Singular foliations, examples and constructions
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1 Singular foliations, examples and constructions Georges Skandalis, joint work with Iakovos Androulidakis Université Paris-Diderot (Paris 7) Institut de Mathématiques de Jussieu - Paris Rive Gauche Cortona, June 29, 2017 Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
2 1. Regular foliations 1.1 Definitions Partition into connected submanifolds local picture: In other words: there is an open cover of M by foliation charts of the form U T where U R p and T R q. T is the transverse direction and U is the leafwise direction. Change of charts: f (u, t) =(g(u, t), h(t)). Vectors tangent to the leaves: subbundle F of the tangent bundle. Integrable subbundle: X, Y vector fields tangent to F,Liebracket[X, Y ] is tangent to F. Conversely Frobenius theorem Every integrable subbundle of the tangent bundle corresponds to a foliation. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
3 Examples of singular foliations 1 R foliated with 3 leaves. ( 1, 0), {0} and (0, +1). 2 R 2,(R n for n 2) foliated with 2 leaves. {(0, 0)} and R 2 \{(0, 0)}. 3 R 3 (R n for n 2) with co-centric spheres: {x 2 R 3 ; kxk 2 = r} (r 2 R + ). Remark. Will not allow any partition into manifolds: foliations have to be defined by vector fields. Not allowed: R 2 partitioned into R {0} and {x} ( 1, 0) and {x} (0, +1) Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
4 Definition of a singular foliations Theorem (Stefan Sussman) Let F C 1 c (M; TM) be a submodule such that 1 F is locally finitely generated 2 if X, Y 2F,then[X, Y ] 2F integrability condition. Then F gives rise to a nice partition into leaves. Definition (Stefan Sussman) A singular foliation is a partition into leaves associated with such a submodule F Cc 1 (M; TM). Di erent submodules can give the same partition into leaves: x@ x or x x... Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
5 Our definition of a singular foliation Definition (Androulidakis-S) A singular foliation is a locally finitely generated, integrable submodule F Cc 1 (M; TM). For us x@ x and x x define di erent foliations. This because we wish to do index theory along the foliation. We need to know: 1 which are the order 1 di erential operators allowed; 2 what is the ellipticity involved. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
6 Very singular foliations Constant rank: regular foliation i.e. integrable subbundle of the tangent bundle. Projective submodule: algebroid with anchor almost injective (\ x : A x! T x M injective on a dense set): nice singular foliation. Debord: integrates to a Lie groupoid. just finitely generated: very singular foliation. Theorem (Androulidakis-S, ) Construction of a holonomy groupoid which is a very singular Lie groupoid (longitudinally smooth Debord) full and reduced C (M, F) Moreover, elliptic operators are regular unbounded multipliers, index takes values in K-theory of C (M, F)... Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
7 Why these constructions? Why the C*-algebra? Contains the resolvants of elliptic di erential operators. Puts together representations on L 2 (M) and on L 2 (leaf ) of say longitudinal laplacian. Essential self-adjointness results, equality of spectra, possible shapes of the spectrum... Why the groupoid? Allows to construct the C*-algebra! f (D) lives on this groupoid if D elliptic self-adjoint order 1 di erential operator, f Schwartz function on R, such that ˆf compact support. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
8 The holonomy groupoid The elements of the holonomy groupoid are germs of (local) di eomorphisms ' in the subgroup exp(f) of Di (M) generated by exponentials of elements of F, divided by an equivalence relation. If just one leaf (F = C c (M; TM)), then G = M M and we only retain the pair ('(x), x). If regular foliation, we just retain the (germ of the) action of ' on the transversal. Definition (x,') is the trivial element if ' 2 exp(i x F) where I x is the ideal of smooth functions vanishing at x. Topology? Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
9 Bi-submersions In the case of a regular foliation: U T and U 0 T 0 foliated charts. h : T! T 0 a holonomy. A chart of the holonomy groupoid: U 0 U T. Source and range maps s(u 0, u, t) =(u, t) and r(u 0, u, t) =(u 0, h(t)). Cannot hope for a manifold with a local di eomorphism to G. We look for manifolds with a submersion to G. Wewillcallthembi-submersions. A bi-submersion is a manifold U, with submersions U r,s M, witha submersion-family from the fibers of s to the leaves of the foliation. Formally Definition A bi-submersion (U, r, s) where U r,s M submersions such that r (F) =s (F) =C 1 c (U;kerdr)+C 1 c (U;kerds). Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
10 Bi-submersions and the holonomy groupoid Inverse of a bi-submersion (U, r, s): (U, s, r). Composition of bi-submersions: fibered product. Existence of bi-submersions: small exponentiation of F around a point. Equivalence relation. (U, r, s) at u 2 U equivalent to (U 0, r 0, s 0 ) at u 0 2 U 0 if 9f : U! U 0 over r and s with f (u) =u 0. Holonomy groupoid: Equivalence quotient. The holonomy groupoid is locally a quotient of a smooth manifold... Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
11 Examples 1 R split in three leaves. Various F n generated by x x.di erent groupoids (but similar C -algebras). Of course, one can legitimately choose F 1. Groupoid RoR +. 2 R 2 foliated with two leaves: {0} and R 2 \{0}. We put F = vector fields that vanish at {0}. Generated by x,@ y far from 0 and I x@ x, y@ x, x@ y, y@ y at 0. Given by the action of GL(2, R) on R 2. Hol(M, F) is a quotient of R 2 o GL(2, R): Hol(M, F) =(R 2 \{0}) (R 2 \{0}) a {0} o GL(2, R) Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
12 Examples (2) Remark. Could have taken vector fields that vanish at higher (k th?) order at 0. Dimension 2(k + 1)... the action of SL(2, R) or of C. Di erent foliations. More generally... Let G GL n (R) be a connected subgroup. The transformation groupoid R n o G defines a foliation. Holonomy groupoid is a quotient of R n o G. If the action on R n \{0} transitive, the holonomy groupoid is (R n \{0}) (R n \{0}) a {0} o G. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
13 If the action of G on R n \{0} is not transitive... 3 Action of SO(3) on R 3. Groupoid SO(3) {0} a {(x, y) 2 (R 3 \ 0) (R 3 \ 0); kxk 2 = kyk 2 }. 4 Let G n = upper triangular matrices with positive diagonal. Invariant subspaces {0} = E 0 E 1 E 2... E n 1 E n = R n. For 0 < k apple n, let k = R n \ E k 1 and Y k = E k \ E k 1.ThesetY k consists of two G orbits. Put Y 0 = {0} and 0 = R n. Let also p k : R n! R n /E k = R n k.letthen G k = {(x, g, y) 2 k G n k k ; p k (x) =gp k (y)} G k. Proposition The holonomy groupoid of the foliation of R n is a union `n j=0 (G j) Yj. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
14 Other examples of the same flavor Remarks 1 Use any parabolic subgroup instead of the minimal parabolic G n. 2 Let P 1, P 2 GL(n, R) or GL(n, C) subgroups, and let P 1 P 2 act on M n (R) or M n (C) by left and right multiplication. I I If P1 = P 2 = GL n classes labeled by rank. If P 1 = P 2 = upper triangular matrices, orbits labeled S n (Bruhat decomposition). 3 One may write projective (or spherical) analogues of these examples. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
15 Baum-Connes conjecture? Why a Baum-Connes conjecture? We constructed a C -algebra which is a receptacle for index problems. Is there a guess of what this K-theory should be? Androulidakis S Set a general framework where such a conjecture can be formulated: we assume the foliation to be nicely described by a sequence of groupoids. Prove it in some cases... Essentially when the groupoids are amenable. Remark Even in the regular case, there are counterexamples to the Baum-Connes conjecture (Higson-La orgue-s), so we cannot hope our BC conjecture to hold... Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
16 Computation of examples 1 The SO(3) action on R 3. Hol(R 3, F) =R + (S 2 S 2 ) ` SO(3). Exact sequence: 0! C 0 (R +) K!C (M, F)! C (SO(3))! 0 Also the mapping cone of C (SO(3))!K(L 2 (S 2 )). K 0 (C (M, F)) is the kernel of the K-theory map Z (N)! Z (K 1 =0). 2 G = SL(n, R) or GL(n, R) or... acting on R n with two orbits 0 and R n \{0}: 0!K!C (M, F)! C (G)! 0 (full C -algebras). Diagram 0 / C 0 (R n \{0}) o G / 0 C 0 (R n ) o G / C (G) / 0 0 / K / C (R n, F) / C (G) / 0 K (C (R n, F)) = Z K (C (G)). Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
17 Thank you for your attention. Androulidakis-S (Paris 7/ IMJ-PRG) Singular foliations, examples and constructions Cortona, June 29, / 17
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