Noncommutative topology and the world’s simplest index theorem

Erik van Erp1

Dartmouth College, Hanover, NH 03755

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved March 6, 2010 (received for review January 29, 2010)

In this article we outline an approach to index theory on the basis solutions—even weak solutions in the sense of distribution the- of methods of noncommutative topology. We start with an explicit ory—are smooth functions. Both equations are hypoelliptic. Hör- index theorem for second-order differential operators on 3- mander pioneered a deep investigation of hypoelliptic equations. manifolds that are Fredholm but not elliptic. This low-brow index It is not true in general that every hypoelliptic operator on a formula is expressed in terms of winding numbers. We then pro- closed manifold is a Fredholm operator, i.e., an operator with a ceed to show how it is derived as a special case of an index theorem well-defined and finite analytic index. But the methods typically for hypoelliptic operators on contact manifolds. Finally, we discuss used to prove hypoellipticity also imply Fredholmness. Given the noncommutative topology that is employed in the proof of this these facts, it seems natural to search for an index theorem along theorem. The article is intended to illustrate that noncommutative the lines of the Atiyah–Singer formula for important classes of topology can be a powerful tool for proving results in classical hypoelliptic Fredholm operators that appear in the literature. analysis and geometry. A positive indication that this is possible was obtained by Hör- mander himself. In 1971, almost 10 years after Atiyah and Singer Introduction: Hypoelliptic Fredholm Index Theory published their result, Hörmander showed that the formula of In Euclidean space, waves propagate freely in all directions. A Atiyah and Singer applies to hypoelliptic operators of type ω standing wave with frequency is a solution of the eigenvalue ðρ; δÞ with 0 ≤ 1 − ρ ≤ δ < ρ ≤ 1 (1). In local coordinates, the ’ problem of Laplace s equation, full symbols of such operators are invertible outside a compact 2 2 2 set, and these symbols can be glued together, by a partition of ∂ u ∂ u ∂ u K0 TM – − − − ¼ ω2u: unity, to a well-defined class in ð Þ to which the Atiyah ∂x2 ∂y2 ∂z2 Singer formula applies. Despite a growing literature devoted to the analysis of hypoel- For any value of ω the solution space of this partial differential liptic operators, for over thirty years no progress was made in hy- equation (PDE) is infinite-dimensional, consisting of superposi- poelliptic index theory since the theorem of Hörmander. Only tions of plane waves traveling in arbitrary directions. The degrees about a decade ago did Epstein and Melrose reopen the inves- of freedom become severely restricted when we place the wave tigation (2, 3). Their goal was to derive an index formula for equation on a closed manifold. Suddenly, solutions exist only for the index of certain hypoelliptic operators associated to contact a discrete set of eigenfrequencies ω, and even with the correct structures. The context for their work was not Hörmander’s the- choice of ω the solution space is always finite-dimensional. The orem, but Boutet de Monvel’s index theorem for Toeplitz opera- 3 reason is intuitively obvious: Unlike the situation in R , waves tors on strictly pseudoconvex CR manifolds (4). Boutet de “wrap around” a closed manifold, and a standing wave can exist Monvel’s theorem is essentially the Atiyah–Singer formula, ex- only if it wraps around in just the right way, always remaining cept that the symbol of an elliptic operator is replaced by an ex- exactly in phase with itself. This lucky coincidence is the excep- pression derived from the symbolic calculus of the Toeplitz tional case, and generically the local propagation law (the PDE) algebra. is in conflict with the obstruction presented by the global topology. Epstein and Melrose’s goal was to extend Boutet de Monvel’s The index theorem of Atiyah and Singer is the ultimate expres- formula to the Heisenberg algebra on a contact manifold. The Hei- sion of this connection between the local features of the PDE and senberg algebra is a nonstandard pseudodifferential theory that the global topology of the manifold. The index of a partial differ- extends the Toeplitz algebra and (like any pseudodifferential the- ential operator P is not exactly the dimension of its solution ory) includes also the algebra of differential operators. The work space. One must correct this dimension by subtracting the num- of Epstein and Melrose (unfortunately largely unpublished) is a ber of linearly independent conditions that are required of the veritable tour de force, and they made considerable progress to- “source” v if the inhomogeneous equation Pu ¼ v is to have solu- ward a solution. Their final formula contains several factors that tions. Thus, the analytic index of P is defined as are not present in the formula of Atiyah and Singer and that are very hard to compute in practice. One of the key differences be- Index P ¼ dim Kernel P − codim Range P: tween the Atiyah–Singer and Boutet de Monvel formulas and the formula of Epstein–Melrose is that the latter is applied to a dif- Atiyah and Singer found a cohomology class ½σðPÞ associated to ferential form (representing the operator) that is not closed and the principal symbol (i.e., the highest-order part) of P and ex- therefore not a proper cohomology class. We suspect that the fail- pressed the analytic index of P in terms of this cohomology class ure to identify a cohomology class is the primary reason for the

by means of a purely topological formula. This topological index appearance of the various mysterious correction factors. MATHEMATICS can often be computed explicitly with the tools of algebraic topol- In the present article we discuss an approach to the hypoellip- ogy, and finding a positive index may be the only way to prove that tic index that has its roots in noncommutative topology. As we will solutions to the PDE exist—which, as we have seen, is not the see, this modern perspective provides a satisfactory resolution of generic situation. A limitation of the theorem of Atiyah and Singer is that it ap- plies only to elliptic operators. Recall that whereas Laplace’s Author contributions: E.v.E. wrote the paper. equation is elliptic, the heat equation is parabolic, and the wave The authors declare no conflict of interest. equation hyperbolic. What Laplace’s equation and the heat equa- This article is a PNAS Direct Submission. tion (but not the wave equation) have in common is that all 1E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1003155107 PNAS ∣ May 11, 2010 ∣ vol. 107 ∣ no. 19 ∣ 8549–8556 Downloaded by guest on September 30, 2021 the problem researched by Epstein and Melrose and shows that, boundary of quaternionic hyperbolic space. In another direction, after all, the Atiyah–Singer formula—just as it is, without bells our theory suggests that Heisenberg manifolds should not be too and whistles—suffices to express the topological index, just as much harder than contact manifolds. The monograph (6) of Beals in the index theorems of Hörmander and Boutet de Monvel. and Greiner is devoted to the analysis of second-order hypoellip- Moreover, our index theorem for contact manifolds appears as tic operators of a slightly more general class than the ones we a special case of a single theorem in noncommutative topology. study here. This is work in progress. It seems that the index pro- Translating this noncommutative index theorem into classical al- blem for Beals–Greiner operators may be just tractable, but our gebraic topology can be done only on a case-by-case basis, result- preliminary results are sufficiently complicated to discourage us ing in theorems that no longer look alike (a kind of “symmetry from trying to push the method any further. breaking,” one could say). ’ We start our exposition with the statement of an index formula The World s Simplest Index Theorem for second-order hypoelliptic PDEs on 3-manifolds. This theo- We start our discussion with a class of second-order PDEs on M rem contains no trace of its origin in noncommutative topology. 3-manifolds. Let be a closed 3-manifold. Consider a scalar P M To those who know elliptic theory, the example will seem strange second-order differential operator on , represented in local and unfamiliar, and it will be hard to guess how this index the- coordinates as orem is related to the formula of Atiyah and Singer. 3 ∂2 3 ∂ “ At the same time, the index formula presented here is as low- P ¼ aij þ bi þ c: ” ∑ ∂xi∂xj ∑ ∂xi brow as we can make it, involving nothing more complicated i;j¼1 i¼1 than the notion of a winding number. I am not aware of an index formula for elliptic differential operators that is this easy to state.* The coefficients aij, bi, and c are smooth functions on M.IfðaijÞ is I hope this convinces the reader that, just because the proof in- a real symmetric matrix with negative eigenvalues—so that P is volves noncommutative topology, the result is not necessarily essentially a Laplacian—then the operator P is elliptic and hence esoteric. Fredholm. However, the Fredholm index In the course of this article we gradually step back from this example, one step at a time. The first step is to understand Index P ¼ dim Ker P − dim Coker P why the specific operators in our example are Fredholm. Here we discuss some classical results from analysis on nilpotent of a Laplacian is always zero, and so the index theory is not very groups. Our treatment of the Heisenberg calculus here is perhaps interesting. somewhat idiosyncratic, because we are mainly interested in ap- Let us see what happens if the matrix ðaijÞ is degenerate, with plications to index problems as opposed to hypoelliptity. It suf- two negative eigenvalues and one equal to zero. We can cover M fices for our purposes here to develop the calculus only for by open sets in which P is represented as

differential operators. 2 2 Once we have a better understanding of the Fredholm theory P ¼ −X − Y þ lower-order terms: of our operators, we can show that the winding numbers in our formula truly arise from an application of the Atiyah–Singer for- Here X and Y are local vector fields on M corresponding to ei- mula. The insight that the Atiyah–Singer formula is all you need genvectors of ðaijÞ. Clearly, the operator P is not elliptic. But a is precisely our main advance since the work of Epstein and theorem of Hörmander (7) provides necessary and sufficient con- Melrose. ditions under which a second-order differential operator (with The next section explains why the Atiyah–Singer formula real coefficients) is hypoelliptic. If the bracket Z ¼½X;Y is lin- early independent of X and Y, then the role of the missing −Z2 works so universally and presents the noncommutative index the- 2 2 orem that is ultimately behind the simple formula involving wind- derivative is in some sense already taken care of by −X − Y . ing numbers presented in the first section. This “mother of all index theorems” says, in essence, that (i) there is only one topo- logical index, namely, the one calculated by Atiyah and Singer, Proposition 1. If the local vector fields X and Y together with their and (ii) the nature of the cohomology class to which the formula bracket ½X;Y span TM (in each open set for which P has a pre- is applied will depend on the details of the calculus. Moreover, sentation as above), and if P has real coefficients, then P is hypoel- finding the right expression of this cohomology class in specific liptic and Fredholm. cases involves a precisely defined computational problem in non- commutative topology. The level of difficulty of this computation But operators that satisfy Hörmander’s condition also have depends on the spectral complexity of the algebra of symbols (we zero index (as follows, for example, from the index formula we will explain this to some degree). The computation is easiest for present below). To get hypoelliptic Fredholm operators with a 0 elliptic operators [one recovers the class in K ðTMÞ defined by nonzero index, a more sophisticated analysis is needed, and we Atiyah and Singer], almost as easy for foliations (carried out in must include the lower-order terms in our discussion. So let us as- ref. 5), and quite a bit harder for contact manifolds (Boutet de sume that P is locally of the form Monvel’s theorem, generalized by Epstein–Melrose to the Her- mite algebra, and now fully extended to the Heisenberg algebra). P ¼ −X2 − Y 2 þ iαX þ iβY þ iγZ þ δ; Now that we have found the source of these theorems, can we produce genuinely new ones? We have explored this question in a where we always have Z ¼½X;Y, and require that X, Y, and Z couple of directions. The structure theory of type I C algebras span TM in their domain of definition. We allow complex values predicts the level of complexity of the index problem in individual for the coefficients, and therefore Hörmander’s bracket condi- cases. We have explored some of the cases that are predicted to tion alone is not sufficient to guarantee Fredholmness. Also re- be most tractable. In one direction, we found that Boutet de member that we do not require that the vector fields X, Y, and Z Monvel’s Toeplitz index formula can be generalized to manifolds can be chosen globally. Both of these facts will turn out to be cru- with higher codimension contact structures, such as appear at the cial for constructing examples of operators with nonzero index. As is easy to verify, the 2-plane bundle ξ spanned by the local vector fields X and Y is a well-defined global vector bundle (it is *One could certainly make the case that the index formula for Toeplitz operators on a P ’ circle is even simpler. We maintain that differential operators are “simpler” than Toeplitz dual to the characteristics of ). Moreover, Hörmander s bracket operators, if for no other reason than to justify the title of the article. condition for local sections in ξ is equivalent to ξ being a contact

8550 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1003155107 van Erp Downloaded by guest on September 30, 2021 structure on M. By a foundational result in contact topology, every the extensive literature on hypoelliptic operators contains no closed orientable 3-manifold admits a contact structure (Marti- mention of the following question: Can we express the Fredholm net’s theorem). In fact, there exist contact structures ξ in every index of P as a “topological index,” in terms of the homotopy type of homotopy type of oriented 2-plane bundles in TM (Lutz’s theo- the contact structure ξ and the map γ? To explain the gap of two or P rem). Thus, operators of the type we are interested in exist on three decades between the analysis of these operators and the every orientable 3-manifold. resolution of the associated index problem, recall that all scalar Orientability of M is a necessary condition, because the triple X;Y;Z M elliptic differential operators have zero index, as do all elliptic dif- provides a well-defined orientation for (it has the same ferential operators (acting on sections in a vector bundle) on odd- orientation as Y, X, ½Y;X¼−Z). In what follows we make the dimensional manifolds. What we have here is a scalar operator on unnecessary but simplifying assumption that ξ is (globally) or- iented as well. an odd-dimensional manifold. If there was some obvious way to We now make an elementary but surprising observation. reduce the index problem for these hypoelliptic operators to an elliptic problem, one would expect to find a zero index in all Proposition 2. The Z coefficient γ in the local presentations of the cases. But now that the problem is solved we know that every operator P is a well-defined global function γ: M → C. 3-manifold (except for a homology sphere) admits hypoelliptic scalar differential operators of the type we are studying here that have nontrivial index. Proof: Suppose that in some open set in M we have two alternative The resolution of the problem can be stated very simply as fol- presentations of P as lows. (A small detail needed to understand the theorem: The Eu- ler class of an oriented contact 2-plane bundle is always even.) P ¼ −X 2 − Y 2 þ iαX þ iβY þ iγZ þ δ Theorem 1. Let L be an oriented link in M such that the 1-cycle −A2 − B2 iα0A iβ0B iγ0C δ; ¼ þ þ þ þ 2½L ∈ H1ðM;ZÞ represents the Poincaré dual of the Euler class eðξÞ ∈ H2ðM;ZÞ. For each odd integer k, the finite collection of Z X;Y C A; B X;Y A; B where ¼½ and ¼½ and where both and loops γ: L → C \ fkg has a winding number are positively oriented. The second-order terms in the two repre- Z sentations can agree only if the pair A; B is obtained from X;Y by 1 dγ an orthogonal transformation IndL;γðkÞ¼ ∈ Z: 2πi L γ − k A ¼ aX þ bY; B ¼ cX þ dY; The Fredholm index of P is a Z-linear combination of these winding with numbers, ab Index P ¼ k · IndL;γðkÞ: ∈ SO 2 : ∑ cd ð Þ k odd

Direct calculation shows that Observe that if all coefficients of P are real valued (as in Pro- γ A2 þ B2 ¼ða2 þ b2ÞX2 þðab þ cdÞðXY þ YXÞþðc2 þ d2ÞY 2 position 1), then is purely imaginary and the map γ: M → C \ foddsg is contractible. Theorem 1 implies that þðaX · b þ cX · dÞY þðbY · a þ dY · cÞX Index P ¼ 0, which is not surprising and can probably be proven 2 2 P ¼ X þ Y þ ξ − terms: by a direct homotopy from to a self-adjoint operator. However, Theorem 1 also implies that if the vector fields X and Y in the and similarly presentation P −X2 − Y 2 iαX iβY iγZ δ C ¼ AB − BA ¼ðad − bcÞðXY − YXÞþðaX · c − cX · a ¼ þ þ þ þ bY c − dY a X aX d − cX b bY d − dY b Y þ · · Þ þð · · þ · · Þ are globally defined, then also Index P ¼ 0, regardless of the γ ¼ Z þ ξ − terms: homotopy type of . I am not aware of an elementary proof of this corollary. Combining these results we see that γ ¼ γ0. Changing Orders to Suit Our Needs To those accustomed to the Fredholm theory of elliptic opera- Before outlining the proof of Theorem 1 we must digress and ex- tors, the following theorem will sound surprising. plain the calculus behind Proposition 3. Why is it that Fredholm- P iγZ Proposition 3. The operator P is Fredholm if the range of the Z coef- ness of depends on the lower-order term ? The answer, in a iγZ P ficient γ does not contain any odd integers. word, is that is included in the highest-order part of , if only The calculus behind Proposition 3 further implies—not sur- we redefine “order” in a suitable way. (References for the calcu- — prisingly once we accept the proposition that the index of lus described in this section are refs. 3, 6, and 10. However, the MATHEMATICS the operator P depends only on the contact structure ξ and perspective taken here is slightly different and influenced by our the homotopy type of the function work on index theory.) The highest-order part of a differential operator like γ: M → C \ f…; −5; −3; −1; 1; 3; 5; …g: 3 ∂2 3 ∂ These curious facts have been known since the late 1970s (see, for P ¼ aij þ bi þ c ∑ ∂xi∂xj ∑ ∂xi example, refs. 8 and 9). Beals and Greiner (6) devoted an entire i;j¼1 i¼1 monograph to the analysis of exactly these operators (in a slightly more general setting). But until the work of Epstein and Melrose at a point m ∈ M,

van Erp PNAS ∣ May 11, 2010 ∣ vol. 107 ∣ no. 19 ∣ 8551 Downloaded by guest on September 30, 2021 3 ∂2 irreducible unitary representations π of the group, except for the P a m ; m ¼ ∑ ijð Þ trivial representation. ∂xi∂xj n i;j¼1 Observe that for the Abelian group R unitary representation theory amounts to Fourier theory, and a Rockland operator on Rn is not well-defined as a differential operator on M. The algebra of is just a homogeneous elliptic constant coefficient operator. In differential operators is only filtered, not graded. Nevertheless, this language, one could define ellipticity by saying that P is el- the highest-order part of P is well-defined and is usually taken liptic if each of the model operators Pm in the highest-order part to be a constant coefficient operator on the tangent fiber 3 of P is a Rockland operator on the Abelian group TmM. This de- TmM≅R , independent of coordinate choices. finition generalizes beautifully to the Heisenberg calculus. All we need to do to include iγZ in the highest-order part of our hypoelliptic operators is to change the filtration on the alge- bra of differential operators. However, in so doing we also must reinvestigate what kind of object the highest-order part of an op- Definition 5: A differential operator P is ξ-elliptic if all the model erator is. operators Pm in its ξ-highest order in the Heisenberg calculus are The Heisenberg calculus is, in essence, nothing more than the Rockland operators. working out of the consequences of an alternative filtration on the algebra of differential operators on M. The filtration is de- Proposition 6. Let ðM; ξÞ be a closed contact manifold, and P a sca- fined as follows: All vector fields in the direction of the contact lar differential operator on M. If for all m ∈ M the ξ-highest-order field ξ (like X and Y) have ξ-order one, as usual, but any vector part Pm is a Rockland operator, then P is a hypoelliptic Fredholm field that is not everywhere tangent to ξ (like Z) has ξ-order two. operator. 2 2 The highest-order part of P in the Heisenberg calculus will then Let us apply this to our example: When is Pm ¼ −X − Y þ clearly include the iγZ term. iγðmÞZ a Rockland operator? We need explicit formulas for the But we must also investigate what “highest-order part” really representations of the Heisenberg group. Because we are inter- means in this calculus. Abstractly, for any filtered algebra, the ested in the representation of the differential operator Pm,itis notion of highest-order part refers to an element in the associated most convenient to express our formulas in terms of the Lie al- graded algebra. So we need to find an analytic model for the as- gebra. The Heisenberg group has two families of irreducible re- sociated graded algebra of the algebra of differential operators presentations. First, there is an infinite family of scalar with the Heisenberg filtration. x; y Observe that in the associated graded algebra, smooth func- representations with two continuous parameters ð Þ, given by f A tions commute with all vector fields , because the commutator π X ix; π Y iy; π Z 0 ½A; f¼A · f is of order zero. It follows that elements in the ðx;yÞð Þ¼ ðx;yÞð Þ¼ ðx;yÞð Þ¼ . graded algebra can be localized at points m ∈ M. The result of this localization is, formally, When we apply these to Pm, we find 2 2 π P x2 y2; Pm ¼ −XðmÞ − YðmÞ þ iγðmÞZðmÞ: ðx;yÞð mÞ¼ þ

As before, the highest-order part of P is not an operator on M, which is invertible unless ðx; yÞ¼ð0; 0Þ (the trivial representa- but it has a natural interpretation as a smooth family Pm, m ∈ M tion). We see that invertibility of the nontrivial scalar representa- T M of operators in the tangent fiber m . The difference is that in tions of Pm amounts to “partial ellipticity” of P in the ξ-directions, P the Heisenberg calculus the operator m is not a constant coeffi- and it is built into our definition of ξ from P. cient operator, but an operator that is left invariant for a nilpotent T M The second family of irreducible representations of the Hei- group structure on the tangent fiber. One identifies m (or, senberg group is labeled by a single parameter t ∈ R \ f0g more precisely, ξm ⊕ TmM∕ξm) with the Heisenberg group, by im- (t ¼ 0 corresponds to the entire collection of scalar representa- posing the commutation relations tions). The representation space is the Bargman–Fock space. It is ½X;Y¼Z; ½X;Z¼½Y;Z¼0. the completion of the space of holomorphic polynomials in one complex variable z ∈ C. The monomials zq∕q! form an orthonor- z Then the ξ-highest-order part of P is the smooth family fPm;m∈ mal basis. The normalization is chosen such that the operators Mg of left invariant homogeneous operators on the (graded) and ∂∕∂z are adjoints. Heisenberg group If t>0, the representation πt is defined by rffiffiffi rffiffiffi Gm ¼ ξm ⊕ TmM∕ξm≅TmM: t ∂ t ∂ π X i z ; π Y z − ; tð Þ¼ 2 þ ∂z tð Þ¼ 2 ∂z If one wants a more explicit formula for the model operator Pm, one could substitute πtðZÞ¼it; ∂ 1 ∂ ∂ 1 ∂ ∂ whereas π−t is the conjugate representation of πt. We find, for XðmÞ¼ − y ;YðmÞ¼ þ x ;ZðmÞ¼ : ∂x 2 ∂z ∂y 2 ∂z ∂z t>0,

2 2 Whereas this substitution is common in the analytic literature, it πtðPmÞ¼−πtðXÞ − πtðYÞ þ iγðmÞπtðZÞ is not useful for our purposes. We prefer to retain the simpler P X Y Z ∂ algebraic expression of m in terms of , , and . ¼ t 2z þ 1 − γðmÞ : We can now state the main results about hypoelliptic operators ∂z in the Heisenberg calculus. As one easily verifies, the basis vectors zq∕q! are eigenvectors for the operator πtðPmÞ with eigenvalues 2q þ 1 − γðmÞ. It follows Definition 4: A ξ-homogeneous invariant operator on a graded nil- that πtðPmÞ is invertible for all t>0 iff γðmÞ is not a positive potent group is a Rockland operator if πðPmÞ is invertible for all odd integer.

8552 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1003155107 van Erp Downloaded by guest on September 30, 2021 Observe that, because of the homogeneity of P, we have To explain our formula we need to clarify the meaning of the quotient π P tπ P ;t>0; tð mÞ¼ þ1ð mÞ op −1 1 ½πðPmÞπðPm Þ ∈ K ðMÞ; as can be seen explicitly in the formula above. Likewise, which appears under the Chern character. Rather than explain π P tπ P ;t>0 −tð mÞ¼ −1ð mÞ . this in general, we will illustrate its meaning by deriving Theo- rem 1 from Theorem 2 by explicitly computing the appropriate π P For this reason, it suffices to verify invertibility of þ1ð mÞ and K-theory class. π P −1ð mÞ. A convenient way of dealing with the conjugate repre- So assume again that P is a ξ-elliptic scalar operator on a π sentation −1 is as follows. Consider the antiautomorphism of the closed contact 3-manifold, locally presented as Lie algebra of the Heisenberg group given by

2 2 X op ¼ X; Yop ¼ Y; Zop ¼ −Z: P ¼ −X − Y þ iαX þ iβY þ iγZ þ δ:

This Lie algebra antiautomorphism induces an antiautomorph- op We have calculated π 1ðPmÞ and π 1ðPm Þ, and we see that the ism of the algebra of invariant operators on G. Therefore invert- þ þ π P π Pop quotient of these two operators is a diagonal operator with eigen- ibility of −1ð mÞ is equivalent to invertibility of þ1ð m Þ. Thus, π P π Pop values all we need to do is verify that þ1ð mÞ and þ1ð m Þ are inver- tible. The advantage of this description is that we can work with 2q 1 − γ m π þ ð Þ : the single representation þ1, which will prove useful when we get 2q 1 γ m to our index formula. þ þ ð Þ To finish the discussion, simply reversing the sign of γ in π P q þ1ð mÞ we obtain These scalars act on the basis vector z ∕q!, which is canonically interpreted as a section in the qth tensor power ðξ1;0Þ⊗q. Here ξ1;0 ∂ π Pop 2z 1 γ m : denotes the oriented 2-plane bundle ξ thought of as a complex þ1ð m Þ¼ þ þ ð Þ ∂z line bundle. Recall that, in general, a cocycle in K1ðMÞ is represented by a This operator is invertible iff γ m is not a negative odd integer. ð Þ pair ½E; σ consisting of a vector bundle E on M and an auto- So, by Definition 5, P is ξ-elliptic precisely if γðmÞ is not an odd morphism σ of E. In the case of our ξ-elliptic operator P, we find integer for any m ∈ M. Then Proposition 3, which seemed bizarre from the perspective of elliptic theory, appears as a special case of Proposition 6. Once appropriately generalized, what seemed     π P ∞ 2q 1 − γ strangely different is seen to be a perfect analogy. ð mÞ ξ1;0 ⊗q; þ ∈ K1 M : op ¼ ð Þ ð Þ πðPm Þ ∑ 2q þ 1 þ γ The Atiyah–Singer Formula (sort of) q¼0 The previous discussion suggests that our hypoelliptic index pro- blem should be rephrased, in the context of the Heisenberg cal- The left-hand side appears in Theorem 2, and the right-hand culus, as follows: Find a topological index that expresses the P ξ side explains its meaning in our example. Observe that the right- Fredholm index of in terms of its -highest-order part hand side only appears to be an infinite sum, because for suffi- ciently large q the automorphisms are close to (and therefore σξðPÞ¼fPm;m∈ Mg: homotopic to) the trivial automorphism. It is this computation of a K-theory class associated to P that is This was, in essence, the problem that Epstein and Melrose set unique to the Heisenberg calculus for contact manifolds. Once we out to solve. Contrary to what is suggested by the complicated have the right K-theory class, we can apply the formula of Atiyah formula they found (announced in ref. 2), it turns out that the ξ and Singer. By using the fact that the Chern character is a ring topological index for -elliptic operators is no different from K the topological index computed by Atiyah and Singer. homomorphism (from theory to deRham cohomology), we find

∞ 2q 1 − γ P ξ 1;0 q þ Theorem 2. (See ref. 11). Let be a -elliptic differential operator on ChðσξðPÞÞ ¼ Chðξ Þ ∧ Ch : M;ξ ∑ 2q þ 1 þ γ a closed contact manifold ð Þ. Then q¼0 Z ξ1;0 Index P ¼ ChðσξðPÞÞ ∧ TdðMÞ; Because is a complex line bundle on a 3-manifold, we have M 1;0 q 1;0 q Chðξ Þ ¼ð1 þ c1ðξ ÞÞ ¼ 1 þ q · eðξÞ; where MATHEMATICS   where eðξÞ denotes the Euler class of the oriented real 2-plane π P σ P ð mÞ ∈ K1 M : field ξ. The (odd) Chern character of an invertible function ½ ξð Þ ¼ op ð Þ πðPm Þ u: M → C is given by 1 u d u: Chð Þ¼2πi log This is the well-known cohomological formula of Atiyah and Singer, except that the principal symbol σðPÞ of an elliptic opera- Because dim M ¼ 3, we have Tdðξ1;0Þ¼1 þ eðξÞ∕2, and the tor is replaced by the Heisenberg symbol σξðPÞ. Atiyah–Singer formula gives

van Erp PNAS ∣ May 11, 2010 ∣ vol. 107 ∣ no. 19 ∣ 8553 Downloaded by guest on September 30, 2021 Z ½fPm;m∈ Mg ∈ K0ðC ðTMÞÞ: Index P ¼ ChðσξðPÞÞ ∧ TdðMÞ M Z ∞ Of course, the Fourier transform in the fibers TmM of the tangent 2q þ 1 ¼ eðξÞ ∧ d logð2q þ 1 − γÞ space gives an algebra isomorphism ∑ M 2 q¼0 C TM ≅C TM : 2q 1 ð Þ 0ð Þ − þ e ξ ∧ d 2q 1 γ ; 2 ð Þ logð þ þ Þ After Fourier transform the family of operators fPm;m∈ Mg is which is equivalent to Theorem 1.† replaced by the function σðPÞ on TM that is usually referred to as the “principal symbol”, and the analytic K-theory element A Factory for Hypoelliptic Index Theorems ½fPm;m∈ Mg corresponds exactly to the topological K-theory We have explained in what sense Theorem 1 is an application of class ½σðPÞ defined by Atiyah and Singer, the topological index of Atiyah and Singer. From the point of view of classical analysis or topology it is very surprising that 0 K0ðC ðTMÞÞ≅K ðT MÞ∶½fPm;m∈ Mg ↦ ½σðPÞ: the Atiyah–Singer formula can be modified in such a straightfor- ward manner. Our next objective is to explain why this is so. As What is the point of this translation? After all, the definition of before, what appears strange from one point of view is clarified the topological class ½σðPÞ seems much simpler than that of the when seen from a higher level of abstraction. In the present case, equivalent analytic class ½fPm;m∈ Mg. It surely is. The point of we must understand the problem in the context of noncommuta- the analytic construction is that it applies much more generally. In — C tive topology specifically, the theory of algebras and analytic particular, it applies without essential modification to the highest- K theory. order part fPm;m∈ Mg of a ξ-elliptic operator P on a contact The idea of analytic K theory is rooted in Atiyah’s representa- P K0 X X manifold. This time, the operators m are translation invariant tion of ð Þ for compact as the set of homotopy classes of for a nilpotent group structure in the fibers of TM. Let us denote maps from X into F, by TξM the tangent space understood as the smooth family of 0 T M≅ξ ⊕ R K ðXÞ¼½X;F; Heisenberg groups m m that underly the Heisenberg calculus. As before, we can form the convolution algebra ∞ where F is the space of all Fredholm operators on a given (in- Cc ðTξMÞ and its completion C ðTξMÞ. These algebras are non- finite-dimensional, separable) Hilbert space. Concretely, if commutative, but we do not have to worry about that. The high- fPx;x∈ Xg is a continuous family of Fredholm operators Px over est-order part fPm;m∈ Mg of any operator P in the Heisenberg ∞ X, then the family of kernels Ex of Px defines a vector bundle E calculus can be conceived as an operator on Cc ðTξMÞ that is ∞ over X (possibly of nonconstant fiber dimension), and so does the right-Cc ðTξMÞ linear. If, in addition, every Pm is a Rockland op- family Fx of cokernels. The “index” of this family can then be erator (which makes P ξ-elliptic), then one can prove that this thought of as the formal difference family is indeed a Fredholm operator in the generalized sense of analytic K theory. Therefore, as before, ½E − ½F ∈ K0ðXÞ: ½fPm;m∈ Mg ∈ K0ðC ðTξMÞÞ: This “family index” is Atiyah’s isomorphism. The idea that K-theory elements can be represented by fa- So, at the very least, we have a K-theory element, albeit an ana- milies of Fredholm operators has been generalized and devel- lytically defined K-theory element. The goal is, then, to compute oped into analytic K theory of C algebras. The limited space of the corresponding topological K cocycle. Analytic K theory is as this article does not allow us to explain these ideas here in any much noncommutative topology as it is analysis. In the present detail. However, the example of Atiyah’s theorem may help the case, a noncommutative version of the Thom isomorphism in reader accept the following facts without much explanation. We K theory (the Connes–Thom isomorphism) implies that have seen that the highest-order part of an elliptic operator P on M P ;m∈ M 0 is a family f m g of elliptic operators. These operators K0 C TξM ≅K T M : ∞ ð ð ÞÞ ð Þ are not Fredholm, because they act on functions Cc ðTmMÞ on T M the Euclidean space m , which is not compact. But they define Therefore, at least theoretically, our analytically constructed a “Fredholm operator” in a generalized sense. First, one can K K ∞ -theory class can be interpreted as a topological -theory class, think of the family as a single differential operator on Cc ðTMÞ that differentiates only in the direction of the fibers. The transla- tion invariance of the operators Pm can be expressed algebraically σ P ∈ K0 TM : ∞ ½ ξð Þ ð Þ by treating Cc ðTMÞ as a convolution algebra, where functions on TM are multiplied by taking their convolution product in each In fact, the construction generalizes further. Suppose that M is fiber. Then the family fPm;m∈ Mg—taken as a single operator ∞ equipped with an arbitrary subbundle H ⊆ TM, of arbitrary codi- on Cc ðTMÞ—commutes with the right module action of ∞ mension. The bundle H could be a foliation, or a contact struc- Cc ðTMÞ on itself. Ellipticity of the individual operators Pm then M C∞ TM ture, it could be related to a sub-Riemannian structure on ,orit implies that this c ð Þ-linear operator satisfies the axioms of a could be a random bundle with no geometric significance what- suitably generalized notion of Fredholm operator. One completes C∞ TM C C TM soever. We can study hypoelliptic operators that are elliptic in the c ð Þ in a suitable norm to obtain the algebra ð Þ. H directions encoded by and of lower order in the transversal The C ðTMÞ-linear Fredholm operator fPm;m∈ Mg then de- directions. Such operators will be the elliptic elements in a gen- fines an (unbounded) element in the analytic K theory of eralized Heisenberg calculus associated to H (where transversal CðTMÞ, vector fields are given order two). The highest-order part for H-elliptic operators in this calculus is a family of invariant Rock- †In a personal communication, Charles Epstein pointed out that the index formula for the land operators Pm on nilpotent groups Gm ¼ Hm ⊕ TmM∕Hm, special case of second-order differential operators can also be derived by a deformation and this family represents an analytic K-theory class of the Heisenberg symbol of the differential operator to the symbol of a Toeplitz opera- ’ tor, to which Boutet de Monvel s index theorem applies. It is not clear if or how this trick P ;m∈ M ∈ K C T M : extends to the general case. ½f m g 0ð ð H ÞÞ

8554 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1003155107 van Erp Downloaded by guest on September 30, 2021 Once again this group is naturally isomorphic to K0ðTMÞ, and so problem—not the index of P, but the index of its highest-order we have part (which, as we have indicated, is a generalized Fredholm op- erator). The tractability of this problem depends on the specifics 0 ½σH ðPÞ ∈ K ðT MÞ: of the geometric structure ðM;HÞ, as we will see in our next and final section. Thus, by using the tools of noncommutative topology, one can always construct a symbol class in K0ðTMÞ for this type of hy- Offspring: Foliations, Contact Manifolds, and More poelliptic operators. Then a very general index problem presents To derive Theorem 2 from Theorem 3 and thereby give a proof of ’ itself: Find the topological index the world s simplest index theorem (Theorem 1), we need to find an explicit formula for the isomorphism 0 IndH : K ðT MÞ → Z 0 K0ðC ðTξMÞÞ≅K ðT MÞ for H-elliptic operators. A priori, this topological index could de- pend on the geometric structure H. But it does not. A beautiful if ξ is a contact structure. The computation depends on a careful argument involving deformation theory of C algebras (this is analysis of the structure of the C algebra C ðTξMÞ. where Lie groupoids play a key role) implies that, once we have In the general case, if H ⊆ TM is an arbitrary bundle, the struc- identified the symbol of a hypoelliptic operator as a class in ture of the C algebra C ðTH MÞ is clarified by the theory of type I 0 C C K ðTMÞ, there is only one topological index. algebras. A algebra is of type I if it can be decomposed, by a series of successive short exact sequences, into commutative C K Theorem 3. Let M be a closed manifold equipped with a subbundle algebras or algebras that are -theoretically equivalent to com- H ⊆ TM. Let P be an H-elliptic operator on M—elliptic in the di- mutative algebras. This reduction to commutative algebras is pre- rections H and of lower order transversally. Then the highest-order cisely what establishes the connection between noncommutative K part σH ðPÞ¼fPm; m ∈ Mg in the generalized Heisenberg calculus and classical topology and will allow us to use topological the- K for ðM; HÞ defines a K-theory class ory to compute an analytic cocycle. Thus, the structure theory of C algebras tells us something significant about the geometric 0 M;H ½σH ðPÞ ∈ K0ðC ðTH MÞÞ≅K ðT MÞ; structure ð Þ and about the difficulty of the related index pro- blem. It is reasonable to expect that the more terms there are in C T M and the Fredholm index of P is computed by the topological index of the decomposition series of ð H Þ (it is always finite), the Atiyah and Singer, more intractable the index problem will be. H TM— Z In the trivial case where ¼ the case of elliptic opera- tors—the computation is fairly straightforward, because of the P Ch σ P ∧ Td M : Index ¼ ð H ð ÞÞ ð Þ isomorphism C ðTMÞ¼C0ðT MÞ established by Fourier theory. TM In general, C ðTH MÞ is commutative if the groups Gm are Abe- lian, which is the case precisely if ½H;H ⊆ H, i.e., if ðM;HÞ is a This theorem generalizes Boutet de Monvel’s theorem for foliated manifold. For foliations, the computation of the class σ P ∈ K0 TM H Toeplitz operators, and it is also the source of our proof of The- H ð Þ ð Þ for -elliptic operators (these are not elliptic) orem 2. Nevertheless, Theorem 3, by itself, is too abstract to be is not much harder than it is for elliptic operators. We have car- useful. To turn it into a computable formula—say, something as ried out this computation and by this general methodology derive — a hypoelliptic index theorem that is similar in spirit to the one concrete as Theorem 1 two obstacles need to be overcome. ρ; δ The first obstacle is a purely analytic problem in the Heisen- proved by Hörmander for hypoelliptic operators of type ð Þ, P except that hypoelliptic operators (and their parametrices) in berg calculus. Given an explicit differential operator , it is not 1 ; 1 too hard to verify whether its H symbol consists of Rockland op- the Heisenberg calculus are of type ð2 2Þ, a case not covered P by Hörmander’s theorem (see ref. 5). erators m. But that is not sufficient in a general setting. One P P From the perspective of the structure theory of C ðTH MÞ, con- must still prove that if each m is Rockland, then has an inverse H ξ in the generalized Heisenberg calculus. This implication is known tact structures are the next best thing, after foliations. If ¼ is H a contact structure, there is a single short exact sequence that de- to be true if the manifold (with its structure ) can be locally C T M identified with the graded nilpotent model group—such as is composes ð ξ Þ, as follows: ’ the case for foliations (Frobenius s theorem) and contact struc- × 0 → C0 M × R ⊗ K → C TξM → C0 ξ → 0. tures (Darboux’s theorem). But in general it can be quite tricky. ð Þ ð Þ ð Þ How hard the problem is in general can be gleaned by skimming ξ TM∕ξ through the monograph of Beals and Greiner (6). The bulk of As before, we assume here that is cooriented; i.e., is a trivial line bundle. Then R× ¼ R \ f0g refers to the punctured fi- that publication is devoted to proving just this fact for second-or- TM∕ξ≅M × R der H-elliptic operators in the case that H is a subbundle of co- ber of the dual of the line bundle . The elementary K C dimension one in TM. To my knowledge, it is not known whether algebra is the (abstract) algebra of compact operators on a separable Hilbert space. this fact (if all Pm are Rockland, then P has an inverse in the cal- To understand this sequence, it is helpful to focus on a single culus) is true in absolute generality. fiber of the bundle of algebras C ðTξMÞ. For every m ∈ M, the The second obstacle is a problem in noncommutative topology. fiber is the group C algebra G ðGmÞ [the completion of the con- In order to compute the Chern character of σH ðPÞ, we must have ∞ volution algebra Cc ðGmÞ in a suitable norm] of the Heisenberg an explicit understanding of the Connes–Thom isomorphism MATHEMATICS group Gm ¼ Hm × R. This group algebra decomposes as K C T M ≅K0 TM : 0ð ð H ÞÞ ð Þ ϕ 0 → C0ðR \ f0gÞ ⊗ K → C ðGmÞ→C0ðHmÞ → 0. Unfortunately, there is no general method for finding a topolo- gical representation of the analytically defined class. Interest- The quotient map ϕ is obtained by assembling all the scalar ingly, the map from analytic to topological K theory is, in representations of the group Gm, parametrized by points in essence, a generalized index—as in the example of Atiyah’s ana- Hm. The remaining irreducible representations of G are labeled 0 lytic representation of elements in K ðXÞ. Therefore, this com- by the dual of the center R ⊂ Gm—excluding zero, which putation of the symbol class in K0ðTMÞ is itself an index corresponds to the scalar representations. The decomposition

van Erp PNAS ∣ May 11, 2010 ∣ vol. 107 ∣ no. 19 ∣ 8555 Downloaded by guest on September 30, 2021 for C ðTξMÞ is obtained from the decomposition of its fibers, and The result of this analysis is that we can identify the bottom- it is important to observe how the representation theory of the row map on the left as the diagonal embedding Heisenberg group is reflected in the structure theory of the sym- 1 1 1 bol algebra C ðTξMÞ. K ðMÞ → K ðMÞ ⊕ K ðMÞ∶a ↦ a ⊕ a: Now that we understand how to decompose the symbol algebra C ðTξMÞ, we can study its K theory. A quotient map is the alge- This knowledge fixes the bottom-row map on the right, braic analog of an inclusion of topological spaces. And just as a X;Y Y ⊆ X 0 × pair of spaces ð Þ with gives rise to a long exact se- K ðM × R Þ → K0ðC ðTξMÞÞ quence in cohomology, so does a short exact sequence of C al- ≅ ↓↓≅ gebras. In the present case, the part of the K-theory sequence that 1 1 1 K ðMÞ ⊕ K ðMÞ → K ðMÞ is relevant to our computation is K1 M 1 0 × up to an automorphism of ð Þ. It turns out to be the map K ðξ Þ → K ðM × R Þ → K0ðC ðTξMÞÞ → 0. K1ðMÞ ⊕ K1ðMÞ → K1ðMÞ∶ða; bÞ ↦ a · b−1: Let us analyze the three terms in this sequence. Starting from the ξ K left, because is a symplectic bundle (and hence -orientable), P ;m∈ M K C T M the Thom isomorphism gives Now we must lift the family f m g from 0ð ð H ÞÞ to K0ðM × R×Þ. Here we must remember the explicit connection be- C T M K1ðξÞ≅K1ðMÞ: tween the decomposition of ð H Þ and the representation theory of the Heisenberg group. Very roughly (I am cheating a R× ≈ R∪R bit here), the Heisenberg symbol lifts to the family of operators From we obtain × Pm;t parametrized by ðm; tÞ ∈ M × R , where Pm;t ¼ πtðPmÞ for op K0ðM × R×Þ≅K0ðM × RÞ ⊕ K0ðM × RÞ≅K1ðMÞ ⊕ K1ðMÞ: t>0, whereas Pm;−t ¼ πtðPm Þ. Chasing this element through the diagram, we find Finally, by combining various isomorphisms, K0 M × R× → K1 M ⊕ K1 M ∶ P ↦ P ;P ð Þ ð Þ ð Þ f m;tg ð m;þ1 m;−1Þ K C T M ≅K0 TM ≅K0 ξ × R ≅K1 M : 0ð ð ξ ÞÞ ð Þ ð Þ ð Þ π P ; π Pop ; ¼ð þ1ð mÞ þ1ð m ÞÞ Thus, our K-theory sequence can be expanded as follows: and then K1 ξ → K0 M × R× → K C T M → 0 ð Þ ð Þ 0ð ð ξ ÞÞ K1 M ⊕ K1 M → K1 M ∶ π P ; π Pop ≅ ↓ ≅ ↓↓≅ ð Þ ð Þ ð Þ ð þ1ð mÞ þ1ð m ÞÞ K1 M → K1 M ⊕ K1 M → K1 M → 0 ↦ π P π Pop −1: ð Þ ð Þ ð Þ ð Þ . þ1ð mÞ þ1ð m Þ

Notice that all the ingredients in this diagram are topological, ex- The result is an explicit formula for the isomorphism between the C C T M cept for the algebra ð ξ Þ. In order to compute the vertical K theories of the noncommutative symbol algebra C ðTξMÞ and arrow at the right of the diagram [the desired isomorphism TM 1 the topological space . K0ððTξMÞÞ≅K ðMÞ], it suffices to work out explicitly what the other maps in the diagram are. Then we will lift the element Proposition 7. Let P be a ξ-elliptic operator on a closed contact mani- σ P P ;m∈ M K C T M K0 M × R× ξð Þ¼f m g from 0ð ð ξ ÞÞ to ð Þ and fold ðM; ξÞ. With the canonical identifications chase it through the diagram. This diagram chase is how we arrive at Theorem 2 as a special case of Theorem 3. 0 1 K0ðC ðTξMÞÞ≅K ðT MÞ≅K ðMÞ; Given the expository nature of this note, we could stop here. But for the reader who has read this far, it seems unfair not to give some final hints as to how the calculation is completed. Glossing we have over many technical details (among other things, the complica- P ; m ∈ M π P π Pop −1 : tions arising from the fact that we are dealing with unbounded ½f m g¼½ þ1ð mÞ þ1ð m Þ K-theory elements), we sketch the main idea. Observe, before we proceed, that the vertical maps in the dia- K gram (apart from the one on the right, which is the one we want to This computation in theory is how we derive Theorem 2 compute) are topologically well understood. What we need to from Theorem 3. In turn, as we have seen, the world’s simplest identify are the two maps in the bottom row by carefully analyzing index theorem 1 is a special case of Theorem 2. We hope that this the maps in the top row. This translation from the top row into discussion of the solution of this problem reveals something of the bottom row is achieved by the techniques of quantization,or the power and role of noncommutative topology as a bridge be- rather its reverse: “passage to the classical limit.” tween classical analysis and topology.

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