In Bourbaki exposé no. 551 (1981), Alain Connes associated to a compact foliated manifold (X,Y)(X,Y) the C*-algebra C∗(X,Y)C^*(X,Y) of the holonomy groupoid, observed that the general computation of the K-theory group K0(X,Y)=K0(C∗(X,Y))K_0(X,Y)=K_0(C^*(X,Y)) and of the analytic index arrow Ind⁡:K1(SY∗(X))→K0(X,Y)\operatorname{Ind}:K^1(S^*_Y(X))\to K_0(X,Y) was 'still far from the mark' beyond fibrations, free R\mathbb{R}-actions, and the Kronecker foliation (where the Pimsner-Voiculescu exact sequence computes K0(Aθ)K_0(A_\theta)), and closed with the program: 'Il reste à élaborer la théorie qui ... remplacera l'utilisation de la structure C∞C^\infty dans le sens transverse dans le cas du feuilletage de Kronecker' -- the emblematic obstruction being type III foliations such as the right-triangular foliation of F\SL(2,R)F\backslash SL(2,\mathbb{R}) for discrete cocompact FF, whose von Neumann algebra is an Araki-Woods type III factor with no trace, so that a differential calculus must be rebuilt from operator densities and the modular automorphism group.

Background

A foliation YY of a compact smooth manifold XX (the source writes (X,Y)(X,Y); modern notation (M,F)(M,\mathcal{F})) is a decomposition of XX into connected immersed submanifolds, the leaves, all of the same dimension, locally looking like 'leaf direction times transverse direction'. Even for simple foliations the topological quotient X/YX/Y, the space of leaves, is pathological: for the Kronecker foliation of the 2-torus T2=R2/Z2\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2 by the solutions of dx1=θdx2dx_1=\theta dx_2 with θ\theta irrational, every leaf is dense and the quotient topology is trivial. Connes' program replaces the leaf space by a noncommutative algebra. The holonomy groupoid GG of the foliation is the smooth groupoid whose arrows are leafwise paths, quotiented by holonomy (the germ of the first-return map along a loop in a leaf); its convolution C*-algebra C∗(X,Y)C^*(X,Y) (a C*-algebra is a norm-closed self-adjoint algebra of operators on Hilbert space; here, the completion of the compactly supported smooth half-densities on GG under the regular representations on the L2L^2-spaces of the holonomy coverings of the leaves) is the basic invariant of the leaf space. When the foliation comes from a submersion p:X→Bp:X\to B, C∗(X,Y)C^*(X,Y) is stably isomorphic to C(B)C(B), so the construction generalizes the function algebra of a genuine quotient; for the Kronecker foliation it is stably the irrational rotation C*-algebra AθA_\theta, generated by unitaries U1,U2U_1,U_2 with U2U1=e2πiθU1U2U_2U_1=e^{2\pi i\theta}U_1U_2. For a C*-algebra AA, K0(A)K_0(A) is the Grothendieck group of finitely generated projective AA-modules -- the noncommutative analogue of vector bundles, whose ranks are detected by traces -- and K1(A)K_1(A) records invertibles. The exposé defines K0(X,Y):=K0(C∗(X,Y))K_0(X,Y):=K_0(C^*(X,Y)), notes it matters beyond index theory (a compact transversal VV determines a finite projective module [V][V], and for a free Lie-group action every integrable representation determines one), and constructs the analytic index exactly as Atiyah-Singer do for manifolds: the filtered algebra of pseudodifferential operators along the leaves (operators that are elliptic on each leaf and vary continuously transversally) yields an exact sequence whose boundary map is Ind⁡:K1(SY∗(X))→K0(C∗(X,Y)),\operatorname{Ind}:K^1(S^*_Y(X))\to K_0(C^*(X,Y)), where SY∗(X)S^*_Y(X) is the cosphere bundle of the foliation, the quotient of the non-zero vectors of the tangent bundle TYTY of the foliation by the radial R+\mathbb{R}^+-action, and K1K^1 is topological K-theory. Two structures organize the problem. First, transverse measures: a holonomy-quasi-invariant measure Λ\Lambda defines a trace on C∗(X,Y)C^*(X,Y), hence a dimension homomorphism dim⁡Λ\dim_\Lambda on K0(X,Y)K_0(X,Y); the exposé's measured index theorem (proved via Breuer-Fredholm theory, the extension of Fredholm theory to operators affiliated with a semi-finite von Neumann algebra, applied to the semi-finite von Neumann algebra of the foliation) computes dim⁡Λ∘Ind⁡\dim_\Lambda\circ\operatorname{Ind} by an Atiyah-Singer-type formula and interprets the Euler pairing with the Ruelle-Sullivan current of Λ\Lambda (the de Rham current on XX dual to the transverse measure) through L2L^2-Betti numbers of the leaves. Second, the smooth transverse structure: in the Kronecker case, the smooth vectors for the two commuting derivations δ1,δ2\delta_1,\delta_2 of AθA_\theta form a dense smooth subalgebra Aθ∞A_\theta^\infty of Schwartz matrices; every finite projective AθA_\theta-module descends to it, it supports connections ∇1,∇2\nabla_1,\nabla_2 of constant curvature, a Chern character with values in the cohomology of R2\mathbb{R}^2, a pseudodifferential calculus, and an index theorem whose pairings are integral multiples of ∣p−qθ∣|p-q\theta| (the underlying K0(Aθ)≅Z2K_0(A_\theta)\cong\mathbb{Z}^2 is computed by the Pimsner-Voiculescu six-term exact sequence, the long exact sequence relating the K-groups of a crossed product A⋊ZA\rtimes\mathbb{Z} to those of AA and the action of the generator; the free R\mathbb{R}-action case of the index computation rests on Connes' Thom-analogue isomorphism, whose KK-theoretic version is due to Fack-Skandalis). The third structure is the obstruction motivating the type III part. The von Neumann algebra of the foliation (the weak closure of C∗(X,Y)C^*(X,Y) in the representation defined by a transverse measure class) can be a factor of type III -- a factor is a von Neumann algebra with trivial center, and type III means it admits no semi-finite trace at all, so no dimension functional is available on it. In the exposé's main example -- X=F\SL(2,R)X=F\backslash SL(2,\mathbb{R}) for a discrete cocompact subgroup FF, foliated by the right orbits of the lower triangular subgroup (this is the weak-stable foliation of the geodesic flow, of Anosov type: an Anosov flow is one whose tangent bundle splits into a flow direction plus transverse directions that are uniformly expanded and contracted) -- the density bundle Ω\Omega over the leaf space (the line bundle of order-1 densities defined by the leafwise flat transverse connection) is measurably non-trivial with non-reducible structure group, the obstruction being the Krieger ratio set (the set of values taken by Radon-Nikodym derivatives of the transverse measure class under the holonomy action, which classifies the type of the associated von Neumann algebra), computed for Anosov foliations by Bowen; the associated factor is an Araki-Woods type III factor (obtained by the Araki-Woods construction of infinite tensor products of matrix algebras, the model of the type III phenomenon). The exposé's Appendix 4 develops the replacement of the trace: random operators form the foliation von Neumann algebra; positive operator densities HH define all normal semi-finite weights φH(T)=∫Trace⁡(HT)\varphi_H(T)=\int\operatorname{Trace}(HT); and the modular automorphism group σtφH\sigma_t^{\varphi_H} of such a weight (the one-parameter automorphism group of Tomita-Takesaki theory, trivial precisely in the tracial case) is computed leafwise. The closing sentence of the exposé asks for the theory that will replace, in general and in this example, the use of the transverse C∞C^\infty structure that works for the Kronecker foliation. The later literature: the longitudinal index theorem of Connes-Skandalis (1984) gives the arrow Ind⁡\operatorname{Ind} a topological formula; cyclic cohomology (the noncommutative analogue of de Rham cohomology, in which the Chern character of finite projective modules takes its values) and the noncommutative calculus (Connes 1985), and the transverse Hopf-algebraic index theorem (Connes-Moscovici 1998) elaborate the smooth side for arbitrary codimension; the Baum-Connes assembly program (which computes the K-theory of a groupoid C*-algebra as the image of a topologically defined K-homology under an assembly map), surveyed in Connes' 'Noncommutative geometry year 2000', is the modern framework for computing K-groups of groupoid C*-algebras, proven for amenable foliations by Tu; and spectral triples (a representation of the algebra on Hilbert space with an unbounded Dirac-type operator whose commutators with the algebra are bounded -- the current notion of noncommutative Riemannian manifold) are built around traces and states, while the type III regime has no comparable general theory.

Problem Statement

The problem, at the source's generality, is to complete the noncommutative geometry of the leaf space of an arbitrary compact foliated manifold (X,Y)(X,Y) to the point where it matches what the Kronecker foliation affords through the irrational rotation algebra: a differential calculus plus computable K-theory and index. Two coupled deliverables:

(1) The general computation of the K-theory of the foliation C*-algebra, K0(X,Y)=K0(C∗(X,Y))K_0(X,Y)=K_0(C^*(X,Y)) (with its K1K_1 companion), and of the analytic index arrow Ind⁡:K1(SY∗(X))→K0(X,Y)\operatorname{Ind}:K^1(S^*_Y(X))\to K_0(X,Y), beyond the cases settled at the time of the source: fibrations (K0(B)K_0(B)), free R\mathbb{R}-actions (Connes' Thom-analogue isomorphism; KK-version by Fack-Skandalis), and the Kronecker foliation (Pimsner-Voiculescu). Compute these groups and this arrow for arbitrary compact foliated manifolds, or establish a general framework giving an effective procedure that, from the data of a compact foliated manifold, returns these groups and this arrow -- one that does not assume amenability or exactness of the holonomy groupoid and so copes with the Higson-Lafforgue-Skandalis counterexample phenomenon. The target also covers non-trivial cases with non-amenable holonomy, in particular the right-triangular foliation of F\SL(2,R)F\backslash SL(2,\mathbb{R}).

(2) The elaboration of 'the theory which will replace the use of the transverse C∞C^\infty structure' of the Kronecker case: a transverse differential calculus for the noncommutative leaf space -- smooth subalgebra, the derivations or their modular replacements, connections and Chern character on finite projective modules, pseudodifferential calculus, and an index theorem -- valid for foliations with non-trivial holonomy. On the tracial/type II side this has been substantially elaborated (cyclic cohomology and the noncommutative calculus of Connes 1985; the longitudinal index theorem of Connes-Skandalis 1984; the transverse Hopf-algebraic index theorem of Connes-Moscovici 1998); the surviving type III core is a transverse calculus in place of the trace and the commuting derivations δ1,δ2\delta_1,\delta_2 of AθA_\theta, together with index-type invariants for type III foliations such as the F\SL(2,R)F\backslash SL(2,\mathbb{R}) example.

The two deliverables form one objective: the index arrow in (1) is exactly the quantity the calculus in (2) is meant to compute, and the type III phenomenon in (2) is precisely the regime where the known computational tools of (1) -- traces and dimensions -- break down.

References

  1. Alain Connes, Feuilletages et algèbres d'opérateurs, Séminaire Bourbaki, 32e année (1979/80), exposé no. 551, in Séminaire Bourbaki vol. 22 (exposés 543-560), Lecture Notes in Mathematics, Springer, Berlin, 1981, pp. 139-155. DOI 10.1007/bfb0089932, https://doi.org/10.1007/bfb0089932; full open-access text at https://numdam.org/item/SB_1979-1980__22__139_0/
  2. Mihai Pimsner and Dan-Virgil Voiculescu, Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras, Journal of Operator Theory 4 (1980), no. 1, 93-118. zbMATH 0474.46059, https://api.zbmath.org/v1/document/0474.46059
  3. Thierry Fack and Georges Skandalis, L'analogue de l'isomorphisme de Thom pour les groupes de Kasparov, Comptes Rendus de l'Académie des Sciences, Paris, Série A 291 (1980), 579-581. zbMATH 0472.46049, https://api.zbmath.org/v1/document/0472.46049
  4. Rufus Bowen, Anosov foliations are hyperfinite, Annals of Mathematics (2) 106 (1977), no. 3, 549-565. DOI 10.2307/1971066, https://doi.org/10.2307/1971066
  5. Alain Connes and Georges Skandalis, The longitudinal index theorem for foliations, Publications of the Research Institute for Mathematical Sciences 20 (1984), no. 6, 1139-1183. DOI 10.2977/prims/1195180375, https://doi.org/10.2977/prims/1195180375
  6. Alain Connes, Non-commutative differential geometry, Publications Mathématiques de l'IHÉS 62 (1985), 41-144. DOI 10.1007/BF02698807, https://doi.org/10.1007/BF02698807
  7. Alain Connes and Henri Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem, Communications in Mathematical Physics 198 (1998), no. 1, 199-246. DOI 10.1007/s002200050477, https://doi.org/10.1007/s002200050477
  8. Alain Connes and Henri Moscovici, Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry, in Essays on Geometry and Related Topics (Mémoires dédiés à André Haefliger), Vol. 1, Monographies de l'Enseignement Mathématique 38, L'Enseignement Mathématique, Geneva, 2001, pp. 217-255. arXiv:math/0102167, https://arxiv.org/abs/math/0102167
  9. Alain Connes, Noncommutative geometry year 2000, arXiv:math/0011193, https://arxiv.org/abs/math/0011193
  10. Jean-Louis Tu, La conjecture de Baum–Connes pour les feuilletages moyennables, K-Theory 17 (1999), no. 3, 215-264. DOI 10.1023/A:1007744304422, https://doi.org/10.1023/A:1007744304422
  11. Nigel Higson, Vincent Lafforgue, and Georges Skandalis, Counterexamples to the Baum-Connes conjecture, Geometric and Functional Analysis 12 (2002), no. 2, 330-354. DOI 10.1007/s00039-002-8249-5, https://doi.org/10.1007/s00039-002-8249-5
  12. Iakovos Androulidakis and Georges Skandalis, A Baum-Connes conjecture for singular foliations, Annals of K-Theory 4 (2019), no. 4, 561-620. DOI 10.2140/akt.2019.4.561, https://doi.org/10.2140/akt.2019.4.561
  13. Hitoshi Moriyoshi and Paolo Piazza, Eta cocycles, relative pairings and the Godbillon-Vey index theorem, Geometric and Functional Analysis 22 (2012), no. 6, 1708-1813. DOI 10.1007/s00039-012-0197-0, arXiv:1102.2876, https://arxiv.org/abs/1102.2876
  14. Francesco Fidaleo and Luca Suriano, Type III representations and modular spectral triples for the noncommutative torus, Journal of Functional Analysis 275 (2018), no. 6, 1484-1531. DOI 10.1016/j.jfa.2018.06.009, arXiv:1806.08253, https://arxiv.org/abs/1806.08253
  15. Fabio Ciolli and Francesco Fidaleo, Modular spectral triples and deformed Fredholm modules, Annals of Functional Analysis 13 (2022), no. 4, paper no. 67. DOI 10.1007/s43034-022-00200-w, arXiv:2206.14762, https://arxiv.org/abs/2206.14762
  16. Moulay Tahar Benameur and James L. Heitsch, Higher relative index theorems for foliations, Journal of Functional Analysis 289 (2025), no. 9, article 111098. DOI 10.1016/j.jfa.2025.111098, arXiv:2402.10316, https://arxiv.org/abs/2402.10316
  17. Hiroshi Takai, C*-algebras of anosov foliations, in Operator Algebras and their Connections with Topology and Ergodic Theory (Busteni, 1983), Lecture Notes in Mathematics 1132, Springer, Berlin, 1985, pp. 509-516. DOI 10.1007/bfb0074906, https://doi.org/10.1007/bfb0074906