that item (2) above has in independent interest, and should be compared with the literature onbounded cohomology of Lie groups, c.f. [13, 21]The geometric applications stated in Theorem 1.5 are then a direct consequence of the G homotopy invarianceof the signature index class, established by Fukumoto in [10] and, for the higher b A -genera, of the vanishing of theindex class Ind C ∗ r ( G ) ( ð ) ∈ K ∗ ( C ∗ r ( G )) of the spin Dirac operator ð of a G -spin G -proper manifold endowed witha G -metric of positive scalar curvature, established by Guo, Mathai and Wang in [11]. In the odd dimensionalcase we argue by suspension. Notice that for (certain) 2-degree classes α , the G -proper homotopy invariance ofthe higher signatures σ ( M, α ) had been already established by Fukumoto. -PROPER MANIFOLDS 3
Acknowledgements.
Part of this research was carried out during visits of HP to Sapienza Universit`a di Romaand of PP to University of Amsterdam. Financial support for these visits was provided by
Istituto Nazionaledi Alta Matematica (INDAM) , through the
Gruppo Nazionale per le Strutture Algebriche e Geometriche e loroApplicazioni (GNSAGA), by the
Ministero Istruzione Universit`a Ricerca (MIUR) , through the project
Spazi diModuli e Teoria di Lie , and by NWO TOP grant nr. 613.001.302.We thank Andrea Sambusetti, Filippo Cerocchi, Nigel Higson, Varghese Mathai, Xiang Tang and Hang Wangfor many informative and useful discussions.2.
Preliminaries: Proper actions and cohomology
Proper actions.
In this section we introduce the geometric setting for this paper, and list some basic toolsthat we will need at several points later on. Let G be a Lie group with finitely many connected components.Recall that a smooth left action of G on a manifold M is called proper if the associated map G × M → M × M, ( g, x ) ( x, gx ) , g ∈ G, x ∈ M, is a proper map. This implies that the stabilizer groups G x of all points x ∈ M are compact and that thequotient space M/G is Hausdorff. The action is said to be cocompact if the quotient is compact.The class of manifolds equipped with a proper action of G can be assembled into a category where themorphisms are given by G -equivariant smooth maps. It is a basic fact that this category has a final object EG meaning that any proper G -action on M is classified by a G -equivariant map ψ : M → EG , unique up to G -equivariant homotopy. This EG is called the classifying space for proper G -actions , and in fact we can take EG := G/K , where K is a maximal compact subgroup. Then, by writing S := ψ − ( eK ) we see that the S isin fact a global slice: it is a K -stable submanifold for which there is a diffeomorphism G × K S ∼ = M, [ g, x ] gx, g ∈ G, x ∈ S. The existence of such a global slice for proper Lie group actions with finitely many connected components wasfirst proved in [1]. When the action is cocompact, S is compact as well. Closely related to the global slice isthe existence of a cut-off function: this is a smooth function χ ∈ C ∞ ( M ) satisfying Z G χ ( g − x ) dg = 1 , for all x ∈ M. Here we have chosen, for the rest of the paper, a Haar measure which we normalized so that the volume of themaximal compact subgroup K ⊂ G is equal to 1. When the action of G is cocompact, we can even choose χ to have compact support. The cut-off function is constructed as follows from the global slice S ⊂ M : choose asmooth function h ∈ C ∞ ( M ) which is equal to 1 on S and zero outside an open neighborhood of S in M . Thenthe function χ ( x ) = (cid:18)Z G h ( g − x ) dg (cid:19) − h ( x ) , is a cut-off function for the action of G . Choosing a G -invariant riemannian metric g on M we can refine thisconstruction as follows: choose the initial function h to have support inside the tube of distance 1 in M around S . Then, rescaling by ǫ > S , we obtain a family of functions h ǫ satisfying h ǫ ( x ) = ( x ∈ S d ( x, S ) > ǫ. Using this as input for the construction of the cut-off function above gives a family of cut-off functions χ ǫ satisfying: Lemma 2.1.
The family of cut-off functions χ ǫ , ǫ > satisfies lim ǫ ↓ χ ǫ = χ S , distributionally. PAOLO PIAZZA AND HESSEL B. POSTHUMA
Proof.
We begin by remarking that pointwiselim ǫ ↓ χ ǫ ( x ) = ( x ∈ S x S. This is because for fixed x ∈ S the family h ǫ ( g − x ) of functions on G converges pointwise to the characteristicfunction of K ⊂ G and therefore by dominated convergence we havelim ǫ ↓ Z G h ǫ ( g − x ) dg = Z G lim ǫ ↓ h ǫ ( g − x ) dg = Z K dg = 1 , by our normalization of the Haar measure on G . With this pointwise limit of χ ǫ ( x ) we have, once again bydominated convergence thatlim ǫ ↓ Z M χ ǫ ( x ) f ( x ) dx = Z M lim ǫ ↓ χ ǫ ( x ) f ( x ) dx = Z S f ( x ) dx, for any test function f ∈ C ∞ c ( M ). (cid:3) Invariant cohomology and the van Est map.
The main point of this subsection is to define the vanEst map associated to a proper action of a Lie group G on M , and to reinterpret this map as the pull-back incohomology along the classifying map ψ M : M → G/K .Let M be a smooth manifold equipped with a smooth proper action of G . We defineΩ • inv ( M ) := { ω ∈ Ω • ( M ) , g ∗ ω = ω, ∀ g ∈ G } , the vector space of invariant differential forms. The de Rham differential restricts to this space of invariant formsand its cohomology, called the invariant cohomology , is denoted by H • inv ( M ). Taking the invariant cohomologydefines a contravariant functor on the category of proper G -manifolds with an equivariant map f : M → N acting on cohomology by pull-back of differential forms as usual. It is not difficult to see that the induced map f ∗ : H • inv ( N ) → H • inv ( M ) depends only on the G -homotopy class it is in. Given the choice of a cut-off function χ , it is shown in [27] that the integral Z M χα of a closed form α ∈ Ω dim( M )inv , cl ( M ), only depend on the cohomology class [ α ] ∈ H dim( M )inv ( M ).For any manifold M equipped with a proper action of G , the van Est map is a map H • diff ( G ) → H • inv ( M ),where H • diff ( G ) is the so-called smooth group cohomology of G . Let us first recall the definition of this smoothgroup cohomology. For G a Lie group, the space of smooth homogeneous group k -cochains is given by C k diff ( G ) := { c : G × ( k +1) → C smooth , c ( gg , . . . , gg k ) = c ( g , . . . , g k ) , forall g, g , . . . , g k ∈ G } . The differential δ : C k diff ( G ) → C k +1diff ( G ) is defined as(2.2) ( δc )( g , . . . , g k +1 ) := k +1 X i =0 ( − i c ( g , . . . , ˆ g i , . . . , g k +1 ) , where the ˆ means omission from the argument of the function. The cohomology of the resulting complex iscalled the smooth group cohomology, written as H • diff ( G ).With this, the van Est map is constructed as follows: given a smooth group c ∈ C k diff ( G ), define the differentialform(2.3) ω χc := ( d · · · d k f c ) | ∆ where d i means taking the differential in the i’th variable of the function f c ∈ C ∞ ( M × ( k +1) ) defined as(2.4) f c ( x , . . . , x k ) := Z G × ( k +1) χ ( g − x ) · · · χ ( g − k x k ) c ( g , . . . , g k ) dµ ( g ) · · · dµ ( g k ) . -PROPER MANIFOLDS 5 Proposition 2.5.
The map c ω χc defines a morphism of complexes Φ χM : ( C • diff ( G ) , δ ) −→ (Ω • inv ( M ) , d dR ) . On the level of cohomology, it is independent of the choice of cut-off function χ . Remark 2.6.
Because of this last property, we will often omit the superscript χ and write ω c and Φ M whenthe context only refers to the cohomological meaning of the differential form and the van Est map. Proof.
We start by giving the abstract cohomological definition of the map Φ M following [7] using a doublecomplex, after which we show how to obtain the explicit chain morphism by constructing a splitting of the rows.The double complex is given as follows. We define C p,q := C ∞ ( G × ( p +1) , Ω q inv ( M )) . The vertical differential δ v : C p,q → C p,q +1 is simply given by the de Rham differential, leaving the G -variablesuntouched. As for the horizontal differential δ h : C p,q → C p +1 ,q : this is given by differential computing thesmooth groupoid cohomology of the action groupoid G × M ⇒ M with coefficients in V q T ∗ M , viewed as arepresentation of this groupoid. Since the G -action is proper, the groupoid G × M ⇒ M is proper by definition.Therefore, the vanishing theorem for the groupoid cohomology of proper Lie groupoids in [7] applies, andwe see that the rows in this double complex are exact. There are obvious inclusions C • diff ( G ) ֒ → C • , , andΩ • inv ( M ) ֒ → C , • , and now we see that by finding the appropriate splittings we can ”zig-zag” from one end tothe other in the double complex: . . . . . . . . . . . . Ω ( M ) d O O / / C , δ v O O δ h / / C , δ h O O s f f ❦❴❙ δ h / / C , δ h O O δ h / / . . . Ω ( M ) d O O / / C , δ v O O δ h / / C , δ h O O δ h / / C , δ h O O s f f ❦❴❙ δ h / / . . .C ( G ) O O δ / / C ( G ) O O δ / / C ( G ) O O δ / / . . . So it remains to find an appropriate splitting s p : C p, • → C p +1 , • . Given a choice of cut-off function χ , theformula ( s p α )( g , . . . , g p − ) := d x Z G χ ( g − x ) α ( g, g , . . . , g p − ) (cid:12)(cid:12)(cid:12)(cid:12) ∆ , α ∈ C p,q , does the job: a straightforward computation shows that δ h ◦ s + s ◦ δ h = id . With this choice of contraction map, one obtains exactly equation (2.3) for the invariant differential formassociated to a group cochain. The preceeding argument therefore shows that the map c ω c is indeed amorphism of cochain complexes. (cid:3) Remark 2.7 (The van Est isomorphism) . The main theorem of [7] states that if M is cohomologically n -connected, the map Φ M induces an isomorphism in cohomology in degree ≤ n and is injective in degree n + 1.In the universal case for the action of G on G/K , which is contractible, we therefore find an isomorphism H • diff ( G ) ∼ = H • inv ( G/K ). This is one version of the classical van Est theorem [30]. In this case we have, by lefttranslation(2.8) Ω • inv ( G/K ) ∼ = • ^ ( g / k ) ∗ ! K , PAOLO PIAZZA AND HESSEL B. POSTHUMA under which the de Rham differential identifies with the Chevalley–Eilenberg differential computing the relativeLie algebra cohomology H • CE ( g ; K ). With this, the van Est isomorphism is written as(2.9) H • diff ( G ) ∼ = H • CE ( g ; K ) . Proposition 2.10.
Let f : M → N be an equivariant smooth map between proper G -manifolds. Then thefollowing diagram commutes: H • diff ( G ) Φ N / / Φ M ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ H • inv ( N ) f ∗ (cid:15) (cid:15) H • inv ( M ) Proof.
Let χ M be a cutt-off function for the G -action on M . Then the pull-back f ∗ χ M is a cut-off function forthe G -action on N . For this cut-off function we obviously have ω f ∗ χ M c = f ∗ ω χ M c . The result now follows fromthe fact that the van Est map is independent of the choice of cut-off function. (cid:3) Corollary 2.11.
Under the van Est isomorphism H • diff ( G ) ∼ = H • inv ( G/K ) , the van Est map is identified withthe pull-back along the classifying map ψ M : M → G/K , i.e., Φ M = ψ ∗ M . Group cocycles of polynomial growth.
In a later stage of the paper, in the discussion of the extensionproperties of cyclic cocycles associated to smooth group cocycles, it will be important to control the growth ofthe group cocycles. To this end, we shall prove below a criterium guaranteeing that we can represent classes in H • diff ( G ) by cocycles that have at most polynomial growth. For this, we begin by recalling Dupont’s inverse [8]of the van Est map Φ G/K establishing the isomorphism (2.9). Choose an Ad K -invariant inner product h , i on g ,which, by left translations, induces a G -invariant Riemannian metric on G/K . This metric defines an orthogonaldecomposition g = p ⊕ k with p ∼ = T eK ( G/K ). Since K is maximal compact, the (riemannian) exponential mapinduces an isomorphism p ∼ = G/K (with inverse denoted by log), and we can define the contraction ϕ s ( x ) := exp( s log( x )) , of G/K to its basepoint eK ∈ G/K , i.e., ϕ = id G/K , and ϕ ( x ) = eK . Now, given k + 1 points g K, . . . , g k K ∈ G/K , also denoted ¯ g , . . . , ¯ g k , we can consider the geodesic simplex ∆ k ( g K, . . . , g k K ) ⊂ G/K defined in-ductively as the cone of ∆ k − (¯ g , . . . , ¯ g k ) with tip point ¯ g . More precisely, define the singular simplex σ k (¯ g , . . . , ¯ g k ) : ∆ k → G/K , where ∆ k := { ( t , . . . , t k ) ∈ R k +1 , t i ≥ , P i t i = 1 } , by σ k ( g K, . . . , g k K )( t , . . . , t k ) := g ϕ t (cid:18) σ k − ( g − g K, . . . , g − g k K ) (cid:18) t − t , . . . , t k − t (cid:19)(cid:19) , and σ ( gK ) := gK . We write ∆ k ( g K, . . . , g k K ) for the image of this simplex. By construction, this k -simplexis G -invariant: g ∆ k (¯ g , . . . , ¯ g k ) = ∆ k ( g ¯ g , . . . , g ¯ g k ). With these simplices we define a map(2.12) J : Ω • inv ( G/K ) −→ C • diff ( G ) , α J ( α )( g , . . . , g k ) := Z ∆ k ( g K,...,g k K ) α, which is easily checked to be a morphism of cochain complexes. Since Φ G/K ◦ J = id, J is a quasi-isomorphism. Theorem 2.13.
Let G be a Lie group with finitely many connected components. Let K be a maximal compactsubgroup and assume that G/K is of non-positive sectional curvature with respect to the G-invariant metricinduced by a AdK-invariant inner product h , i on g . Then the group cocycle associated to a closed α ∈ Ω k inv ( G/K ) has polynomial growth. More precisely, if we write d ( g ) for the distance from eK to gK in G/K , there existsa constant
C > such that the following estimate holds true: | J ( α )( g , . . . , g k ) | ≤ C (1 + d ( g )) k · · · (1 + d ( g k )) k . Proof.
We denote by || α || the norm of the Lie algebra cocycle α ∈ C k CE ( g ; K ) = Ω k inv ( G/K ) defined by the K -invariant metric on the Lie algebra g of G that defines the metric on G/K . Since α is a G -invariant differentialform we obviously have the inequality | J ( α )( g , . . . , g k ) | ≤ || α || Vol(∆ k (¯ g , . . . , ¯ g k )) . -PROPER MANIFOLDS 7 We will now prove that, under the assumptions of the Lemma, the volume of the geodesic k -simplex on G/K has at most polynomial growth in the geodesic distance of its vertices, thus completing the proof the Lemma.For this we adapt an argument from [16]; we thank Andrea Sambusetti for bringing this article to our attention.First remark that as ϕ s ( gK ) is the geodesic connecting gK to the base point eK , the simplex ∆ k ( g K, . . . , g k K )has, by construction, the property that for any interior point x ∈ ∆ k ( g K, . . . , g k K ) there are k geodesics γ ( s ) , . . . , γ k ( s k ), each connecting two points in the boundary ∂ ∆ k ( g K, . . . , g k K ) passing through x whosevelocities span T x ∆ k ( g K, . . . , g k K ). Consider the one-parameter family of simplices s ∆ k ( eK, ϕ s ( g K ) . . . ϕ s ( g k K )) , s ∈ [0 , k ( eK, g K, . . . , g k K ) to the basepoint eK . This contraction is generated by a vector field Y whichhas the property that it is a Jacobi field with respect to each geodesics γ i ( s i ) mentioned above passing throughthe point x ∈ ∆ k ( eK, ϕ s ( g K ) , . . . ϕ s ( g k K )). The Jacobi equation satisfied by Y therefore gives d ds i || Y ( s i ) || = 2 ||∇ ∂/∂s i Y || − (cid:28) R (cid:18) Y ( s i ) , dγ i ( s i ) ds i (cid:19) dγ i ( s i ) ds i , Y ( s i ) (cid:29) ≥ , since the sectional curvature of G/K is non-positive. The maximum principle gives that || Y ( s i ) || attains itsmaximum at one of the points s i = 0 ,
1. Since this holds true for any i , we conclude that the maximum is attainedon ∂ ∆ k ( eK, . . . , ϕ s ( g k K )), and, proceeding inductively on k , in one of the vertices ϕ s ( g i K ) , i = 1 , . . . , k . Buton these vertices, s ϕ s ( g i K ) is simply the geodesic connecting eK with g i K , which is generated by theEuler vector field P i X i ∂∂X i on p which has polynomial growth of degree 1 in the geodesic distance. Since theexponential map exp : p → G/K is a radial isometry, the same holds true on
G/K , and we can conclude thatthe vector field Y has polynomial growth of degree 1.Remark that the generating vector field Y is tangent to ∆ k ( eK, g K, . . . , g k K ) to all the boundary facesexcept for ∆ k − ( g K, . . . , g k K ) ⊂ ∂ ∆ k ( eK, g K, . . . , g k K )). The standard variational formula for the volumetherefore gives dds (cid:0) Vol(∆ k ( eK, ϕ s ( g K ) , . . . , ϕ s ( g k K ))) (cid:1) = Z ∆ k ( eK,ϕ s ( g K ) ,...,ϕ s ( g k K )) div( Y ) d vol ∆ k = Z ∂ ∆ k ( eK,ϕ s ( g K ) ,...,ϕ s ( g k K )) ( Y · n ) d vol ∂ ∆ k = Z ∆ k − ( ϕ s ( g K ) ,...,ϕ s ( g k K )) ( Y · n ) d vol ∆ k − ≤ Y i (1 + d ( g i ))Vol(∆ k − ( ϕ s ( g K ) , . . . , ϕ s ( g k K )) , where n denotes the vector field normal to the boundary, and we have used the fact that Y · n is zero on allfaces except ∆ k − ( g K, . . . , g k K ) ⊂ ∂ ∆ k ( eK, g K, . . . , g k K )). We now use induction: for k = 1, the geodesicsimplex ∆ ( g K, g K ) is simply the geodesic line segment connecting g K and g K , so the estimate in theLemma obviously holds true. Assume now that the estimate holds true for all degrees up to k −
1. Then, bythe mean value theorem:Vol(∆ k ( eK, g K, . . . , g k K )) = dds (cid:0) Vol( ϕ s (∆ k ( eK, g K, . . . , g k K )) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) s = s , for some s ∈ [0 , , ≤ C k Y i =1 (1 + d ( g i )) . The estimate for the general simplex ∆ k ( g K, . . . , g k K ) follows by translation over g (which is an isometry)together with the triangle inequality. (cid:3) Example 2.14.
As an example, consider the abelian group G = R with maximal compact group given by thetrivial group { } ⊂ R . In this abelian case we have that H • inv ( R ) = V • R , and a generator in degree 2 is PAOLO PIAZZA AND HESSEL B. POSTHUMA given by the area form dx ∧ dy , so that we find(2.15) J ( dx ∧ dy )( x, y, z ) = Area R (∆ ( x, y, z )) , which evidently grows polynomially in the norm of x, y and z . Remark 2.16. i ) When G is a connected semisimple Lie group, G/K is a non-compact symmetric space and has non-positive sectional curvature [14]. Therefore the curvature assumptions in the Lemma are automaticallysatisfied in this case. In fact, the conjecture in [9] is that for semisimple Lie groups all these cocyclesare bounded. For recent work on this conjecture, see [13, 21]. In this last reference, different simplicesare used, given by the barycentric subdivision of the geodesic ones, to prove boundedness of the topdimensional cocycle for general connected semisimple Lie groups. ii ) In general, the polynomial bounds of the Lemma above are not sharp, as expected from the conjecturementioned in i ). For example, when G = SL (2 , R ), the maximal compact subgroup is given by K = SO (2) so that G/K = H , the hyperbolic 2-plane. Again, we have H ( H ) = R , with generator thehyperbolic area form. This leads to a smooth group cocycle given by the same formula as (2.15) above,replacing the Euclidean area by the hyperbolic one, but this time the cocycle is bounded because thearea of a hyperbolic triangle does not exceed π , confirming the boundedness in top-degree mentionedin i ). 3. Algebras of invariant kernels
Smoothing kernels of G -compact support. Let M as above, a closed smooth manifold carrying asmooth proper action of a Lie group G with | π ( G ) | < ∞ and with compact quotient. We choose an invariantcomplete Riemannian metric, denoted h , with associated distance function denoted by d M ( x, y ) for x, y ∈ M ,and volume form d vol( x ). We fix a left-invariant metric on G and we denote by d G the associated distancefunction. Definition 3.1.
Consider a G -equivariant smoothing kernel k ∈ C ∞ ( M × M ) ; thus k is an element in C ∞ ( M × M ) G × G . We say that k is of G -compact support if the projection of supp( k ) ⊂ M × M in ( M × M ) /G , with G acting diagonally, is compact. We denote by A cG ( M ) the set of G -equivariant smoothing kernels of G -compact support. It is well known that A cG ( M ) has the structure of a Fr´echet algebra with respect to the convolution product( k ∗ k ′ )( x, z ) = Z M k ( x, y ) k ′ ( y, z ) d vol( y )It is also well known that each element k ∈ A cG ( M ) defines an equivariant linear operator S k : C ∞ c ( M ) → C ∞ c ( M ), the integral operator associated to the kernel k , and that S k ◦ S k ′ = S k ∗ k ′ . Moreover, S k extends to anequivariant bounded operator on L ( M ). We have therefore defined a subalgebra of B ( L ( M )) that we denoteas S cG ( M ); by definition(3.1) S cG ( M ) := { S k , k ∈ A cG ( M ) } . The case in which there is an equivariant vector bundle E on M is similar, in that we start with G -equivariantelements in C ∞ ( M × M, E ⊠ E ∗ ) and then proceed analogously, defining in this way the Fr´echet algebra A cG ( M, E )and S cG ( M, E ) := { S k , k ∈ A cG ( M, E ) } , a subalgebra of B ( L ( M, E )).
Notation.
Keeping with a well establised abuse of notation, we shall often identify A cG ( M, E ) with S cG ( M, E )thus identifying a smoothing kernel k in A cG ( M, E ) with the corresponding operator S k ∈ S cG ( M, E ).3.2.
Holomorphically closed subalgebras.
Using the remarks at the end of the previous subsection we seethat that S cG ( M, E ) is in an obvious way a subalgebra of the reduced Roe C ∗ -algebra C ∗ ( M, E ) G . Recallthat C ∗ ( M, E ) G is defined as a the norm closure in B ( L ( M, E )) of the algebra C ∗ c ( M, E ) G of G -equivariantbounded operators of finite propagation and locally compact. In fact, S cG ( M, E ) ⊂ C ∗ c ( M, E ) G . The Roe algebrais canonically isomorphic to K ( E ), the C ∗ -algebra of compact operators of the Hilbert C ∗ r ( G )-Hilbert module -PROPER MANIFOLDS 9 E obtained by closing the space of compactly supported sections of E on M , C ∞ c ( M, E ), endowed with the C ∗ r G -valued inner product(3.2) ( e, e ′ ) C ∗ r G ( x ) := ( e, x · e ′ ) L ( M,E ) , e, e ′ ∈ C ∞ c ( M, E ) , x ∈ G .
See for example [15] where the Morita isomorphism K ∗ ( K ( E )) = K ∗ ( C ∗ ( M, E ) G ) M −−→ K ∗ ( C ∗ r G )is explicitly discussed. We shall come back to this important point in a moment.The subalgebra S cG ( M, E ) is not holomorphically closed in C ∗ ( M, E ) G . On the other hand, such a subalgebraof C ∗ ( M, E ) G is implicitly constructed in [15, Section 3.1] by making use of the slice theorem. We recall theessential ingredients, following [15, Section 3.1] (we also extend the context slightly for future use).As already remarked in the previous section, under our assumptions on G , there exists a global slice forthe action of G on M : thus if K is a maximal compact subgroup of G there exists a K-invariant compactsubmanifold S ⊂ M such that the action map [ g, s ] → gs , g ∈ G , s ∈ S , defines a G -equivariant diffeomorphism G × K S α −→ M where S is compact because the action is cocompact. Corresponding to this diffeomorphism we have an isomor-phism E ∼ = G × K ( E | S ) and thus isomorphisms C ∞ c ( M, E ) ∼ = ( C ∞ c ( G ) ˆ ⊗ C ∞ ( S, E | S )) K , C ∞ ( M, E ) ∼ = ( C ∞ ( G ) ˆ ⊗ C ∞ ( S, E | S )) K . See [15, Section 3.1]. Here we are taking the projective tensor product ˆ ⊗ π of the two Fr´echet algebras; however,since C ∞ ( S, E | S ) is nuclear, the injective ˆ ⊗ ǫ and projective ˆ ⊗ π tensor products coincide, which is why we donot use a subscript. Consider now Ψ −∞ ( S, E | S ), also a nuclear Fr´echet algebra, and let e A cG ( M, E ) := ( C ∞ c ( G ) ˆ ⊗ Ψ −∞ ( S, E | S )) K × K e A cG ( M, E ) is a Fr´echet algebra, with product denoted by ∗ . Let e k ∈ e A cG ( M, E ) and consider the operator T e k on L ( M, E ) given by(3.3) ( T e k e )( gs ) = Z G Z S g e k ( g − g ′ , s, s ′ ) g ′ − e ( g ′ s ′ ) ds ′ dg ′ This is a bounded G -equivariant operator with smooth G -equivariant Schwartz kernel given by κ ( gs, g ′ s ′ ) = g e k ( g − g ′ , s, s ′ ) g ′ − where the g and g ′ , − on the right hand side are used in order to identify fibers on the vector bundle E . Theassignment e k → T e k is injective and satisfies T e k ◦ T e k ′ = T e k ∗ e k ′ . Consider the subalgebra of the bounded operators on L ( M, E ) given by { T e k , e k ∈ e A cG ( M, E ) } endowed with the Fr´echet algebra structure induced by the injective homomorphism ˜ k → T ˜ k . It is easy to seethat this algebra is precisely equal to the algebra we have considered in the previous subsection, S cG ( M, E ) := { S k , k ∈ A cG ( M, E ) } . Thus, in formulae,(3.4) S cG ( M, E ) = { T e k , e k ∈ e A cG ( M, E ) } . Summarizing: using the slice theorem we have realized S cG ( M, E ) as a projective tensor product of convolutionoperators on G and smoothing operators on S . This preliminary result puts us in the position of enlarging thealgebra S cG ( M, E ) and obtain a subalgebra dense and holomorphically closed in C ∗ ( M, E ) G .To this end we give the following definition. Definition 3.2.
Let A ( G ) a set of functions on G . We shall say that A ( G ) is admissible if the followingproperties are satisfied:(1) A ( G ) is a Fr´echet space and there are continuous inclusions C ∞ c ( G ) ⊂ A ( G ) ⊂ L ( G ) ; (2) the action by convolution defines a continuous injective map A ( G ) ֒ → C ∗ r ( G ) which makes A ( G ) asubalgebra of C ∗ r ( G ) ;(3) A ( G ) is holomorphically closed in C ∗ r ( G )We can then consider e A G ( M, E ) := ( A ( G ) ˆ ⊗ Ψ −∞ ( S, E | S )) K × K a Fr´echet algebra and for e k ∈ e A G ( M, E ) the bounded operator T e k on L ( M, E ) given by(3.5) ( T e k e )( gs ) = Z G Z S g e k ( g − g ′ , s, s ′ ) g ′ − e ( g ′ s ′ ) ds ′ dg ′ The operator T e k is an integral operator with G -equivariant Schwartz kernel κ given by κ ( gs, g ′ s ′ ) = g e k ( g − g ′ , s, s ′ ) g ′ − . Since A ( G ) ֒ → C ∗ r ( G ), with A ( G ) acting by convultion, we see that T e k is L -bounded. Definition 3.3.
We define A G ( M, E ) as the subalgebra of the bounded operators on L ( M, E ) given by A G ( M, E ) := { T e k , e k ∈ e A G ( M, E ) } . We endow A G ( M, E ) with the structure of Fr´echet algebra induced by the injective homomorphism ˜ k → T ˜ k . Proposition 3.6.
Under the assumptions (1)–(3) for A ( G ) appearing in Definition 3.2 the following holds: (i) We have a continuous inclusion of Fr´echet algebras (3.7) S cG ( M, E ) ⊂ A G ( M, E )(ii) A G ( M, E ) is a dense subalgebra of C ∗ ( M, E ) G and it is holomorphically closed.Proof. (i) The continuous inclusion of Fr´echet algebras S cG ( M, E ) ⊂ A G ( M, E ) follows immediately from (3.4).(ii) The fact that A G ( M, E ) is a dense subalgebra of C ∗ ( M, E ) G is proved precisely as in [15, Lemma 3.6];the property of being holomorphically closed follows easily from the hypothesis that A ( G ) is holomorphicallyclosed in C ∗ r G and the well known fact that Ψ −∞ ( S, E | S ) is holomorphically closed in the compact operatorsof L ( S, E | S ). (cid:3) Definition 3.4.
Let G be a Lie group and let L be a length function on G . We consider (3.8) H ∞ L ( G ) = { f ∈ L ( G ) : Z G (1 + L ( x )) k | f ( x ) | dx < + ∞ ∀ k ∈ N } endowed with the Fr´echet topology induced by the sequence of seminorms (3.9) ν k ( f ) := || (1 + L ) k f || L . We shall say that the pair ( G, L ) satisfies the Rapid Decay property (RD) if there is a continuous inclusion H ∞ L ( G ) ֒ → C ∗ r ( G ) . For conditions equivalent to the one given here, see [4]. We also recall that if G satisfies (RD) then G isunimodular. See [17]. Proposition 3.10.
Let G be a Lie group with | π ( G ) | < ∞ ; we can and we shall choose L to be the lengthfunction associated to a left-invariant Riemannian metric. Assume additionally that G satisfies (RD) (withrespect to this L ). Then (3.11) H ∞ L ( G ) = { f ∈ L ( G ) : Z G (1 + L ( x )) k | f ( x ) | dx < + ∞} satisfies the properties (1) (2) (3) given in Definition 3.2. Consequently, for G with | π ( G ) | < ∞ and with the(RD) property, there exists a subalgebra of C ∗ ( M, E ) G , denoted S ∞ G ( M, E ) , which consists of integral operators,is dense and holomorphically closed in C ∗ ( M, E ) G and contains S cG ( M, E ) as a subalgebra.Proof. The fact that H ∞ L ( G ) is not only contained in C ∗ r ( G ), via convolution, but in fact a subalgebra of it,follows from [19]. Hence H ∞ L ( G ) satisfies the properties (1) and (2) given in Definition 3.2. The fact thatthis subalgebra is holomorphically closed is proved as in [18]. The rest of the proposition then follows fromProposition 3.6. (cid:3) -PROPER MANIFOLDS 11 Example 3.12.
Examples of Lie groups that satisfy property RD, and to which our theory applies, are:1. The abelian group R n satisfies (RD). In this case the algebra H ∞ L ( R n ) associated to the length functiondefined by the Euclidean metric is the algebra of rapidly decaying functions on R n .2. Connected semisimple Lie groups satisfy property (RD), c.f. [4], for example G = SL (2 , R ). In this casethe algebra H ∞ L ( G ) is closely related to Harish–Chandra’s Schwartz algebra C ( G ), see below. Remark 3.13.
We have just seen that if G is semisimple then by choosing A ( G ) = H ∞ L ( G ) we obtain aholomorphically closed subalgebra S ∞ G ( M, E ) ⊂ C ∗ ( M, E ) G . Notice that there are other algebras that can beconsidered. For example, we can consider as in [15] the Harish-Chandra Schwartz algebra C ( G ) ⊂ C ∗ r ( G ). Thisis a holomorphically closed subalgebra of C ∗ r ( G ), c.f. [22], which is made of smooth functions acting by convo-lution. The corresponding algebra C G ( M, E ) ⊂ C ∗ ( M, E ) G is a subalgebra of C ∗ ( M, E ) G with elements thatare in fact smoothing operators. One can prove, see [31, § II.9], that C ( G ) ⊂ H ∞ L ( G ) and thus, consequently, C G ( M, E ) ⊂ S ∞ G ( M, E ). Notice that Hochs and Wang have proved that the heat operator exp( − tD ) is anelement in C G ( M, E ). Hence exp( − tD ) ∈ S ∞ G ( M, E ).4.
Index classes
From now on we shall make constant use of the identification A cG ( M, E ) ≡ S cG ( M, E ).4.1.
The index class in K ∗ ( C ∗ ( M, E ) G ) . We consider as before a closed even-dimensional manifold M witha proper cocompact action of G . Let D a G -equivariant odd Z -graded Dirac operator. Recall, first of all,the classical Connes-Skandalis idempotent. Let Q σ be a G -equivariant parametrix of G -compact support withremainders S ± ; here the subscript σ stands for symbolic. Consider the 2 × P σ := (cid:18) S S + ( I + S + ) QS − D + I − S − (cid:19) . This produces a class(4.2) Ind c ( D ) := [ P σ ] − [ e ] ∈ K ( A cG ( M, E )) with e := (cid:18) (cid:19) To understand where this definition comes from, see for example [6].Recall now that A cG ( M, E ) ⊂ C ∗ ( M, E ) G . Definition 4.1.
The C ∗ -index associated to D is the class Ind C ∗ ( M,E ) ( D ) ∈ K ( C ∗ ( M, E ) G ) obtained by takingthe image of the Connes-Skandalis projector in K ( C ∗ ( M, E ) G ) .Unless absolutely necessary we shall denote this index class simply by Ind( D ) . Remark 4.3.
If we are in the position of considering a dense holomorphically closed subalgebra A G ( M, E )of C ∗ ( M, E ) G as in the previous section, then we can equivalently take the image of the Connes-Skandalisprojector in K ( A G ( M, E )) (recall that, by construction, A cG ( M, E ) ⊂ A G ( M, E ) ⊂ C ∗ ( M, E ) G ). For example,if G satisfies (RD) and | π ( G ) | < ∞ , then we can take the C ∗ -index class as the image of the Connes-Skandalisprojector in K ( S ∞ G ( M, E )).
Remark 4.4.
There are other representatives of Ind( D ) ∈ K ( C ∗ ( M, E ) G ) that can be of great interest.For example, as in Connes-Moscovici [6], we can choose the parametrix (which is not of G -compact support) Q V =:= I − exp( − D − D + ) D − D + D + obtaining I − Q V D + = exp( − D − D + ), I − D + Q V = exp( − D + D − ).This particular choice of parametrix produces the idempotent(4.5) V D = e − D − D + e − D − D + (cid:16) I − e − D − D + D − D + (cid:17) D − e − D + D − D + I − e − D + D − ! We call this the Connes-Moscovici idempotent. One can also consider the graph-projection [ e D ] − [ e ] ∈ K ( C ∗ ( M, E ) G ) with e D given by(4.6) e D = (cid:18) ( I + D − D + ) − ( I + D − D + ) − D − D + ( I + D − D + ) − D + ( I + D − D + ) − D − (cid:19) . Finally, following Moscovici and Wu [25], we can consider the projector(4.7) P ( D ) := (cid:18) S S + ( I + S + ) P S − D + I − S − (cid:19) . with P = u ( D − D + ) D − , S + = I − P D + , S − = I − D + P and u ( x ) := u ( x ) with u ∈ C ∞ ( R ) an even functionwith the property that w ( x ) = 1 − x u ( x ) is a Schwartz function and w and u have compactly supported Fouriertransform. One proves easily that P ( D ) ∈ M × ( A cG ( M, E )) (with the identity adjoined). It is not difficult toprove that Ind( D ) := [ P σ ] − [ e ] = [ V D ] − [ e ] = [ e D ] − [ e ] = [ P ( D )] − [ e ] in K ( C ∗ ( M, E ) G ) . The advantage of using the Connes-Moscovici projection, the graph projection or the Moscovici-Wu projectionis that Getzler rescaling can be used in order to prove the corresponding higher index formulae. This is crucialif one wishes to pass, for example, to manifolds with boundary. However, in this paper we shall concentratesolely on closed manifolds and will rather use the approach to the index theorem given in [27]; this employs thealgebraic index theorem in a fundamental way.4.2.
The index class in K • ( C ∗ r ( G )) . There is a canonical Morita isomorphism M between K ∗ ( C ∗ ( M, E ) G )and K ∗ ( C ∗ r ( G )). This is clear once we bear in mind that C ∗ ( M, E ) G is isomorphic to K ( E ); however, for reasonsconnected with the extension of cyclic cocycles, we want to be explicit about this isomorphism. We assumethe existence of a dense holomorphically closed subalgebra A ( G ) ⊂ C ∗ r ( G ) and follow Hochs-Wang [15]. Let A G ( M, E ) be the dense holomorphically dense subalgebra of C ∗ ( M, E ) G corresponding to A ( G ), as defined inSubsection 3.2. Define a partial trace map Tr S : A G ( M, E ) → A ( G ) associated to the slice S as follows: if f ⊗ k ∈ ( A ( G )) ˆ ⊗ Ψ −∞ ( S, E | S )) K × K thenTr S ( f ⊗ k ) := f Tr( T k ) = f Z S tr k ( s, s ) ds, with T k denoting the smoothing operator on S defined by k and Tr( T k ) its functional analytic trace on L ( S, E | S ).It is proved in [15] that this map induces the Morita isomorphism M between K ∗ ( C ∗ ( M, E ) G ) and K ∗ ( C ∗ r ( G )).We denote the image through M of the index class Ind( D ) ∈ K ( C ∗ ( M ) G ) in the group K ( C ∗ r ( G )) byInd C ∗ r ( G ) ( D ). There are other, well-known descriptions of the latter index class: one, following Kasparov,see [20], describes the C ∗ r ( G )-index class as the difference of two finitely generated projective C ∗ r ( G )-modules,using the invertibility modulo C ∗ r G )-compact operators of (the bounded-tranform of) D ; the other descriptionis via assembly and KK -theory, as in [2]. All these descriptions of the class Ind C ∗ r ( G ) ( D ) ∈ K ( C ∗ r ( G )) areequivalent. See [29] and [28, Proposition 2.1].5. Cyclic cocycles and pairings with K -theory Cyclic cohomology.
In this subsection we shortly review the basic complex computing cyclic cohomology.Let A be a unital algebra. The space of reduced Hochschild cochains is defined as C • red ( A ) := Hom C ( A ⊗ ( A/ C • , C )and is equipped with the Hochschild differential b : C k red ( A ) → C k +1red ( A ) given by the standard formula bτ ( a , . . . , a k +1 ) := k X i =0 ( − i τ ( a , . . . , a i a i +1 , . . . , a k ) + ( − k +1 τ ( a k a , . . . , a k − ) . -PROPER MANIFOLDS 13 The cyclic bicomplex is given by . . . . . . . . .C ( A ) B / / b O O C ( A ) B / / b O O C ( A ) b O O C ( A ) B / / b O O C ( A ) b O O C ( A ) b O O where B : C k red ( A ) → C k − ( A ) denotes Connes’ cyclic differential Bτ ( a , . . . , a k − ) := k − X i =0 ( − ( k − i τ (1 , a i , . . . , a k − , a , . . . , a i − ) . We denote the total complex associated to this double complex by CC • ( A ). When A is not unital, we considerthe unitization ˜ A = A ⊕ C , and compute cyclic cohomology from the complex CC • ( A ) := CC • ( ˜ A ) /CC • ( C ).Finally, let us close by mentioning that the structure underlying the definition of cyclic cohomology is thatof a cocyclic object: this is a cosimplicial object ( X • , ∂ • , σ • ) equipped with an additional cyclic symmetry t n : X n → X n of order n + 1 satisfying well-known compatibility conditions with respect to the coface operators ∂ and degeneracies σ , c.f. [24]. For the cyclic cohomology of an algebra the underlying cosimplicial object isgiven by X k = C k ( A ) with coface and degeneracies controlling the Hochschild complex. The additional cyclicsymmetry t underlying cyclic cohomology is simply the operator which in degree k cyclically permutes the k + 1entries in a cochain τ ∈ C k ( A ).5.2. The van Est map in cyclic cohomology.
Let G be a unimodular Lie group with | π ( G ) | < ∞ . In thissubsection we describe, following [26, 27], how to obtain cyclic cocycles from smooth group cocycles. In this, wecan work with two algebras: C ∞ c ( G ), the convolution algebra of the group, and A cG ( M ), the algebra of invariantsmoothing operators with cocompact support. In order to simplify the notation we take the vector bundle E tobe the product bundle of rank 1.We start with the following well-known remark: inspection of the differential (2.2) shows that the cochaincomplex ( C • diff ( G ) , δ ) computing smooth group cohomology H • diff ( G ) comes from an underlying cosimplicialstructure given by coface maps ∂ i and codegeneracies σ j defined on the vector space of homogeneous smoothgroup cochains C • diff ( G ). This simplicial vector space can be upgraded to a cocyclic one by the cyclic operator t : C • → C • given by ( tf )( g , . . . , g k ) = f ( g k , g , . . . , g k − ) , f ∈ C k diff ( G ) . As seen above, the Hochschild theory of this cocyclic complex is just the smooth group cohomology. Theassociated cyclic theory is given by L i ≥ H •− i diff ( G ).Let us now describe the associated cyclic cocycles on the convolution algebra C ∞ c ( G ). Instead of using the fullcomplex of smooth group cochains, we shall restrict to the quasi-isomorphic subcomplex C • diff ,λ ( G ) ⊂ C • diff ( G )of cyclic cochains, i.e., cochains c ∈ C k diff ( G ) satisfying c ( g , . . . , g k ) = ( − k c ( g k , g , . . . , g k − ) . Let c ∈ C k diff ( G ) be a smooth homogeneous group cochain. Define the cyclic cochain τ c ∈ C k ( C ∞ c ( G )) by(5.1) τ Gc ( a , . . . , a k ) := Z G × k c ( e, g , g g , . . . , g · · · g k ) a (( g · · · g k ) − ) a ( g ) · · · a k ( g k ) dg · · · dg k . Next up is the algebra A cG ( M ) of invariant smoothing operators with cocompact support. Again given a smoothhomogeneous group cochain c ∈ C k diff ( G ), we now define a cyclic cochain on this algebra by the formula τ Mc ( k , . . . , k n ) := Z G × k Z M × ( k +1) χ ( x ) · · · χ ( x n ) k ( x , g x ) · · · k n ( x n , ( g · · · g n ) − x ) c ( e, g , g g , . . . , g · · · g n ) dx · · · dx n dg · · · dg n . (5.2) Proposition 5.3.
The following holds true: i ) The map c τ Gc defined above is a morphism of cochain complexes and therefore induces a map Ψ G : H • diff ( G ) → HC • ( C ∞ c ( G )) .ii ) The map c τ Mc defined above is a morphism of cocyclic complexes and therefore induces a map Ψ M : H • diff ( G ) → HC • ( A cG ( M )) . Proof.
Both of the statements are already known: for the first one, see [26, § § (cid:3) Example 5.4.
In Example 2.14 we discussed the smooth group 2-cocycles for G = R , G = SL (2 , R ), associatedto the area forms of the homogeneous space G/K , equal to R and H respectively. Let us now consider thecyclic cocycles defined by these forms via the construction (5.1) above. For G = SL (2 , R ) this gives the followingcyclic 2-cocycle on C ∞ c ( SL (2 , R ): τ SL (2 , R ) ω ( f , f , f ) := Z SL (2 , R ) Z SL (2 , R ) f (( g g ) − ) f ( g ) f ( g )Area H (∆ (¯ e, ¯ g , ¯ g )) dg dg This is exactly the cyclic cocycle considered by Connes in [5, § G = R we get a cyclic 2-cocycleon C ∞ c ( R ) (with convolution product) given by the same formula with the hyperbolic area replaced by theEuclidean area, and integrations being over R instead of SL (2 , R ), again considered in [5, § f ˆ f this cocycle takes the usual form τ ω ( f , f , f ) = Z R ˆ f d ˆ f ∧ d ˆ f , for f , f , f ∈ C ∞ c ( R ) . Extension properties.
In the previous subsection we have constructed cyclic cocycles τ Gc on C ∞ c ( G ) and τ Mc on A cG ( M ) from a homogeneous smooth group cocycle c . (Recall, once again, that for notational conveniencewe are taking E to be the product rank 1 bundle.) In § G ensuringthat these algebras embed into holomorphically closed subalgebras A ( G ) and A G ( M ) of the reduced group C ∗ -algebra and of the Roe algebra. Now we want to discuss the extension properties of these cocycles. Assume,quite generally, that we are given a subalgebra A ( G ) as in Definition 3.2, with associated algebra of operatorson L ( M ) denoted, as usual, as A G ( M ). First we have: Proposition 5.5.
Let c ∈ C k diff ,λ ( G ) be a smooth group cocycle. Then we have: τ Gc extends to A ( G ) ⇐⇒ τ Mc extends to A G ( M ) Proof.
Recall that the algebra A G ( M ) is constructed from the choice of A ( G ) ⊂ C ∗ r ( G ) by the slice theorem:an invariant kernel k belongs to A G ( M ) if the function˜ k ( g, s , s ) := k ( s , gs )belongs to ( A ( G ) ˆ ⊗ Ψ −∞ ( S, E | S )) K × K . These functions ˜ k i ( g i , x i , x i +1 ), i = 0 , . . . n − k n (( g · · · g n ) − , x n , x )are used in the formula (5.2) for the cocycle τ Mc . Since the cut-off function χ has compact support, performingthe integrations over M in equation (5.2), we end up with the pairing of an element in A ( G ) ⊗ ( k +1) with thegroup cocycle c as defined in (5.1). But then it is clear that τ Mc is well-defined on A G ( M ) if and only if τ Gc iswell-defined on A ( G ). (cid:3) For the following, recall from § α ∈ Ω k inv ( G/K ) to a smooth group cocycle J ( α ) ∈ C k diff ( G ). For notational convenience, we willdrop the J in the description of the associated cyclic cocycles, writing τ Gα and τ Mα instead of τ GJ ( α ) and τ MJ ( α ) . -PROPER MANIFOLDS 15 Proposition 5.6.
Let G be a Lie group with finitely many connected components and satisfying the rapid decayproperty (RD). Assume that G/K is of non-positive sectional curvature. Then the cocycle τ Gα associated to aclosed invariant differential form α ∈ Ω k inv ( G/K ) extends continuously to H ∞ L ( G ) . Consequently, the cycliccocycles τ Mα extends to S ∞ G ( M ) .Proof. Recall the definition of the smooth group cocycle J ( α ) ∈ C k diff ( G ) defined in (2.12), satisfying thepolynomial estimates of Theorem 2.13. This, together with the rapid decay property of G , ensures we canfollow the line of proof of [6, Prop. 6.5], where the analogous extension property is proved for certain discretegroups. To show that the cyclic cocycle τ α extends continuously to the algebra H ∞ L ( G ), we need to show thatit is bounded with respect to the seminorm ν k in (3.9) defining the Fr´ech`et topology, for some k ∈ N . Let a , . . . , a k ∈ H ∞ L ( G ), and write ˜ a := | a | , ˜ a i ( g ) := (1 + d ( g )) k | a i ( g ) | , i = 1 , . . . , k . Then we can make thefollowing estimate: | τ Gα ( a , . . . , a k ) | ≤ C Z G × k (1 + d ( g )) k · · · (1 + d ( g k )) k | a (( g · · · g k ) − ) | · | a ( g ) | · · · | a k ( g k ) | dg · · · dg k = C (˜ a ∗ . . . ∗ ˜ a k )( e ) ≤ C || ˜ a ∗ . . . ∗ ˜ a k || C ∗ r ( G ) ≤ C || ˜ a || C ∗ r ( G ) · · · || ˜ a k || C ∗ r ( G ) ≤ CD k +1 ν p (˜ a ) · · · ν p (˜ a k ) = CD k +1 ν p + k ( a ) · · · ν p + k ( a k ) . In this computation we have used the fact that the Plancherel trace a a ( e ) on the convolution algebra has acontinuous extension to C ∗ r ( G ), together with the rapid decay property: || a || C ∗ r ( G ) ≤ D || (1 + d ) p a || L , for some p . Altogether, this proves the proposition. (cid:3) Pairing with K -theory. Cyclic cohomology was originally developed by Connes to pair with K -theoryvia the Chern character. Let us recall this construction: let τ = ( τ , τ , . . . , τ k ) ∈ CC k ( A ) be a cyclic cocycleof degree 2 k on a unital algebra A , and [ p ] − [ q ] an element in K ( A ) represented by idempotents p, q ∈ M N ( A ).The number h [ p ] − [ q ] , τ i := k X n =0 ( − n (2 n )! n ! (cid:18) τ n (cid:18) tr( p − , p, . . . , p ) (cid:19) − τ n (cid:18) tr( q − , q, . . . , q ) (cid:19)(cid:19) , where tr : M N ( A ) ⊗ ( n +1) → A ⊗ ( n +1) is the generalized matrix trace, is well-defined and depends only on the(periodic) cyclic cohomology class of τ . Proposition 5.7.
Let c , A ( G ) and A G ( M ) as in Proposition 5.5, and assume that τ Gc , and therefore τ Mc ,extends. Then, under the Morita isomorphism M : K ( C ∗ ( M, E ) G ) ∼ = −→ K ( C ∗ r ( G )) , we have the equality: (cid:10) [ p ] − [ q ] , τ Mc (cid:11) = (cid:10) M ([ p ] − [ q ]) , τ Gc (cid:11) . Proof.
Recall that the isomorphism M : K ( C ∗ ( M, E ) G ) → K ( C ∗ r ( G )) is implemented by the partial trace mapTr S : A G ( M, E ) → A ( G ) on the respective dense subalgebras. By the abstract Morita isomorphism M , itsuffices to consider a simple idempotent e = e ⊗ e ∈ M n ( A G ( M, E )) so that Tr S ( e ) = Tr S ( e ) e yields anidempotent in M n ( A ( G )), where we have extended Tr S to matrix algebras in the usual way by combining withthe matrix trace.Because we know that the cyclic cohomology class of ˜ τ c is independent of the choice of a cut-off function,the pairing with K -theory does not depend on this choice either so we can choose the family χ ǫ constructed in Lemma 2.1 and take the limit as ǫ ↓ (cid:10) [ e ] , τ Mc (cid:11) = lim ǫ ↓ (2 k )! k ! Z G × k Z M × ( k +1) χ ǫ ( x ) · · · χ ǫ ( x n ) e ( x , g x ) · · · e ( x n , ( g · · · g n ) − x ) c ( e, g , g g , . . . , g · · · g n ) dx · · · dx n dg · · · dg n , = (2 k )! k ! Z G × k Z S × ( k +1) e ( x , g x ) · · · e ( x n , ( g · · · g n ) − x ) c ( e, g , g g , . . . , g · · · g n ) dx · · · dx n dg · · · dg n , = (2 k )! k ! Tr S ( e · · · e ) Z G × k e ( g ) · · · e (( g · · · g n ) − ) c ( e, g , g g , . . . , g · · · g n ) dg · · · dg n , = (cid:10) [ M ( e )] , τ Gc (cid:11) , where, to go to the last line, we have used the fact that e = e is an idempotent. This completes the proof. (cid:3) Higher C ∗ -indices and geometric applications Higher C ∗ -indices and the index formula. Let M and G be as above, with M even dimensional. Hence G is a unimodular Lie group with | π ( G ) | < ∞ . (For the time being we do not put additional hypothesis on G .) Let E be an equivariant complex vector bundle. Consider an odd Z -graded Dirac type operator D actingon the sections of E . We have then defined the compactly supported index class Ind c ( D ) ∈ K ( A cG ( M, E )).Let α ∈ H evendiff ( G ) and let Ψ M ( α ) ∈ HC even ( A cG ( M, E )) be the cyclic cohomology class corresponding to α . Weknow that, in general, we have a pairing(6.1) K ( A cG ( M, E )) × HC even ( A cG ( M, E )) −→ C We thus obtain, through Ψ M : H • diff ( G ) → HC • ( A cG ( M, E )), a pairing(6.2) K ( A cG ( M, E )) × H evendiff ( G ) −→ C In particular, by pairing Ind c ( D ) ∈ K ( A cG ( M, E )) with α ∈ H evendiff ( G ) we obtaining the higher indices Ind c,α ( D ) := h Ind c ( D ) , Ψ M ( α ) i , α ∈ H evendiff ( G ) . On the other hand, we can also take the image of α through the van Est map Φ M : H • diff ( G ) → H • inv ( M ); recallthat this is nothing but the pull-back through the classifying map ψ M : M → G/K once we identify H • diff ( G )with H • inv ( G/K ). The following theorem is proved in [27]:
Theorem 6.3 (Pflaum-Posthuma-Tang, c.f. [27]) . Let M , G and D as above. In particular, M is even dimen-sional. Let α ∈ H evendiff ( G ) . Then the following identity holds true: (6.4) Ind c,α ( D ) = Z M χ M ( m ) AS( M ) ∧ Φ M ( α ) with AS( M ) the Atiyah-Singer integrand on M : AS( M ) := ˆ A ( M, ∇ M ) ∧ Ch ′ ( E, ∇ E ) .Equivalently, (6.5) Ind c,α ( D ) = Z M χ M ( m ) AS( M ) ∧ ψ ∗ M ( α ) if we identify H • diff ( G ) and H • inv ( G/K ) via the van Est isomorphism, c.f. Remark 2.7. We now make the fundamental assumption that G satisfies the rapid decay property and that G/K is ofnon-positive sectional curvature. Let S ∞ G ( M, E ) ⊂ C ∗ ( M, E ) G be the dense holomorphically closed subalgebradefined by the rapid decay algebra H ∞ L ( G ) ⊂ C ∗ r ( G ). Thanks to the results of the previous Section we canextend the pairing (6.2) to a pairing(6.6) K ( S ∞ G ( M, E )) = K ( C ∗ ( M, E ) G ) × H evendiff ( G ) −→ C obtaining in this way the higher C ∗ -indices of D , denoted Ind α ( D ). These numbers are well defined andcan be computed by choosing a suitable representative of the class Ind( D ) ∈ K ( C ∗ ( M, E ) G ). Choosing the -PROPER MANIFOLDS 17 Connes-Skandalis projector we can apply again the index formula of Pflaum-Posthuma-Tang, obtaing for each α ∈ H evendiff ( G ) the C ∗ -index formula(6.7) Ind α ( D ) = Z M χ M ( m ) AS( M ) ∧ Φ M ( α ) . Notice that we also have a pairing(6.8) K ( C ∞ c ( G )) × HC even ( C ∞ c ( G )) −→ C and thus, through the homomorphism Ψ G : H • diff ( G ) → HC ∗ ( C ∞ c ( G )), a pairing(6.9) K ( C ∞ c ( G )) × H evendiff ( G ) −→ C According to the results of the previous section this pairing extends to a pairing(6.10) K ( C ∗ r ( G )) × H evendiff ( G ) −→ C if G satisfies (RD). In particular, we can define the C ∗ r ( G )-indeces Ind C ∗ r ( G ) ,α ( D ) by pairing Ind C ∗ r ( G ) ( D ) ∈ K ( C ∗ r ( G )) with α ∈ H evendiff ( G ). Moreover, from Proposition 5.7 we get the following equality:(6.11) h Ind( D ) , Ψ M ( α ) i = (cid:10) Ind C ∗ r ( G ) ( D ) , Ψ G ( α ) (cid:11) which means that(6.12) Ind C ∗ r ( G ) ,α ( D ) = Ind α ( D ) ∀ α ∈ H evendiff ( G )and thus, thanks to (6.7), we can state the following fundamental result: Theorem 6.13.
Let G be a Lie group satisfying the properties stated in the introduction: | π ( G ) | < ∞ , (RD)and EG of non-positive curvature. Let α ∈ H evendiff ( G ) . Then there is a well-defined associated higher C ∗ r ( G ) -index Ind C ∗ r ( G ) ,α ( D ) and the following formula holds: (6.14) Ind C ∗ r ( G ) ,α ( D ) = Z M χ M ( m ) AS( M ) ∧ Φ M ( α ) . Equivalently, if we identify H • diff ( G ) and H • inv ( G/K ) ≡ H • inv ( EG ) via the van Est isomoprhism, then Ind C ∗ r ( G ) ,α ( D ) = Z M χ M ( m ) AS( M ) ∧ ψ ∗ M α. For α = 1, the associated cyclic cocycle (5.1) is just the Plancherel trace τ G ( f ) = f ( e ) on C ∗ r ( G ), andthe Theorem reduces to the L -index theorem first proved by Wang in [32]. Remark that in this case thetrace extends to C ∗ r ( G ) without problems, so the assumptions on the curvature of G/K and property (RD) areunnecessary.6.2.
Higher signatures and their G -homotopy invariance. Let M and N two orientable G -proper mani-folds and let f : M → N be a G -homotopy equivalence. Let us denote by D sign M and D sign M the correspondingsignature operators. Then, according to the main result in [10] we have that(6.15) Ind C ∗ r ( G ) ( D sign M ) = Ind C ∗ r ( G ) ( D sign N ) in K ( C ∗ r ( G )) . Consequently, from (6.14), we obtain the following result, stated as item (i) in Theorem 1.5 in the Introduction:
Theorem 6.16.
Let G be a Lie group satisfying the properties stated in the introduction: | π ( G ) | < ∞ , (RD)and EG of non-positive curvature. Let M and N are two orientable G -proper manifolds and assume thatthere exists an orientation preserving G -homotopy equivalence between M and N . Let us identify H • diff ( G ) and H • inv ( G/K ) ≡ H • inv ( EG ) via the van Est isomorphism. Then. for each α ∈ H • inv ( EG ) : Z M χ M ( m ) L ( M ) ∧ ψ ∗ M α = Z N χ N ( n ) L ( N ) ∧ ψ ∗ N α Proof.
For even dimensional manifolds, this follows immediately from the previous discussion. For the odd-dimensional case we argue by suspension. Thus, let M be an orientable odd dimensional G -proper manifold.We endow M with a G -invariant riemannian metric g M . Consider R and the natural action of Z on it bytranslations (this is a free, proper and cocompact action). Taking the product of M and R we obtaining theeven dimensional ( G × Z )-proper manifold M × R ; it has compact quotient equal to M/G × S . We endow M × R with the ( G × Z )-invariant metric g M + dt . Consider the dual group T := Hom( Z , U (1)). The signature operatoron M × R defines an index classe in the group K ( C ∗ ( M × R ) G × Z ), which is isomorphic to K ( C ∗ ( G ) ˆ ⊗ C ( T )).Consider the generator d ′ of H ( Z ; Z ) ⊂ H ∗ ( Z ; C ) and let d := √− π d ′ ∈ H ∗ ( Z ; C ). We know that H ∗ ( Z ; C )can be identified with H ∗ Z ( E Z ; C ) and that E Z = R ; we denote by Ξ : H ∗ ( Z ; C ) → H ∗ Z ( R ; C ) = H ( S ) thisisomorphism. Consider EG × E Z ≡ EG × R ≡ G/K × R . To α ∈ H odddiff ( G ) ≡ H oddinv ( EG ) ≡ H oddinv ( G/K ) weassociate β := α ⊗ Ξ( d ) ∈ H oddinv ( G/K ) ⊗ H Z ( R ) = H oddinv ( G/K ) ⊗ H ( S ) . Now, on the one hand, we have natural homomorphismsΨ G × Z : H oddinv ( G/K ) ⊗ H ( S ) → HC even ( C ∞ c ( G ) ˆ ⊗ C ∞ ( S ))and Ψ M × R : H oddinv ( G/K ) ⊗ H ( S ) → HC even ( A cG × Z ( M × R ))where we remark that A cG × Z ( M × R ) = A cG ( M ) ˆ ⊗A c Z ( R ) and also that A cG × Z ( M × R ) = C ∗ c ( M × R ) G × Z . Onthe other hand the classifying map ψ M and the classifying map for the Z -action on R give together a smooth( G × Z )-equivariant map ψ M × R : M × R → G/K × R . We can apply the Pflaum-Posthuma-Tang index theoremand obtain, for the signature operator, (cid:10) Ind C ∗ c ( M × R ) G × Z ( D M × R ) , Ψ M × R ( β ) (cid:11) = Z G Z S χ M L ( M × R ) ψ ∗ M ( α ) ∧ Ξ( d ) = Z G χ M L ( M ) ψ ∗ M ( α ) = σ ( M, α ) . If G satisfies (RD), then this formula remains true for the C ∗ ( M × R ) G × Z index, because S ∞ G ( M ) ˆ ⊗S Z ( R ), with S Z ( R ) denoting the smooth Z -invariant kernels of R × R of rapid polynomial decay, is a dense holomorphicallyclosed subalgebra of C ∗ ( M × R ) G × Z to which the pairing with Ψ M × R ( β ) extends. Consequently D Ind C ∗ ( G ) ˆ ⊗ C ( S ) ( D M × R ) , Ψ G × Z ( β ) E = σ ( M, α ) . Now, if M and N are G-homotopy equivalent, then M × R and N × R are G × Z homotopy equivalent. Hencethe corresponding signature index classes in K ( C ∗ ( G ) ˆ ⊗ C ( T )) are equal; thus D Ind C ∗ ( G ) ˆ ⊗ C ( S ) ( D M × R ) , Ψ G × Z ( β ) E = D Ind C ∗ ( G ) ˆ ⊗ C ( S ) ( D N × R ) , Ψ G × Z ( β ) E This gives us σ ( M, α ) = σ ( N, α ) . which is what we wanted to prove in odd dimension. (cid:3) Higher b A -genera and G-metrics of positive scalar curvature. Let S be compact smooth manifoldwith an action of a compact Lie group K . In general, the existence of a K -invariant metric of positive scalarcurvature on S is a more refined property than the existence of a positive scalar curvature metric on S ; indeed, asshown by Berard-B´ergery in [3], averaging a positive scalar curvature metric on S might destroy the positivity ofthe scalar curvature. For sufficient conditions on K and on S ensuring the existence of such metrics see [23, 12].If M is a G -proper manifold we can try to built a G -invariant positive scalar curvature metric on M througha K -invariant positive scalar curvature metric on the slice S . This is precisely what is achieved in [11]: Theorem 6.17 (Guo-Mathai-Wang, c.f. [11]) . Let G be an almost connected Lie group and let K be a maximalcompact subgroup of G . If S is a compact manifold with a K -invariant metric of positive scalar curvature, thenthe G -proper manifold G × K S admits a G -invariant metric of positive scalar curvature. -PROPER MANIFOLDS 19 This result shows that the space of positive scalar curvature G -metrics on a G -proper manifold can be non-empty.We can ask for numerical obstructions to the existence of a positive scalar curvature G -metric. Assume that M has a G-equivariant spin structure and let ð be the associated spin-Dirac operator. Then one can show, seeagain [11], that(6.18) Ind C ∗ r ( G ) ( ð ) = 0 in K ∗ ( C ∗ r G ) . The following result was stated as item (ii) in the main Theorem, Theorem 1.5, in the Introduction:
Theorem 6.19.
Let G be a Lie group satisfying the properties stated in the introduction: | π ( G ) | < ∞ , (RD)and EG of non-positive curvature. Let M be a G -proper manifold admitting a G -equivariant spin structure.Let us identify H • diff ( G ) and H • inv ( G/K ) ≡ H • inv ( EG ) via the van Est isomoprhism. If M admits a G -invariantmetric of positive scalar curvature, then b A ( M, α ) := Z M χ M ( m ) b A ( M ) ∧ ψ ∗ M α = 0 for each α ∈ H • inv ( EG ) .Proof. The even dimensional case follows directly from our C ∗ -index formula and from (6.18). In the odddimensional case we argue by suspension, as we did for the signature operator. It suffices to observe that if M is an odd dimensional G -proper manifold admitting a G -equivariant spin structure and a G -invariant metricof positive scalar curvature g M , then M × R is an even dimensional ( G × Z )-proper manifold with a ( G × Z )-equivariant spin structure and with a ( G × Z )-invariant metric g M + dt which is of positive scalar curvaturetoo. Consequently, the analogue of (6.18) holds for the spin Dirac operator on M × R and so, arguing as forthe signature operator, we finally obtain that b A ( M, α ) := R M χ M ( m ) b A ( M ) ∧ ψ ∗ M α = 0 as required. (cid:3) References [1] Herbert Abels. Parallelizability of proper actions, global K -slices and maximal compact subgroups. Math. Ann. , 212:1–19,1974/75.[2] Paul Baum, Alain Connes, and Nigel Higson. Classifying space for proper actions and K -theory of group C ∗ -algebras. In C ∗ -algebras: 1943–1993 (San Antonio, TX, 1993) , volume 167 of Contemp. Math. , pages 240–291. Amer. Math. Soc., Providence,RI, 1994.[3] Lionel B´erard-Bergery. La courbure scalaire des vari´et´es riemanniennes. In
ERA Conferences , volume 4 of
Inst. ´Elie Cartan ,pages 30–45. Univ. Nancy, Nancy, 1981.[4] I. Chatterji, C. Pittet, and L. Saloff-Coste. Connected Lie groups and property RD.
Duke Math. J. , 137(3):511–536, 2007.[5] Alain Connes. Noncommutative differential geometry.
Inst. Hautes ´Etudes Sci. Publ. Math. , (62):257–360, 1985.[6] Alain Connes and Henri Moscovici. Cyclic cohomology, the Novikov conjecture and hyperbolic groups.
Topology , 29(3):345–388,1990.[7] Marius Crainic. Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes.
Comment. Math.Helv. , 78(4):681–721, 2003.[8] Johan L. Dupont. Simplicial de Rham cohomology and characteristic classes of flat bundles.
Topology , 15(3):233–245, 1976.[9] Johan L. Dupont. Bounds for characteristic numbers of flat bundles. In
Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ.Aarhus, Aarhus, 1978) , volume 763 of
Lecture Notes in Math. , pages 109–119. Springer, Berlin, 1979.[10] Y. Fukumoto. G -homotopy invariance of the analytic signature of proper co-compact g-manifolds and equivariant novikovconjecture. Preprint arXiv.1709.05884 .[11] H Guo, V. Mathai, and H. Wang. Positive scalar curvature and Poincar´e duality for proper actions. Preprint arXiv.1609.01404v5 .[12] Bernhard Hanke. Positive scalar curvature with symmetry.
J. Reine Angew. Math. , 614:73–115, 2008.[13] Tobias Hartnick and Andreas Ott. Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups.
Int.Math. Res. Not. IMRN , (9):2068–2093, 2012.[14] Sigurdur Helgason.
Differential geometry, Lie groups, and symmetric spaces , volume 34 of
Graduate Studies in Mathematics .American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original.[15] P Hochs and H. Wang. A fixed point formula and Harish-Chandra’s character formula. Preprint arXiv.1701.08479v2 .[16] Hisao Inoue and Koichi Yano. The Gromov invariant of negatively curved manifolds.
Topology , 21(1):83–89, 1982.[17] Ronghui Ji and Larry B. Schweitzer. Spectral invariance of smooth crossed products, and rapid decay locally compact groups. K -Theory , 10(3):283–305, 1996.[18] Paul Jolissaint. K -theory of reduced C ∗ -algebras and rapidly decreasing functions on groups. K -Theory , 2(6):723–735, 1989.[19] Paul Jolissaint. Rapidly decreasing functions in reduced C ∗ -algebras of groups. Trans. Amer. Math. Soc. , 317(1):167–196,1990. [20] G. G. Kasparov. The operator K -functor and extensions of C ∗ -algebras. Izv. Akad. Nauk SSSR Ser. Mat. , 44(3):571–636, 719,1980.[21] S. Kim and I. Kim. Simplicial volume, barycenter method, and bounded cohomology. Preprint arXiv:1503.02381 .[22] Vincent Lafforgue. K -th´eorie bivariante pour les alg`ebres de Banach et conjecture de Baum-Connes. Invent. Math. , 149(1):1–95,2002.[23] H. Blaine Lawson, Jr. and Shing Tung Yau. Scalar curvature, non-abelian group actions, and the degree of symmetry of exoticspheres.
Comment. Math. Helv. , 49:232–244, 1974.[24] Jean-Louis Loday.
Cyclic homology , volume 301 of
Grundlehren der Mathematischen Wissenschaften . Springer-Verlag, Berlin,second edition, 1998.[25] H. Moscovici and F.-B. Wu. Localization of topological Pontryagin classes via finite propagation speed.
Geom. Funct. Anal. ,4(1):52–92, 1994.[26] M. J. Pflaum, H. Posthuma, and X. Tang. The localized longitudinal index theorem for Lie groupoids and the van Est map.
Adv. Math. , 270:223–262, 2015.[27] Markus J. Pflaum, Hessel Posthuma, and Xiang Tang. The transverse index theorem for proper cocompact actions of Liegroupoids.
J. Differential Geom. , 99(3):443–472, 2015.[28] Paolo Piazza and Thomas Schick. Rho-classes, index theory and Stolz’ positive scalar curvature sequence.
J. Topol. , 7(4):965–1004, 2014.[29] John Roe. Comparing analytic assembly maps.
Q. J. Math. , 53(2):241–248, 2002.[30] W. T. van Est. On the algebraic cohomology concepts in Lie groups. I, II.
Nederl. Akad. Wetensch. Proc. Ser. A. = Indag.Math. , 17:225–233, 286–294, 1955.[31] V. S. Varadarajan. Harmonic analysis on real reductive groups . Lecture Notes in Mathematics, Vol. 576. Springer-Verlag,Berlin-New York, 1977.[32] Hang Wang. L -index formula for proper cocompact group actions. J. Noncommut. Geom. , 8(2):393–432, 2014.
Dipartimento di Matematica, Sapienza Universit`a di Roma
E-mail address : [email protected] Korteweg–de Vries Institute for Mathematics, University of Amsterdam
E-mail address ::