The transverse index theorem on foliations

University of Amsterdam MSc Mathematical Physics Master Thesis

The transverse index theorem on foliations

Author: Abel Stern

Supervisor: Dr. H.B. Posthuma Second reader: Dr. R.R.J. Bocklandt

November 2015

The transverse index theorem on foliations Abel Stern

Abstract We carefully review the transverse index theorem of Connes and Moscovici. In order to construct abstract transversally elliptic operators, we define a metric on the frame bundle that is almost invariant under diffeomorphisms. Using the fibration of this bundle and the metric, we construct a spectral triple that describes noncommutatively the space of leaves of a foliation. The functions and vector fields on the bundle generate a Hopf algebroid which yields cyclic cocycles on the convolution algebra of a prolonged Haefliger groupoid. The Hopf cyclic cohomology is calculated as the Gelfand-Fuchs cohomology and includes the Chern character of our spectral triple.

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Introduction Popular summary Acknowledgements

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Chapter 1. The hypoelliptic signature operator on foliations 1. The almost isometric action of Diff(M ) on the frame bundle F 2. The algebra AP := Cc∞ (P ) ⋊ Γ and its representation 3. The vertical signature operator QV 4. Extra structure provided by a connection ω ∈ Ω1 (F, g) 4.1. The dual trivialization of T F 5. The horizontal signature operator QH 6. The hypoelliptic signature operator 7. The index map Ki (A) − →Z

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Chapter 2. The Hopf algebroid HF and the index theorem 1. The groupoid cocycle γ 2. The Hopf algebroid HF and the characteristic map 2.1. The algebra HF and its representation on AF 2.2. The left bialgebroid structure of HF 2.3. The invariant trace, the twisted antipode and the subalgebra K ≃ U (k) 2.4. Relative cyclic cohomology of Hopf algebras and the characteristic map 3. The periodic cyclic cohomology of HF 3.1. The Haefliger-van Est isomorphism H ∗ (a(ΓF )) ≃ Hd∗ (ΓF ; O) 3.2. The map Φ from the groupoid cohomology of ΓF , with coefficients in O, to the periodic cyclic cohomology of AF 4. The transverse index theorem of Connes and Moscovici

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Bibliography

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Appendix A. 1. Foliations 1.1. Holonomy 1.2. The Bott connection 2. Partial connections on vector bundles 2.1. Exterior covariant derivative 2.2. The induced connection on the exterior dual bundle 3. Cyclic cohomology and K-theory 3.1. The universal differential graded algebra Ω• (A) 3.2. The cup product in cyclic cohomology [Con85, Part II.1] 3.3. Pairing of cyclic cohomology with K-theory 4. The frame bundle as a crossed product

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Contents

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Introduction Imagine one is provided with a smooth manifold N that has been (integrably) foliated. That is, someone has divided N up into a collection of non-intersecting immersed submanifolds of a given codimension m, called the leaves of the foliation. Our mission is to gain a geometric understanding of the space of these leaves. It seems natural to start with a m-dimensional submanifold M that intersects every leaf only once. However, any given leaf may locally be dense in N . Therefore, we must allow M to intersect countably with the leaves. Such a submanifold is called transversal. If we can somehow ’quotient out’ the multiple intersections, we have a local model of the space of leaves. The transport of points in M along the leaves give rise to a discrete pseudogroup Γ ⊂ Diff(M ) of local diffeomorphisms of M , called the holonomy pseudogroup. Thus, we wish to make sense of the space M/Γ. However, even in the simplest examples - consider, for instance, the group Γ of all translations of S 1 by an irrational angle - there is no hope that the topological quotient will be smooth or even locally Hausdorff. The orbits of any point may certainly be dense in M . Therefore, at least for the purposes of differential geometry, we should not just consider the set of equivalence classes of points in M , but rather keep track of all the equivalences between points and of the reason why they are equivalent. This information is nicely encoded in the groupoid M ⋊ Γ. We will use noncommutative geometry to study the convolution algebra Cc∞ (M ) ⋊ Γ of this groupoid, which the physicist might like to view as consisting of fields on M that carry the geometric structure of interest. We wish to define abstract elliptic operators on the groupoid, whose associated index maps will give us K-theoretic topological invariants of the space of leaves. In the first chapter of this thesis, we follow [CM95] in their study of a prolonged groupoid F ⋊ Γ whose operator K-theory is isomorphic1 to that of M ⋊ Γ. We use a choice of connection on F to define a Riemannian metric on F that is almost invariant under Γ. This metric leads to the construction of the hypoelliptic signature operator on a bundle isomorphic to Ω∗ (F ). The convolution algebra, the space of square-integrable sections of the bundle and the hypoelliptic signature operator form a spectral triple 1It is a bit more subtle than that, and is connected to the reason for studying F ⋊ Γ instead of M ⋊ Γ: see section 7. 3

4

INTRODUCTION

in noncommutative geometry. To such any spectral triple we may associate an abstract elliptic operator, that is, a Fredholm module. This Fredholm module represents a nontrivial class in the K-homology of M ⋊ Γ. In principle, index maps in noncommutative geometry, that is, homomorphisms from K-cohomology of a C ∗ -algebra A to Z, are calculated as traces of operators over Hilbert spaces. These traces contain global information about a space and are, in general, not easy to evaluate. Therefore, one would like to have an index theorem, descibing how the global information about the space encoded in the index can be calculated using local geometrical structures. In noncommutative geometry, this means that we wish to express an index homomorphism K∗ (A) − → Z as a pairing with a fixed element of the periodic cyclic cohomology HP ∗ (A). Although there is a general local index theorem in noncommutative geometry, it is not very enlightening in this particular case. Therefore, the second chapter of this thesis deals with an index theorem adapted to our setting, following [CM98]. In order to calculate the index, we construct a Hopf algebroid of differential operators on the prolonged Haefliger groupoid F ⋊ Diff(F ). Using the action of this Hopf algebroid on the convolution algebra and a trace on this algebra, we realize the Hopf cyclic cohomology of the algebroid within the periodic cyclic cohomology of the convolution algebra. On the one hand, this Hopf cyclic cohomology can be shown to be isomorphic to the Gelfand-Fuchs cohomology. On the other hand, it was shown in [CM98], using the general local index theorem, that the Chern character of the hypoelliptic signature operator is contained in the image of the Hopf cyclic cohomology. Thus, the index map is in the end expressed as a pairing with an element of the Gelfand-Fuchs cohomology. In [Rod14] this element was identified for certain pseudogroups Γ ⊂ Diff. The general case, however, remains unsolved. Popular summary Fundamentally, most airports look the same. Next to the runways and the parking space for the planes there is a big building with check-in counters, customs, bathrooms and shops. If not for the noise, the view of the ground falling away and the sinking feeling in the stomach when the plane accelerates downward, the weary traveler might well believe that there exists only one place, the Airport, and that the main result of stepping in and out of an airplane is a strange reshuffling of the various walls, corridors and signs that make up the Airport. If such travelers are sensitive to the architecture of their surroundings, they might wish to draw a map. Surely, this cannot be a conventional map, for every flight would throw it into disarray. At first, the travelers might remove from their map those parts of the Airport that change when taking a plane, because they could be considered nonessential to the Airport or even fictional. However, this would quickly

POPULAR SUMMARY

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leave them with a blank sheet, similar to the situation where there are many essentially human properties that still are not shared by every human. Therefore, a more informed procedure is called for. At every object on the map, the travelers could add a footnote containing a table, indexed by flight numbers, that lists the places on the map where the object would move if one took the corresponding flight. If the object disappears after some flight, that flight is simply not listed with the chair. In mathematics, by the way, such maps with footnotes are a special case of what is called a groupoid, and most of this thesis is concerned with the geometrical study of a particular class of groupoids. When one tries to use such a map for navigation, one would like to know the distance between two points in the Airport. This, however, is difficult to define clearly, because the walking distance between two objects, say a particular bench and the newspaper stand, has changed every time one steps out of an airplane. There is a way to extend the map with extra data, however, such that a consistent notion of distance can be defined. This was dubbed ’background-independent geometry’ by its inventors, Connes and Moscovici, and the first chapter of this thesis investigates their construction. An important part of mathematical geometry is concerned with associating numbers to shapes in such a way that the numbers do not change when the shape is deformed in a way that keeps is essentially the same. Similarly, most cars have four wheels and different cars can vary wildly without this simple fact changing. In the present setting, there is a way to use the backgroundindependent geometry to define many such numbers that capture the shape of the Airport. However, these numbers are defined in a very abstract way and one needs to understand the geometry of the entire Airport well to calculate them. Therefore, one tries instead to see how these ’global’ quantities are related to tiny, local movements on the Airport, like bats guesssing the shape of a cave by sonar. That is the theme of the second chapter of this thesis. Of course, the beauty and inhumanity of mathematical abstraction is that this thesis is not about airports at all, like the geographer in Le Petit Prince who is, in the end, not interested in the land he is charting. It could just as well be about the geometry of spacetime in the presence of gravity, which some people hope to describe in a similar framework, or about the space of possible trajectories of a double pendulum (or in fact of any classical mechanical system). I feel that there is a long road to walk before attaining harmony between the playful and misleading exactness of mathematical thinking and the full realization of our vast ignorance about the world and about ourselves. Therefore, my heart goes out to the continued struggle of the humanities in their venerable endeavour to dazzle us into balance.

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INTRODUCTION

Acknowledgements Om te weten dat er woord voor woord een andere taal is daarom schrijf ik – Jan Arends

I wish to thank Hessel Posthuma for the time and care he has taken to aid me, and for the many insights he provided. I am very grateful to Raf Bocklandt for reviewing my thesis, which most obviously could not have been written without the work of Connes and Moscovici. The front picture was taken from J. McKenzie Alexander, J. Himmelreich, and C. Thompson: ”Epistemic landscapes, optimal search and the division of cognitive labor.” Philosophy of Science, 2014. Finally I wish to thank Laura San Giorgi for her support and advice, and for just being wonderful.

CHAPTER 1

The hypoelliptic signature operator on foliations Let M be a smooth manifold and Γ ⊂ Diff(M ) be a pseudogroup of local diffeomorphisms. The smooth structure of M and the action of Γ are encoded in the crossed product algebra A = Cc∞ (M ) ⋊ Γ, which is isomorphic to the convolution algebra of the groupoid M ⋊ Γ. In the noncommutative analog of Riemannian geometry, a geometric space is modelled by a spectral triple: a representation of A in the Hilbert space L2 (M ) combined with an operator D ∈ B(L2 (M )), analogous to the Dirac operator, that represents the line element. However, in the present setting, the action of Γ may be such that it does not preserve any geometric structure on M at all, except for its smoothness. Thus, such an operator D may not exist. Moreover, as discussed in section 7, the pairing between the K-theory associated to A and it s periodic cyclic cohomology is not everywhere well defined, so even if we are able to construct K-cycles on A, we would have to resort to some additional geometry in order to describe the index maps analytically. The challenge, then, is to find an algebra AP that, together with a representation in a Hilbert space, captures the smooth structure of M and the action of Γ. In [CM95], this part of the problem was solved. Connes and Moscovici constructed a spectral triple canonically associated to M , using a prolonged groupoid equipped with a K-theoretic Thom isomorphism. The associated Fredholm module provides an element of the K-homology of M ⋊ Γ. The present chapter deals with this construction, which is mostly geometric and contains some interesting ingredients, such as a canonical almost-diffeomorphism-invariant metric attached to any G-structure, and the Heisenberg calculus on integrable foliations. 1. The almost isometric action of Diff(M ) on the frame bundle F Because the action of Diff(M ) does not, in general, preserve any Riemannian structure on M , we will pass to a canonically defined bundle over M on which it does. We will construct Riemannian metrics on the frame bundle Fr(M ) over M , such that Γ acts almost isometrically on Fr(M ), in the sense that there is a Γ−invariant subbundle V of Fr(M ) such that Γ acts isometrically on both V and T Fr(M )/V , although V is not Riemannian as a foliation. This will suffice to construct the algebra and Hilbert space, and to form the 7

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1. THE HYPOELLIPTIC SIGNATURE OPERATOR ON FOLIATIONS

vertical part of the hypoelliptic signature operator, that together form the spectral triple encoding the smooth structure of M and the action of Γ. From the perspective of G-structures, the trivial structure on a smooth manifold M is its frame bundle, a principal GLn -bundle Fr(M ). π

Definition 1.1. The frame bundle Fr(M ) − → M is the bundle of linear ∼ maps p : Rn − → Tm M . ϕ

→ V ⊂ Rn , π −1 (U ) ≃ GLn ×V by the map p 7→ (Locally, above M ⊃) U − (dϕ ◦ p), (ϕ ◦ π)(p) , so Fr(M ) is indeed a fiber bundle, with fiber GLn . With the natural right action given by rA (p) := p ◦ A for A ∈ GLn , Fr(M ) is a principal GLn -bundle. Furthermore, the following proposition shows that points on Fr(M ) are naturally related to Riemannian structures on M . π

P M be the quotient Fr(M )/O(n) by the right Proposition 1.2. Let P −−→ action given above. There is a bijective correspondence between sections g:M − → P of πP and Riemannian metrics on M .

( ) Proof. Let G ∈ Γ M, T ∗ M ⊗ T ∗ M be a metric on M . Then, let O be the of all p ∈ Fr(M ) such that xt y = ( subbundle ) of Fr(M ) consisting Gπ(p) p(x), p(y) for all x, y ∈ Rn , that is, the bundle of all orthonormal frames. Clearly, O is a principal O(n)-bundle. Define the section g : M → − P pointwise by m − → Om /O(n). Conversely, let g : M − → P be a section of P . Define ( the bundle O) by Om := {p ∈ Frm (M ) | p · O(n) = gm }. Then, let G ∈ Γ M, T ∗ M ⊗ T ∗ M be ( ) ( )t the metric Gm v, v ′ = p−1 (v) p−1 (v ′ ) for any p ∈ Om . This construction is inverse to the one in the previous paragraph, and vice versa. □ In order to model the action of Γ ⊂ Diff(M ) on M , we must lift it to Fr(M ) in the natural way. Definition 1.3. Let ϕ ∈ Diff(M ). The left action of Diff(M ) on Fr(M ) is lϕ : Fr(M )m − → Fr(M )ϕ(m)

lϕ (p) := (dϕ) ◦ p.

We will consider more generally the case where M carries a Γ-invariant Gstructure, that is, a principal G-subbundle F ⊂ Fr(M ), where G ⊂ GLn is a linear group and F is preserved by {lϕ | ϕ ∈ Γ}. For example, if G = O(n), we have a Riemannian metric preserved by Γ, cf. remarks 1.22 and 1.32 and the proposition above. Another example is G = Spk (R), when dim M = 2k, which is an almost symplectic structure preserved by Γ and where the bundle F consists of all syplectic frames, that is, the fiber Fm consists of all Darboux bases of the symplectic vector space Tm M . For the general theory of G-structures, cf. [KN96]. In case M is just a smooth manifold, set F = Fr(M ).

1. THE ALMOST ISOMETRIC ACTION OF Diff(M ) ON THE FRAME BUNDLE F

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If we are to replace M by F as the object of study, because it naturally carries part of the requisite structure, we must ask how this still bears on the original problem of studying Cc∞ (M )⋊Γ. The answer lies in the following theorem. Theorem 1.4 ([Con86, Lemma 5.3]). Let F be a principal G-bundle over M and let K be the maximal compact subgroup of G. Then, there is a Thom isomorphism ∼

β : Ki (C0 (M ) ⋊ Γ) − → Ki+rank(F/K) (C0 (F/K) ⋊ Γ). Now, we could proceed using P := F/K, as the authors of [CM95] did originally. However, since O(n) is not a normal subgroup of GLn , and in general K is not a normal subgroup of G, the construction in terms of P is geometrically far less elegant, and does not lend itself naturally to the continued treatment of the subject along the lines of [CM98]. Therefore, we follow the latter in constructing everything in terms of F , and restrict to K-invariant objects afterward. Now, lemma 1.5, below, provides a hint that a Γ-invariant Euclidean structure may exist on a subbundle of T F . Let θ ∈ Ω1 (F ; Rn ) be the tautological form θp : Tp F − → Rn , X 7→ p−1 (dπ(X)). Lemma 1.5. Let ϕ ∈ Γ. The tautological form θ has the property lϕ∗ θ = θ Proof. Note that π ◦ lϕ (p) = ϕ(π(p)). Then, (lϕ∗ θ)p (X) = (lϕ p)−1 (dϕ ◦ dπ(X)) = θp (X). □ Remark 1.6. In fact, lemma 1.5 holds in the opposite way as well: if σ ∈ Diff(F ) preserves θ, then σ = dϕ for some local isomorphism of G-structures ϕ ∈ Diff(M ). The pullback of the Euclidean metric gRn through θ is a Γ-invariant section of T ∗ F ⊗ T ∗ F , but it is clearly highly degenerate, with kernel V := ker dπ ⊂ T F. Definition 1.7. The normal bundle of the codimension dim M foliation V on F is the quotient bundle N = T F/V , with quotient map q : T F − → N. It is equipped with the Euclidean structure gN |p := θp∗ gRn . The subbundle V of T F defines an integrable foliation of F , as dπ[v, w] = 0 if dπ(v) = dπ(w) = 0. As the action of Γ preserves V , it descends to the normal bundle, and by the previous lemma that action is isometric. Let g be the Lie algebra of G. The Euclidean metric on V can be pulled back from any nondegenerate form on g using the infinitesimal action of the Lie algebra.

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Proposition 1.8. There is a canonical Euclidean structure on V = ker dπ which is invariant under the actions of Γ and G. ∼

Proof. The map Lp : g 7→ p · g is an isomorphism G − → Fπ(p) . Thus, ∼ the infinitesimal action dLp of g is an isomorphism g − → Vp . Any right G-invariant metric gG on G can then be pulled back to V by ( )∗ gV |p := dL−1 gG . p If R denotes the right action of G on itself, Lp ◦ lϕ = Rq−1 ◦dϕ◦p ◦ Lq for any q ∈ π(lϕ (p)). Therefore, G-invariance of gG implies Γ-invariance of gV . There is at least one right invariant metric on G: as g ⊂ gln (R) by definition, we take the Hilbert-Schmidt norm ⟨x, y⟩g := Tr(x∗ y) and turn into a G-invariant metric ⟨X, Y ⟩TA G := ⟨dRA−1 (X), dRA−1 (Y )⟩g . □ Thus, the foliation V and its normal bundle N carry canonical (i.e. defined with reference only to F ) Euclidean metrics, with respect to which Γ acts isometrically. Finally, the action of G and the tautological form θ yield a volume form on F , depending only on orientations of g and Rn . Proposition 1.9. There is a canonical Γ-invariant volume form on F . Proof. Let ω ∈ Ω1 (F, g) be an auxiliary connection on F . Choosing orthonormal bases of G and Rn , the volume form is ∧ ∧ vol = ωα ∧ θk . α

k

First, note that provide a trivialization of F ∗ . Then, as lϕ∗ preserves V , lϕ∗ ω α − ω α = γ(ϕ)αk θk , and thus lϕ∗ vol = vol. Last, if ω ′ is another connection, ω − ω ′ must vanish on V as ω(dLp (A)) = ω ′ (dLp (A)) = A, and thus ω α − ω ′ α = fkα θk . Thus, vol is independent of the choice of ω. □ {ω α , θk }

π

Thus, we have found a foliation V of F − → M and Euclidean structures on V and N such that Γ acts by isometries. We will proceed by defining those parts of the spectral triple modeling M/Γ that require no structure on F beyond what has now been provided. 2. The algebra AP := Cc∞ (P ) ⋊ Γ and its representation ( ) Clearly, C0 (F/Γ) is in the center of C0 F ⋊ Γ , via pullback along the quotient map F − → F/Γ. If the pseudogroup Γ were to act( freely and ) properly, C0 (F/Γ) would be strongly Morita equivalent to C(0 F ⋊ )Γ , because it would be an abelian subalgebra of finite index, so C0 F ⋊ Γ would satisfy the prequisites of [Rie74, Thm 8.10]. Thus, in the present case of an action that is not proper, we are led to the suggestion that Cc∞ (F ) ⋊ Γ might be a natural candidate to model the algebra of coordinates on F/Γ.

2. THE ALGEBRA AP := Cc∞ (P ) ⋊ Γ AND ITS REPRESENTATION

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Definition 1.10. The algebra)}A := Cc∞ (F )⋊Γ is generated by the symbols { ∗ ( f Uϕ ϕ ∈ Γ, f ∈ Cc∞ dom(ϕ) subject to ( ) Uϕ∗ f = lϕ∗ f Uϕ∗ ,

f g(p) = f (p)g(p),

Uϕ∗ = Uϕ−1 ,

f ∗ (p) = f (p),

equipped with the equivalence relation f Uϕ∗ = gUψ∗ iff f and ϕ agree with g and ψ when restricted to supp f ∪ supp g. Equivalently, A is the convolution algebra of the étale groupoid F ⋊ Γ ⇒ F with the set F of objects and morphisms Γ ∋ ϕ : p → − lϕ p. This is clear because the topology on F ⋊ Γ is generated by the basis {[p, ϕ] | p ∈ dom ϕ, ϕ ∈ Γ}, where [q, ϕ] denotes the arrow defined by the germ of ϕ at q. The right action of G on F descends to a left action by automorphisms using precomposition on C ∞ (F ), which clearly commutes with the right action of Diff(M ). Thus, Definition 1.11. AP ⊂ A is the subalgebra generated by all K-invariant elements of A under the action of A ∈ K on f Uϕ∗ ∈ A: ) ( ∗ ) ( (f ) Uϕ∗ A · f Uϕ∗ = rA Clearly, AP ≃ Cc∞ (P ) ⋊ Γ, which explains the notation. If F had been equipped with a Riemannian metric and Γ had acted isometrically, the natural representation of AP would have been on ΩL2 (F ), the L2 space of forms on F . The space HP presented below is non-canonically isomorphic to ΩL2 (F )(K) , as shown in section 4. Note that the right G-action on N is trivial, because π ◦ rA = π. ( )(K) Definition 1.12. HP := Γ F, Λ• V ∗ ⊗ Λ• N ∗ is the bigraded ( Hilbert ) space of K-invariant forms (on F , with respect to the )action A · ν ⊗ η = ) ( ∗ ν ⊗ η, for A ∈ K, ν ∈ Γ F, Λ• V ∗ , η ∈ Γ F, Λ• N ∗ , endowed with the rA inner product ∫ ′ ′ ⟨ν ⊗ η, ν ⊗ η ⟩ = ⟨ν, ν ′ ⟩V ⟨η, η ′ ⟩N vol . F

In the end, HP is non-canonically isomorphic to Ω• (F )(K) , but the present definition allows us to naturally define the action of AP , below. See also remark 1.30. Definition 1.13. The action of AP on HP is given by ( ) f Uϕ∗ ν ⊗ η = f lϕ∗ ν ⊗ lϕ∗ η. In light of lemmata 1.5 and 1.8, Uϕ∗ acts unitarily.

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3. The vertical signature operator QV ◦

Because V is an involutive foliation of F , the Bott connection ∇ [BGJ72] supplies us with a partial covariant exterior derivative d∇◦ . Combined with the partial covariant exterior derivative d∇ ˜ induced by a choice of connection, see below, this will form the hypoelliptic signature operator Q. By the natural isomorphism V ∗ ⊗ N ∗ ≃ Hom(V, N ∗ ), we may identify HPr,s with Γ(Hom(Λr V, Λs N ∗ ))(K) , that is, the space of K-invariant Λs N ∗ -valued forms on V . Recall that the fiber g of V acts on T F and, thus, on N . Definition 1.14. The vertical differential d∇◦ : HPr,s − → HPr+1,s is the covari◦

ant exterior derivative with respect to the induced Bott connection ∇ (see A.2 and A.3), under the isomorphism HPr,s ≃ Γ(Hom(Λr V, Λs N ∗ ))(K) . ( ) Remark 1.15. In terms of Γ ΛV ∗ ⊗ ΛN ∗ , if h is any section of q : T F → − T F/V , ) ( )( (−1)s d∇◦ α V1 , ... , Vr+1 ; N1 , ... , Ns = r+1 ∑ ( ) (−1)j+1 Vj α V1 , ... , Vˆj , ... , Vr+1 ; N1 , ... , Ns ) j=1

∑ ) ( + (−1)i+j α [Vi , Vj ], V1 , ... , Vî , ... , Vˆj , ... , Vr+1 ; N1 , ... , Ns i 0. Proof. We start by proving that all elements of Gk0 are ( ) of the form k k p p above. Let j = j0 (ψ) ∈ G0 . Let D ψ( ∈ Γ V, S V ∗ ⊗ V ) be the p-th iterated derivative Dp ψ(x)(v 1 , ... , v p ) := vi11 · · · vipp ∂i1 · · · ∂ip ψ (x), so that D(Dp ψ) = Dp+1 ψ. By definition, Dψ(x) ∈ G for all x ∈ dom ψ.( Thus, ) D2 ψ(x) ∈ V ∗ ⊗TDψ(x) G and in general Dp+1 ψ(x) ∈ S p V ∗ ⊗TDp−1 ψ(x) · · · (TDψ(x) G) .

1. THE GROUPOID COCYCLE γ

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Because TDψ(0) G = g and TA g = g for all A ∈ g, we may conclude that Dp+1 ψ(0) ∈ g(p) , and by definition j0k (ψ)(x) = Dψ(0)(x) + · · · +

1 k D ψ(0)(x, ... , x). k!

Now we must prove that all such polynomials are contained in Gk0 . To that end, let i0 be the map that sends a formal polynomial to its evaluation as a function on V . Let j be such a polynomial in consideration and let ψ = i0 (j). b,j ,...,j Then, d(ψ)ab (y) = (c1 )ba + · · · + (ck )a 1 k−1 yj1 · · · yjk−1 so dψ(0) = id, and ∑ d d 1 ,...,jr dψ(x + ty)|t=0 = (xj + tyj1 ) · · · (xjr + tyjr )|t=0 ∈ g, (cr+1 )b,j a dt dt 1 k−1 r=0

so dψ(x) ∈ G for all x. Restricting to a neighbourhood of 0 to ensure invertibility completes the proof that i0 (j) ∈ Diff G □ 0 (V ). Recall the vector fields {Xk , Yα } from section 1 associated to bases of Rn and g, respective ly, using a connection, and the groupoid 1-cocycle γ(ϕ) = ω ◦ dlϕ ◦ X. Now, we are finally in the position to investigate the properties of γ. For p, q ∈ F , let ΓF (p : q) := {ϕ ∈ ΓF | dϕ(p) = q}. Proposition 2.5. The connection ω induces an isomorphism jpr (ΓF (p : q)) ≃ Gr0 for any p, q ∈ F . Proof. Using the affine connection ∇M induced by ω, any p ∈ F allows expπ(p)

p

us to define normal coordinates Rn − → Tπ(p) M −−−−→ M . If x and y are normal around π(p), π(q), the germ of the unique affine map sending p to q is the germ of x−1 ◦ y. Therefore, the map x∗ y ∗ : Diff G → ΓF (p : q) is 0 (V ) − a bijection on germs. Thus, it descends to an isomorphism Gr0 − → jpr (ΓF (p : q)). □ (r+1)

Definition 2.6. For all q, define γp p ∈ F , r ≥ 0, by

: ΓF (p : q) − → Hom(V ⊗r+1 , g), for

( ) γp(r+1) (ϕ) : v0 , ... , vr 7→ LX(vr ) · · · LX(v1 ) γ(ϕ) p (v0 ),

with vi ∈ V , ϕ ∈ ΓF (p : q). (r)

Clearly, γp (ϕ) depends only on jpr (ϕ). Proposition 2.7. By proposition 2.5, γp descends to a map Gr0 → − Hom(V ⊗r+1 , g). Then, (r)

(r)

lies in g(r) . ∑ ⊕ (r) (ii) For all p, q ∈ F , the map kr=1 γp : Gr0 − → kr=1 g(r) is a bijection. (i) The image of γp

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2. THE HOPF ALGEBROID HF AND THE INDEX THEOREM

Proof. (i) We use coordinates as in 2.5. Take χij (f ) := dxi (f (ej )) and υji (f ) := dy i (f (ej )) around p and q, so that χ(p) = υ(q) = id. Then, ωij (q ′ ) = (υ −1 (q ′ ))ki dυkj , θk (p′ ) = (χ−1 (p′ ))kl dxl and d(lx∗∗ y∗ ψ υij )(p′ ) = d(∂i ψ l )χjl (p′ )) + ∂i ψ l d(χjl )(p′ ), so we have j γik (p′ )(x∗ y ∗ ψ) = (χ−1 (p′ ))li ∂µ (∂l ψ m )χjm χµk .

Furthermore, we immediately have j γik|k (p′ )(x∗ y ∗ ψ) = (χ−1 (p′ ))li ∂µr · · · ∂µ1 ∂µ (∂l ψ m )χjm χµk χµk11 ... χµkrr , 1 ,...,kr j γik|k (p)(x∗ y ∗ ψ) = ∂kr · · · ∂k1 ∂k (∂i ψ j ) = D0r+2 (ψ)ji,k,k1 ,...,kr . 1 ,...,kr (r+1)

Therefore, γp in g(r+1) .

(ϕ) is symmetric in {i, k, k1 , ... , kr } and thus lands (r)

(ii) Because we identified γp with D0r+1 , above, this follows easily from proposition 2.4. □ Definition 2.8. For all k ≥ 1, choose a basis {ϵk1 , ... , ϵkdim g(k) | ϵkµ :=

eαµ,i1 ,...,ik eα ⊗ ei1 ⊗ · · · ⊗ eik } of g(k) , with eα ∈ g and eij ∈ V ∗ . Then, define the abelian subalgebra d ⊂ End(AF ) as the span of {δkµ }, where ( ) δkµ (Uϕ∗ ) := γ (k) (ϕ)µ Uϕ∗ ,

and γ (k) (ϕ)µ is, by definition, the smooth function on F given at p ∈ F by (k) the projection onto ϵkµ of γp (ϕ). For brevity, we will write δ1µ = δ µ , so that γkα (ϕ) = eαµ,k γ (1) (ϕ)µ . Because of 2.7, dim d = dim G∞ (0) . In particular, if G is of finite type, meaning that g(k) = 0 for some k (and thus for all l ≥ k), d will be finite-dimensional. 2. The Hopf algebroid HF and the characteristic map In general, when ω need not be flat, there is a natural Hopf algebroid HF of functions and vector fields on F acting on AF , and the characteristic map can be defined on this level. 2.1. The algebra HF and its representation on AF . In differential geometry, the algebra of differential operators is generated by functions and vector fields on a manifold. This motivates the following definition. Definition 2.9. The algebra HF ⊂ End(A ( F ) is )generated by functions ∞ r ∈ r := C (F ) and vector fields Z ∈ Γ F, T F , acting on AF in the following ways: ( ) ( ) s(r) f Uϕ∗ := rf Uϕ∗ t(r) f Uϕ∗ := (lϕ∗ r)f Uϕ∗ Z(f Uϕ∗ ) := Z(f )Uϕ∗ ,

2. THE HOPF ALGEBROID HF AND THE CHARACTERISTIC MAP

29

Clearly, the maps s and t turn HF into a r-bimodule by r · h · r′ := s(r) ◦ t(r) ◦ h, and d ⊂ HF commutes with s and t. The next proposition, due to prop. 2.7, motivates the definition of HF as lying inside End(AF ) as opposed to inside End(Cc∞ (F ) ⋊ Γ), because it simplifies the treatment of HF . The point is that Γ, in general, does not provide sufficiently many jets of diffeomorphisms to turn d into a free algebra, unlike ΓF . Let us call an action of a set B on AF pointwise surjective if for all pairwise distinct h1 , ... , hk ∈ B and all p, q ∈ F the map evp ◦ (h1 , ... , hk ) : AF |(p:q) → − k ∗ C is surjective, where AF |(p:q) = {f Uϕ ∈ AF | lϕ (p) = q}. In particular, this implies (’multi-’)faithfulness (see prop. 2.13). Proposition 2.10. The set B := {δkµ Xl1 · · · Xlb Yα1 · · · Yαc | k ≥ 1, li ≤ li+1 , αi ≤ αi+1 } acts (as above) pointwise surjectively on AF . Proof. We start with d. Here, the result follows immediately from 2.7: ⊕ for any gkµ ekµ ∈ kr=1 g(r) there is a ϕ ∈ ΓF (p : q) such that δkµ (ϕ)(p) = gkµ . Now for ⟨Xk , Yα ⟩. Let ϕ be any local diffeomorphism from a neighbourhood of π(p) to a neighbourhood of π(q), such that lϕ (p) = q, as before. As df = Xk f θk + Yα f ω α , the equations Xk f = fk , Yα f = fα , f (p) = c have a simultaneous local solution f on a neighbourhood U ⊂ dom ϕ of p for any set of smooth functions {fk , fα } and c ∈ C. Then, b := f Uϕ∗ is an element of AF with the required properties. By induction on the number of X and Y appearing, applying the above repeatedly (we can keep ϕ fixed), the result follows for {Xl1 · · · Xlb Yα1 · · · Yαc } ⊂ B. Because the procedure for d (finding a suitable diffeomorphism ϕ) and that for ⟨Xk , Yα ⟩ (finding a suitable function f ) commute, the result follows. □ Proposition 2.11. The set B provides a Poincaré-Birkhoff-Witt basis of the r-bimodule HF . In particular, HF is a free left module over r ⊗ r. Proof. Linear independence of B over r follows from 2.10. Thus, we must prove that B spans HF . Let H be the submodule spanned by {s(r)t(r′ )B | r, r′ ∈ r}. Recall that ( ) Yα (γkβ (ϕ)) = cβαγ γkγ (ϕ) +(ρlαk γlβ (ϕ). Thus, [Γ F, T F , d] ⊂ H. Because ) {Xk , Yα } trivialize T F , Γ F, T F ⊂ H. Furthermore, [Z, s(r)] = s(Z(r)) ∈ H. Finally, [Xk , t(r)]f Uϕ∗ = t(dlϕ Xk (r))(f Uϕ∗ ) = t(γkα (ϕ)Yα (r))(f Uϕ∗ ) and [Yα , t(r)] = t(Yα (r)). Thus, H = HF . □

30


Now we consider the product on AF and its interaction with HF . Proposition 2.12. The representation of HF satisfies Yα (a1 a2 ) = Yα (a1 )a2 + a1 Yα (a2 ) µ

µ

s(r)(a1 a2 ) = s(r)(a1 )a2

µ

δ (a1 a2 ) = δ (a1 )a2 + a1 δ (a2 ) Xk (a1 a2 ) = Xk (a1 )a2 + a1 Xk (a2 ) +

t(r)(a1 a2 ) = a1 t(r)(a2 ) eαµ,k δ µ Yα (a2 )

Proof. This follows directly from Uϕ∗ Y = Y Uϕ∗ , Uϕ∗ X = XUϕ∗ + γY Uϕ∗ , the cocycle property of γ and t(r)(Uϕ∗2 ϕ1 ) = Uϕ∗1 t(r)(Uϕ∗2 ). □ The next proposition is important and prepares the stage for the characteristic map HC ∗ (HF ) − → HC ∗ (AF ). As as(r)(b) = t(r)(a)b, the maps s, t turn AF into a r-bimodule algebra, with r · a · r′ := s(r)(t(r)(a)). Because the bimodule structures of AF and HF commute, in the sense that (r · h · r′ )(a) = r · h(a) · r, there is a natural action of HF ⊗r HF on AF ⊗r AF . rk rk Proposition 2.13. The linear map T : H⊗ − → Hom(A⊗ , AF ) defined by F F

T (h1 ⊗r · · · ⊗r hk )(a1 ⊗r · · · ⊗r ak ) := h1 (a1 ) · · · hk (ak ), is injective. Proof. The map is well defined on tensors because t(r)(a)b = as(r)(b) and multiplication is multilinear. ∑ Assume that i h1i ⊗r · · · ⊗r hki ∈ ker T . By multilinearity of the tensor product, we may assume that each hji is of the form s(rij )t(r′ jj )Bij , where Bij ∈ B. We will use 2.10 repeatedly. Pick p1 , ... , pk in F , and fix i. Then, using 2.10, pick a1 ∈ AF such that Bi1 (a1 ) = fi1 Uϕ∗1 , with fi1 (p1 ) = 1 and i

lϕ1 (p1 ) = p2 , and Bj1 (a1 ) = 0 for Bj1 ̸= Bi1 ∈ B. Repeating this process, for i all p1 , ... , pk and all i we find a1 , ... , ak such that 0 = h1i (a1 ) · · · hki (ak ) = ri1 (p1 )r′ i (p2 )ri2 (p2 ) ... rik (pk−1 )r′ i (pk ). 1

k

This, then, clearly implies that s(ri1 )t(r′ i )h1i (a′1 ) ⊗r · · · ⊗r s(rk )t(r′ k )hki (a′k ) = 0 1

for all {a′j }.

□

Remark 2.14. As K is, in general and in the universally applicable case G =(GLn , not ) a normal subgroup of G, we cannot simply take the quotient of Γ F, T F by its Lie algebra, so the action of HF is most naturally defined on AF , instead of on AP . Instead, as remarked previously, we will have to resort to relative cyclic cohomology of Hopf algebras later on.


31

2.2. The left bialgebroid structure of HF . Definition 2.15. A left bialgebroid (in the sense of [Böh03]) over a ring R is a R-bimodule algebra A (with left and right representations s, t of R) equipped with a coassociative bimodule map ∆ : A → − A ⊗R A such that for all r ∈ R and all a ∈ A, ∆(aa′ ) = a(1) a′(1) ⊗R a(2) a′(2) ,

a(1) t(r) ⊗R a(2) = a(1) ⊗R a(2) s(r),

where we use sumless Sweedler notation, and a homomorphism ϵ : A → − R such that ϵ(as(ϵ(b))) = ϵ(ab) = ϵ(at(ϵ(b))). In order to turn the algebra representation HF − → End(AF ) into a bialgebroid representation, we need a coproduct ∆ : HF − → HF ⊗r HF such that diagram 1 commutes: we need (h(1) a1 )(h(2) a2 ) = h(a1 a2 ), in sumless Sweedler notation. AF ⊗r AF

µ

∆(h)

AF ⊗r AF

AF h

µ

AF

Figure 1. Representation of HF on AF , with product µ The reason that we will obtain a bialgebroid rather than a bialgebra is as follows. By 2.12, we need to set ∆(s(r)) = s(r) ⊗ 1. This, however, is not coassociative. Thus, the coproduct is only associative when defined as ∆ : HF − → HF ⊗r HF . As r does not lie in the center of HF (the subspace that does is only closed under multiplication when ω has constant curvature) , HF is not a bialgebra over r, and must therefore be defined as a bialgebroid. Proposition 2.16. There is a unique bimodule map ∆ : HF → − HF ⊗r HF that makes figure 1 commute. In particular, ∆(Yα ) = Yα ⊗r 1 + 1 ⊗r Yα ,

∆(s(r)) = s(r) ⊗r 1,

∆(δ ) = 1 ⊗r δ + δ ⊗r 1,

∆(t(r)) = 1 ⊗r t(r),

µ

µ

µ

∆(Xk ) = Xk ⊗r 1 + 1 ⊗r Xk + ekµ,α δ µ ⊗r Yα , for all r ∈ r. Proof. Existence and uniqueness follow from 2.13, because µ(∆(h)(a⊗r b)) = T (∆(h))(a⊗r b). The particular formulas follow directly from 2.12. □ Proposition 2.17. The bimodule map ∆ : HF → − HF ⊗r HF defines a coproduct on HF . That is, ∆ is coassociative, for all r ∈ r h(1) t(r) ⊗r h(2) = h(1) ⊗r h(2) s(r) and ∆(hh′ ) = h(1) h′(1) ⊗r h(2) h′(2) .

32


Proof. By the previous proposition and 2.11, coassociativity of ∆ is equivalent to associativity of AF : ( ) T ∆ ⊗r 1 ∆(h)(a1 ⊗r a2 ⊗r a3 ) = ∆(h)(a1 a2 ⊗r a3 ) = h(a1 a2 a3 ) ( ) T 1 ⊗r ∆ ∆(h)(a1 ⊗r a2 ⊗r a3 ) = ∆(h)(a1 ⊗r a2 a3 ) = h(a1 a2 a3 ) Furthermore, T (h(1) t(r)⊗r h(2) )(a1 ⊗r a2 ) = T (∆(h))(a1 ·r⊗r a2 ) = T (∆(h))(a1 ⊗r r·a2 ) = T (h(1) ⊗r s(r)h(2) )(a1 ⊗r a2 ). Finally, T (∆(hh′ )) = hh′ ◦µ = h◦h′ ◦µ = h ◦ µ ◦ ∆(h′ ) = µ ◦ ∆(h) ◦ ∆(h′ ) = T (h(1) h′(1) ⊗r h(2) h′(2) ). By injectivity of T , the statement follows. □ There is a natural action of HF on C ∞ (F ), as HF is generated by vector ∗. fields and smooth functions on F and the subalgebra d acts as 0 on Uid Thus, we can define ϵ : HF − → r by ϵ(h) := h(1). Proposition 2.18. (HF , r, s, t, ∆, ϵ) is a left bialgebroid in the sense of [Böh03]. That is, on top of the previous proposition, ϵ(1) = 1 and ϵ(as(ϵ(b))) = ϵ(ab) = ϵ(at(ϵ(b))). ∏ ∏ Proof. We have ϵ( i s(ri )t(r′ i )) = i ri r′ i for ri , r′ i ∈ r and clearly ϵ = 0 outside U (s(r) ∪ t(r)). In particular ϵ(1) = 1. Moreover, for a = ∏ ∏ ∏ a ′a b ′b a ′a b ′b i s(ri )t(r i ), b = i s(ri )t(r i ), all sides of the last formula equal ij ri r i rj r j . □ 2.3. The invariant trace, the twisted antipode and the subalgebra K ≃ U (k). The geometric constructions in section 1 led to the definition of a Γ-invariant volume form vol on the principal G-bundle F . This yields a trace on AF , given by ∫ ∗ τ (f Uid ) := f vol τ (f Uϕ∗ ) = 0 for ϕ ̸= id F

where the trace property τ (ab) = τ (ba) follows from the Γ-invariance of vol. Proposition 2.19. The bilinear form (a, b) − → τ (ab) is nondegenerate. That is, the map AF − → A∗F , a − → τ (−, a) is injective, and in particular τ (b, a) = 0 for all b implies a = 0. ∑ ∗ Proof. Let a ∈ AF . We have a = ϕ fϕ Uϕ . For any ϕ and any open U ⊂ supp fϕ , let RϕU be the set of all ψ such that U ⊂ supp fψ and ∑ ψ|U = ϕ|U . Then, let gϕU equal ψ∈RU f¯ψ |U and let ϕ′ := ϕ−1 |ϕ(U ) . Then, ϕ ∫ ∑ τ (gϕU Uϕ∗′ , a) = U | ψ∈RU fψ |2 vol. Clearly, if the latter vanishes for all ϕ, U , ϕ so does a. □ Lemma 2.20. Let S : HF − → HF be the antihomomorphism uniquely defined by S(Xk ) = −Xk + eαµ,k δ µ Yα

S(s(r)) = t(r)

S(Yα ) = −Yα + Tr(eα )1

S(t(r)) = s(r)

S(δ ) = −δ . µ

µ


33

Then, (i) τ (h(a)b) = τ (aS(b)) for all a,b ∈ HF . (ii) S 2 = id. Proof. First of all, that S is uniquely defined follows from 2.11. (i) On Xk , we first use 2.12 and then τ (δ µ (ab)) = 0. The latter also leads to the formula for δ µ . Because t(r)(a)b = as(r)(b) and τ (s(r)(a)b) = τ (at(r)(b)), the results for s(r) and t(r) follow. Fid d nally, τ (Yα (f )) = dt τ (re∗teα f ) = dt det(eteα )τ (f ) = Tr(eα )τ (f ) and ∗ Yα commutes with Uϕ . By the antihomomorphism property, the result extends to all of HF . (ii) That S 2 = id is clear on the generators and extends to HF by 2.11. □ Definition 2.21. A Hopf algebroid (in the sense of [Lu96]) is a left bialgebroid (A, R, s, t, ∆, ϵ), equipped with an antihomomorphism S : A − → A and a section ξ : A ⊗R A − → A ⊗ A such that S(t(r)) = s(r) for all r ∈ R and µA (S ⊗ 1)∆ = t ◦ ϵ ◦ S,

µA (1 ⊗ S)ξ ◦ ∆ = s ◦ ϵ.

Proposition 2.22. For any section ξ : HF ⊗r HF − → HF ⊗HF , (HF , r, s, t, ∆, ϵ, S) is a Hopf algebroid in the sense of [Lu96]. Proof. By the bialgebra axiom and the antihomomoprhism property of S, it will suffice to check the properties for the generators of the PBW basis. Clearly the relations hold for s(r) and t(r), r ∈ r. Furthermore, both formulae vanish on ker ϵ, and the choice of ξ is irrelevant because ∆ maps into ∑ s(r )B □ i i ⊗r Bj t(rj ) ⊂ HF ⊗r HF , on which ξ is fully predetermined. ij Remark 2.23. In particular, HF is also a Hopf algebroid in the sense of [BS04]. However, it is easier to prove this indirectly through the less general definition given in [Lu96]. Remark 2.24. The AF -module Hilbert space L2 (F, vol) is generated, under the GNS construction, by τ . That is, if J is the ideal {a ∈ AF |τ (a∗ a) = 0}, then L2 (F, vol) is the Cauchy completion of AF /J under the norm ⟨a, b⟩ = τ (b∗ a). In section 2.4 we will be concerned with Hopf-cyclic classes, that is, cyclic ∑ α 1 α n cocycles ϕ on AF such that ϕ(a0 , ... , an ) = τ (a 0 h1 (a ) ... hn (a )), for α α {hi } ∈ Hn . For now, we can already see that the fundamental class [F/Γ] might be Hopf-cyclic, as ∫ ∗  f l∗ (df ) ∧ · · · ∧ (l ... l 0 ϕ0 1 ϕ0 ϕk−1 ) dfk ϕ0 ... ϕk = 1 [F/Γ](f0 Uϕ∗0 , ... , fk Uϕ∗k ) = F  0 otherwise

34


and lϕ∗ df = df − γkα (ϕ)Yα f θk whereas ∑ sgn(σ)Yα1 fσ1 · · · YαN fσN X1 fσN +1 · · · Xn fσN +n vol, df1 ∧ · · · ∧ dfk = σ∈Sk

where N = dim g, so using the cocycle property of γkα and expanding [F/Γ], all terms are of the form τ (a0 h1 (a1 ) ... hk (ak )). There is a natural monomorphism of Hopf algebroids U (g) − → HF by eα 7→ Yα , where the Hopf algebra structure of U (g) is the natural one ∆(eα ) = eα ⊗r 1 + 1 ⊗r eα , S(eα ) = −eα . Let k be the Lie algebra of K. By the PBW theorem, the natural homomorphism U (k) − → U (g) is also injective. Denote its image in HF by K. (K)

The action of K on AF ≃ AP is particularly simple, because clearly (K) (K) kAF = {0}, so k(a) = ϵ(k)a for all k ∈ K if a ∈ AF . If K is connected, the converse holds as well. 2.4. Relative cyclic cohomology of Hopf algebras and the characteristic map. The following lemma, combined with the properties 2.20 of τ , motivated the definition of Hopf cyclic cohomology. rp Lemma 2.25. The map χ : H⊗ − → C p (AF ), F

χ(h1 ⊗r · · · ⊗r hp )(a0 , ... , ap ) := τ (a0 h1 (a1 ) · · · hp (ap )), rp is a monomorphism of vector spaces, and the image of ⊕p H⊗ is a cyclic F p submodule of ⊕p C (AF ). In particular, this defines the Hopf (periodic) cyclic cohomology of HF , by restriction from C ∗ (AF ).

Proof. If ψ ∈ C p (AF ) is contained in the image of χ, it can be put into the form ∑ ψ(a0 , ... , ap ) = τ ( a0 hi1 (a1 ) · · · hip (ap )) i i 1 i p i h1 (a ) · · · hp (a ) ⊗p

∑

by 2.20. By 2.19 the element is uniquely determined 1 p by ψ(−, a , ... , a ) and the resulting map AF − → AF factors through the quotient map πr because the r-bimodule structure of HF and the r-bimodule algebra so ψ uniquely determines the map ∑ structure of AF are compatible, ∑ T ( i hi1 ⊗r · · · ⊗r hip ) : a1 ⊗r · · · ⊗r ap 7→ i h1i (a1 ) · · · hpi (ap ). By injectivity ∑ of T (2.13), ϕ therefore uniquely determines the element i hi1 ⊗r · · · ⊗r hip , so χ is indeed injective. The cyclic structure on C q (AF ) is generated by the following operators δi ψ(a0 , ... , ap ) = ψ(a0 , ... , ai ai+1 , ... , ap ) 0

p

p 0

δp ψ(a , ... , a ) = ψ(a a , ... , a

p−1

0≤i≤p

)

σi ψ(a0 , ... , ap ) = ψ(a0 , ... , ai , 1, ai+1 , ... , ap ) τp ψ(a0 , ... , ap ) = ψ(ap , a0 , ... , ap−1 ) and by 2.20 and 2.12 those preserve the image of χ.

□

3. THE PERIODIC CYCLIC COHOMOLOGY OF HF

35

iK

Let K ,−→ GLn be a compact subgroup. The cyclic complex of Cc∞ (F/K) ⋊ ΓF is identified with the subcomplex {C p (AF , K) := i∗k C p (AF )}. Definition 2.26. The relative (periodic) cyclic cohomology of HF with respect to K ,− → GLn is the pullback of the (periodic) cyclic cohomology of p {C (AF , K)} through χ. The relative characteristic map is the restriction of χ to the relative (periodic) cyclic cohomology. 3. The periodic cyclic cohomology of HF We may regard HF as the algebra of differential operators on the Lie étale groupoid ΓF := F ⋊ ΓF . This groupoid has base space F and morphisms ΓF (p, q) = {germp (ϕ) | lϕ (p) = q}. That is, the morphisms from p to q are the germs at p of ΓF (p : q). The smooth structure is induced by the étale map s : ΓF − → F, ΓF (p, q) 7→ p, so a base for its topology consists∪of sets {germp (ϕ) | p ∈ U ⊂ dom ϕ} of germs of local diffeomorphisms ϕ ∈ q ΓF (p : q). The Hopf algebroid cohomology should then be viewed as a kind of differentiable cohomology of ΓF . This calls to mind the isomorphism [Hae76] Hd∗ (ΓF ; O) ≃ H ∗ (a(ΓF )), where Hd∗ (ΓF ; O) is the differentiable cohomology of the orientation sheaf O of F , computed by the complex of smooth maps from the composable (p) product ΓF to smooth currents on F , with the differentials δ and d given by (p) (p−1) the projections ΓF → − ΓF and the (pullback by the) de Rham differential, respectively (see 2.32), and H ∗ (a(ΓF )) is the Lie algebra cohomology of jets in Diff G (Rn ). Now, by [Cra99, 4.14], there is an isomorphism ⊕ H k (ΓF ; O) ,→ HP ∗ (AF )[1] k+dim F ≡∗

onto the localization of the periodic cyclic cohomology of the convolution algebra by the space of units. We show that the image of Hd∗ (ΓF ; O) is isomorphically mapped onto χ∗ HP ∗ (HF ), so what we obtain H ∗ (a(ΓF )) ≃ HP ∗ (HF ) ,→ HP ∗ (AF )[1] . 3.1. The Haefliger-van Est isomorphism H ∗ (a(ΓF )) ≃ Hd∗ (ΓF ; O). The van Est theorem states that the differentiable cohomology Hd∗ (G; V ), for a Lie group G (with a finite number of connected components) and a finite dimensional smooth representation V , is isomorphic to H ∗ (g, K; V ), where g is the Lie algebra of G and K is its maximal compact subgroup. This is implemented through the realization that the latter is the cohomology of G-invariant V -valued forms on G/K. This theorem can be extended to

36


certain smooth groupoids [Hae76]. Here, we are interested in the groupoid ΓF , and the correct analogy of a Lie algebra turns out to be the algebra a(ΓF ) of formal G-vector fields. Definition 2.27. Let a(ΓF ) be the ( Lie algebra) of formal G-vector fields on Rn . That is, a vector field a ∈ Γ U ⊂ Rn , T U is called a G-vector field if the flow ϕat ∈ Diff G (Rn ) for all t. Let ak (ΓF ) be the algebra of k-jets at 0 of G-vector fields. Then, a(ΓF ) := lim ak (ΓF ), ←− ∑ i with elements v = i v (x)∂i , where v i are formal power series in x, and the induced Lie bracket is given by [v i (x)∂i , v j (x)∂j ] = v i (x)(∂i v j (x))∂j − v j (x)(∂j v i (x))∂i . To see the relation to the previous discussion on jets of diffeomorphisms, recall that Gk0 consists of jets of diffeomorphisms in Diff G (Rn ) that preserve the origin to first order, and the coefficients of such jets are spanned by ⊕k (i) i=1 g , whereas a(ΓF ) consists of jets of the⊕infinitesimal generators of (i) and {v ∈ a(Γ ) | diffeomorphisms in Diff G (Rn ). That is, both F i≥1 g G i 2 n v (0) = 0+O(x )} can be regarded as tangent spaces to Diff 0 (R ), where the former contains deformations of jets at the identity and the latter contains jets of deformations of the identity. ⊕k (i) − there is an injection → ak (ΓF ) onto the jets of G-vector fields i=0 g preserving the origin. The multilinear alternate forms on a(ΓF ) generate the Gelfand-Fuchs algebra C ∗ (a(ΓF )): Definition 2.28. The Gelfand-Fuchs algebra C ∗ (a(ΓF )) is the algebra of forms c : Λ• a(ΓF ) − → R such that c(v1 , ... , vp ) = c(j0k (v1 ), ... , j0k (vp )) for some finite k. It is equipped with the differential ∑ dc(v0 , ... , vp ) = (−1)i+j c([vi , vj ], ... , vî , ... , vˆj , ... ), i