Foliations

Manifold V and smooth subbundle F of TV

a) Every x ∈ V is contained in a submanifold

L of V such that

Ty(L) = Fy ∀ y ∈ L .

b) Every x ∈ V is in the domain U ⊂ V of a

submersion p : U → Rq (q = CodimF ) with

Fy = Ker(p∗)y ∀ y ∈ U .

c) C∞(F ) = X ∈ C∞(TV ) , Xx ∈ Fx ∀x ∈V is a Lie algebra.

d) The ideal J(F ) of smooth exterior differ-

ential forms which vanish on F is stable by

exterior differentiation.

1

Leaves

The leaves of the foliation (V, F ) are the

maximal connected submanifolds L of V with

Tx(L) = Fx, ∀x ∈ L

1) Though V is compact, the leaves Lα, α ∈ Xcan fail to be compact.

2) The space X of leaves Lα, α ∈ X can fail

to be Hausdorff and in fact the quotient

topology can be trivial (with no non trivial

open subset).

2

Holonomy

3

Invariant measure for flows

Assume dimF = 1 and F is oriented.

Take X ∈ C∞(F+). The leaves are then the

orbits of the flow ψt = exp tX.

Measure = 0-dimensional current

〈µ, ω〉 = ∫ω dµ, ∀ω ∈ C∞(V )

Invariance for X

ψt µ = µ⇐⇒ ∂X µ = 0⇐⇒ d(iX µ) = 0

4

Transverse measure for 1-dimensional

foliations

Assume dimF = 1 and F is oriented.

X,X ′ ∈ C∞(F+) are related by X ′ = ϕX,

ϕ ∈ C∞(V )+.

ψ′t(v) = ψT(t,v)(v) ∀ t ∈ R , v ∈ V .

ψt µ = µ⇐⇒ ψ′t µ′ = µ′ µ′ = ϕ−1µ

C = iX µ

iX ′µ′ = iX µ

5

Ruelle-Sullivan Current

1) C is closed, i.e. dC = 0

2) C is positive in the leaf direction, i.e. if

ω is a smooth 1-form whose restriction to

leaves is positive then 〈C,ω〉 = 0.

〈µ, f〉 = 〈C, ω〉 , ∀ω ∈ C∞(Λ1 T ∗C) , ω(X) = f .

6

Transverse Measure

A transverse measure Λ for the foliation (V, F )

is a σ-additive map B → Λ(B) from Borel

transversals (i.e. Borel sets in V with V ∩ Lcountable for any leaf L) to [0,+∞] such that

1) If ψ : B1 → B2 is a Borel bijection and ψ(x)

is on the leaf of x for any x ∈ B1, then

Λ(B1) = Λ(B2).

2) Λ(K) < ∞ if K is a compact subset of a

smooth transversal.

7

The Ruelle-Sullivan cycle and the Euler

number of a measured foliation

(V, F ) equipped with a transverse measure Λ

dC = 0 thus C defines a cycle [C] ∈ Hk(V,R)

Let now e(F ) ∈ Hk(V,R) be the Euler class of

the oriented real vector bundle F on V

χ(F,Λ) = 〈e(F ), [C]〉

Poincare and H. Hopf:

χ(M) =∑

p∈ZeroX

ω(X, p)

χ(F,Λ) =

∫

p∈ZeroXω(X, p) dΛ(p)

8

The degree of a vector field

In local coordinates, X = Σ ai ∂∂xi

, the matrix

∂ ai

∂xj(p) is non degenerate and the local degree

is the sign of its determinant

9

Continuous Dimensions

Let B,B′ be Borel transversals, then if the two

bundles of Hilbert spaces (HL)L∈V/F , HL =

`2(L∩B); H ′L = `2(L∩B′) are measurably iso-

morphic, one has Λ(B) = Λ(B′).

Type II von Neumann algebra

It is the algebra of random operators (modulo

null sets for Λ):

M = (TL)L∈V/F |TL ∈ End L2(L) , ∀L ∈ V/F

L is the holonomy covering of L.

10

Real Betti numbers

Theorem (ac 1978)

a) For each j = 0,1,2, . . . ,dimF , there exists

a Borel transversal Bj such that the bundle

(Hj(L,C))L∈V/F of j-th square integrable har-

monic forms on L is measurably isomorphic to

(`2(L ∩B))L∈V/F .

b) The scalar βj = Λ(Bj) is finite, indepen-

dent of the choice of Bj, of the choice of the

Euclidean structure on F .

c) One has Σ(−1)j βj = χ(F,Λ).

11

Dimension two leaves

dimF = 2

H(0)L = square integrable harmonic 0-forms

on L

As harmonic 0-forms are constant, there are

two cases:

If L is not compact, one has H(0)L = 0.

If L is compact, one has H(0)L = C.

Corollary

If the set of compact leaves of (V, F ) is Λ-

negligible then the integral∫K dΛ of the in-

trinsic Gaussian curvature of the leaves is 5 0.

Proof 12π

∫K dΛ = β0 − β1 + β2 = −β1 5 0.

Q.E.D.

12

Longitudinal elliptic operators

One starts with a pair of smooth vector bun-

dles E1, E2 on V together with a differential

operator D on V from sections of E1 to sec-

tions of E2 such that:

1) D restricts to leaves, i.e. (Dξ)x only de-

pends on the restriction of ξ to a neigh-

borhood of x in the leaf of x (i.e. D only

uses partial differentiation in the leaf direc-

tion).

2) D is elliptic when restricted to any leaf.

13

The index theorem for measured foliations

Theorem (ac 1978)

a) There exists a Borel transversal B (resp. B′)such that the bundle (`2(L∩B))L∈V/F is mea-

surably isomorphic to the bundle (KerDL)L∈V/F(resp. to (KerD∗L)L∈V/F ).

b) The scalar Λ(B) <∞ is independent of the

choice of B and noted dimΛ(Ker (D)).

c) dimΛ(Ker (D))−dimΛ(Ker (D∗)) =

. ε 〈ch σDTd(FC), [C]〉

(ε = (−1) k(k+1)2 , k = dimF , Td (FC) = Todd

genus, σD = symbol of D) .

14

`2 Betti numbers for measured equivalence

relations

Theorem (D. Gaboriau 2000)

The Betti numbers βj(F,Λ) of a foliation with

contractible leaves are invariants of the mea-

sured equivalence relation R = (x, y) | y ∈ leaf(x).

0∂0← C

(2)0

∂1← C(2)1

∂2← · · · ∂n← C(2)n

∂n+1← C(2)n+1

∂n+2← · · ·

H(2)n (Σ,R, µ) := Ker∂n/ Im∂n+1

βn(Σ,R) := dimΛ(H(2)n (Σ,R, µ))

is independent of the choice of a bounded n-

connected R-complex Σ.

(Cheeger-Gromov in the case of discrete groups

after Atiyah’s `2 Betti numbers for covering

spaces).

15

II1 Factors and discrete groups

Group Γ II1 Factor M

Representation Correspondence(M-bimodule )

Trivial rep. Standard form L2(M)

Regular rep. Coarse : L2(M)HS= L2(M)⊗L2(M)

Amenable L2(M) ⊂weakly L2(M)HS

Property T (k) L2(M) isolated (ac + vj)

16

II1 Factors and Betti numbers (ac + ds)

By results of W. Luck for discrete groups Γ,

β(2)∗ (Γ) = dimL(Γ)H∗(Γ;L(Γ))

where H∗ stands for the algebraic group ho-

mology.

L2-homology of a von Neumann algebra M :

H(2)k (M) = Hk(M ;M⊗Mo).

Here Hk stands for the algebraic Hochschild

cohomology of M .

H(2)0 (M, τ) 6= 0 if and only if M is hyperfinite.

One is led to consider the L2-Betti numbers,

β(2)k (M) = dimM⊗MoH

(2)k (M)

Problem For which groups does one have

β(2)∗ (L(Γ)) = β

(2)∗ (Γ)

17

Foliation 7→ von Neumann algebra

It is the algebra of random operators:

M = (TL)L∈V/F |TL ∈ End L2(L) , ∀L ∈ V/F

||T || = EssSup ||TL||(modulo null sets for the smooth (Lebesgue)

measure)

Examples

• Reeb Foliation 7→ Type I∞

• Kronecker Foliation 7→ Type II∞ hyperfinite

• Anosov Foliation 7→ Type III1 hyperfinite

• Flat connection 7→ Type II∞ non-hyperfinite

(genus > 1)

18

Invariants of von Neumann algebras

TT : φ→ σφt

Theorem (ac 1972)

δ : R→ OutM = AutM/IntM

(σψt = Adut σ

φt )

Invariants

• S(M) = ∩Spec∆ϕ ⊂ R+

• T (M) = Ker δ ⊂ R

• W (M) = Flow of weights.

(⇒ hyperfinite non-ITPFI factor)

19

Weights on foliation algebra

For manifolds the positive measures in the “smooth”

class are given by 1-densities,

Theorem (ac 1976)

• Faithful normal weights ϕ on M correspond

to positive (unbounded) random operator

one-densities

(TL,v)L∈V/F |TL,λ v = λ TL,v , ∀v ∈ ∧qT (V/F )

• Independently of v one has

σϕt = AdT itL,v

• Independently of v one has

(Dψ : Dϕ)t = SitL,v T−itL,v ∈M

20

The Godbillon-Vey invariant

Let (V, F ) be a transversally oriented compact

foliated manifold, of codimension 1.

F = Ker ω , ω ∈ C∞(V, T ∗(V ))

dω = β ∧ ω , α = dβ ∧ β

GV := Class α ∈ H3(V,R)

Theorem (ac 1983)(refining previous work of

Heitsch-Hurder)

Let M be the associated von Neumann algebra,

and W (M) be its flow of weights. Then if the

Godbillon-Vey class of (V, F ) is different from

0, there exists an invariant probability measure

for the flow W (M).

(⇒ type III)

21

Homology of traces

Let B be a unital Banach algebra, view the dual

space B∗ as a bimodule over B with 〈aϕb, x〉 =〈ϕ, bxa〉, ∀ a, x, b ∈ B.

Lemma Let δ be a densely defined derivation

of B with values in B∗. Assume that the unit

1B belongs to the domain of the adjoint δ∗ of

δ, then:

a. τ = δ∗(1) is a trace on B.

b. The map of K0(B) to C given by τ is equal

to 0.

22

Homology ∼ 0 gives 1-traces

Definition Let B be a Banach algebra. By a

1-trace on B we mean a densely defined deriva-

tion δ from B to B∗ such that

〈δ(x), y〉 = −〈δ(y), x〉 ∀x, y ∈ Dom δ .

Lemma Let δ be a 1-trace on B, then:

a) δ is closable.

b) There exists a unique map of K1(B) to C

such that, for any u ∈ GLn(Dom δ) (closure

of δ) one has:

ϕ(u) = 〈u−1, δ(u)〉

〈x, δ(a)〉 =∫ ∑

Γ

xg g(dag−1) ∀x ∈ C(S1)oΓ

implies K1(C(S1)) ⊂ K1(C(S1) o Γ).

23

Higher jet bundle J+k (S1)

J+k = bundle of positive frames of order k on

the oriented manifold S1.

Jet or order k, jk(f), at 0 ∈ R of a local orienta-

tion preserving diffeomorphism f of a neighbor-

hood of 0 in R to a neighborhood of y = f(0)

S1 = R/Z with y the corresponding coordinate.

Then natural coordinates in J+k are (y, y1, . . . , yk),

y ∈ R/Z, y` ∈ R, y1 > 0.

f(t) = y+ ty1 + t2y2 + . . .+ tkyk , t ∈ R .

J+k is a principal Gk bundle over S1, where Gk is

the Lie group of k jets of orientation preserving

diffeomorphisms of R which fix 0 ∈ R.

For any f ∈ J+k , g ∈ Gk, the product is just the

composition f g of the jets.

24

The one-trace dLog(ϕ′) on A = C0(J+1 ) o Γ

Γ acts on S1 by diffeomorphisms.

`(g) ∈ C∞(S1) is the logarithm of the Jacobian

of the diffeomorphism associated to g ∈ Γ:

`(g) = Logdg(y)

dy

1-cocycle ω ∈ Z1(Γ,Ω2J+1

) given by:

ω(g) = d`(g) ∧ dy1y1

δ(∑

Ug yg)=∑

Ug yg ωg

where for f ∈ C∞c (J+1 ), fωg is the element of

A∗ given by

〈h, fωg〉 =∫

J+1

h1 fωg , ∀h =∑

hg Ug ∈ A

25

Anabelian one-traces

Definition Let δ be a 1-trace on B, then δ is

anabelian iff the domain of the adjoint δ∗ con-

tains the center Z(B∗∗) and δ∗ = 0 on Z(B∗∗).

Then for any h ∈ Z(B∗∗) the product h δ is also

an anabelian one-trace.

Let δ = dLog(ϕ′) on A = C0(J+1 ) o Γ, then

a) δ is an anabelian one-trace.

b) δ is invariant under the action of G1 = R∗+.

c) Any u ∈ K1(A) defines a G1-invariant nor-

mal linear form on W (M) by

L(h) = 〈u−1, h δ(u)〉 , ∀h ∈W (M) ⊂ Z(A∗∗)

26

Assembly map (ac + pb)

Geometric group

K∗(V,Γ) is the K homology of the pair (Bτ, Sτ)

of the unit ball, unit sphere bundle of τ = TV

over VΓ.

A cycle (N,F, g) is a triple where N is a com-

pact manifold without boundary, F ∈ K∗(N)

a K-theory class, and g is a continuous map

from N to VΓ = V ×ΓEΓ, which is K-oriented,

i.e., such that the bundle TN ⊕ g∗τ is gifted

with a Spinc structure.

µ : K∗(V,Γ)→ K∗(C0(V )× Γ)

is the basic index construction of K-theory classes.

27

Geometric examples

Let (V, F ) be a foliated manifold, special cases

of the assembly map include

• Closed transversals N ⊂ V give elements of

K0(C∗(V, F )).

• Analytic longitudinal index

Index(DL) ∈ K(C∗(V, F ))

(is well defined in full generality, no trans-

verse measure (ac + gs)).

28

Chern Character

Let τ = TV and Hτ∗ (VΓ,Q) be the ordinary sin-

gular homology of the pair (Bτ, Sτ) over VΓ,

and with coefficients in Q. The Chern charac-

ter

ch : K∗(V,Γ)→ Hτ∗ (VΓ,Q)

is a rational isomorphism.

Thom isomorphism

Φ : Hτq+n(VΓ,Q)→ Hq(VΓ,Q) (n = dimV )

(where Φ(z) = p∗(U∩z), ∀ z ∈ Hq+n((Bτ, Sτ),Q)

and where p is the projection from Bτ to the

base VΓ).

Thus Φ ch is a rational isomorphism:

Φ ch : K∗(V,Γ)→ H∗(VΓ,Q) .

29

Index pairing

Let δ = dLog(ϕ′) on A = C0(J+1 )oΓ, then for

any x ∈ K∗(J+1 ,Γ) one has

〈µ(x), δ〉 = 〈Φchx, (Bπ)∗GV〉

where GV ∈ H3(WO1) is the Godbillon-Vey

class, and π : J+1 o Γ → G1 the natural homo-

morphism to the Haefliger groupoid.

One has the same formula replacing every-

where J+1 by S1 and the 1-trace δ by the fol-

lowing cyclic 2-cocycle τ on C∞c (S1 o Γ):

τ(f0, f1, f2) =∑∫

f0(γ0) f1(γ1) f

2(γ2)ω(g1, g2)

ω(g1, g2) = d`(g1g2) `(g2)− `(g1g2) d`(g2)

30

Hochschild cohomology H∗(A,A∗)

bτ(a0, a1, . . . , an+1) =

∑(−1)j τ(a0, . . . , ajaj+1, . . . , an+1)+

(−1)n+1 τ(an+1a0, . . . , an)

Cyclic cohomology HCn(A)

τ(a1, . . . , an, a0) = (−1)n τ(a0, . . . , an) , ∀ ai ∈ A

Exact Triangle

H∗(A,A∗)B I

HC∗(A)S−→ HC∗(A)

31

Densely defined cocycles on Banach alge-

bras

Definition

Let B be a Banach algebra. By an n-trace on

B we mean an n + 1 linear functional τ on a

dense subalgebra A of B such that

a) τ is a cyclic cocycle on A.

b) For any ai ∈ A, i = 1, . . . , n there exists

C = Ca1,...,an <∞ such that for all xj ∈ A,

|τ((x1da1)(x2da2) . . . (xndan))| ≤ C ‖x1‖ . . . ‖xn‖ ∀xi ∈ A .

32

Chern character for Banach algebras

Lemma

Let τ be an n-trace on a Banach algebra B.

Then there exists a map ϕ of Ki(B) (i = n(2))

to C such that:

a) If n is even and e ∈ ProjMq(Domain τ) then

ϕ([e]) = τ ⊗Tr (e, . . . , e) .

b) If n is odd and u ∈ GLq(Domain τ) then

ϕ([u]) = τ ⊗Tr (u−1, u, u−1, u, . . . , u−1, u) .

33

Forms and Currents

Space X Algebra A

Vector bundle Finite projectivemodule

Differential form Hochschild cycle

DeRham current Hochschild cocycle

DeRham homology Cyclic cohomology

Chern Weil theory Pairing 〈K(A), HC(A)〉

34

Hopf algebra H1 (ac + hm)

As an algebra H1 is the universal enveloping

algebra of the Lie algebra X, Y, δn ;n ≥ 1 and

brackets

[Y,X] = X , [Y, δn] = n δn , [X, δn] = δn+1

[δk, δ`] = 0 , n, k, ` ≥ 1 .

As a Hopf algebra, the coproduct ∆ : H1 →H1 ⊗H1 is determined by

∆Y = Y ⊗ 1 + 1⊗ Y , ∆ δ1 = δ1 ⊗ 1 + 1⊗ δ1

∆X = X ⊗ 1 + 1⊗X + δ1 ⊗ Y

and the multiplicativity property

∆(h1 h2) = ∆h1 ·∆h2 , h1, h2 ∈ H1

35

Action of H1 on A = C0(J+1 ) o Γ

J1+(S1) ' S1 × R+

j(t) = y+ t y1 + · · · , y1 > 0 ,

ϕ(y, y1) = (ϕ(y), ϕ′(y) · y1) .

The action of H1 is then given as follows:

Y (fU∗ϕ) = y1∂f

∂y1U∗ϕ , X(fU∗ϕ) = y1

∂f

∂yU∗ϕ

δn(fU∗ϕ) = yn1

dn

dyn

(log

dϕ

dy

)fU∗ϕ

The volume formdy ∧ dy1

y21on J1

+(S1) is in-

variant under Diff+(S1) and gives rise to the

following trace τ : A→ C,

τ(fU∗ϕ) =

∫J1+(S1) f(y, y1)

dy∧dy1y21

if ϕ = 1 ,

0 if ϕ 6= 1 .

(1)

36

The trace τ is ν-invariant under H1

The trace τ is ν-invariant with respect to the

action H1⊗A→ A and with the modular char-

acter ν ∈ H∗1 , determined by

ν(Y ) = 1, ν(X) = 0, ν(δn) = 0 ;

The invariance property is given by the identity

τ(h(a)) = ν(h) τ(a) , ∀h ∈ H1

The fact that S2 6= 1 is automatically cor-

rected by twisting with ν. Indeed, S = ν ∗ Ssatisfies

S2 = 1 .

37

Hopf cyclic cohomology HC∗Hopf(H)

Hopf algebra H, character ν with

S2 = 1 S = ν ∗ S

Cn(H) = H⊗n

Face operators ∂i : Cn−1(H)→ Cn(H) are

∂0(h1 ⊗ . . .⊗ hn−1) = 1⊗ h1 ⊗ . . .⊗ hn−1,

∂j(h1 ⊗ . . .⊗ hn−1) = h1 ⊗ . . .⊗∆hj ⊗ . . .⊗ hn−1 , 1 ≤ j ≤ n− 1 ,

∂n(h1 ⊗ . . .⊗ hn−1) = h1 ⊗ . . .⊗ hn−1 ⊗ 1 ;

Degeneracy operators σi : Cn+1(H) → Cn(H),

are

σi(h1 ⊗ . . .⊗ hn+1) = h1 ⊗ . . .⊗ ε(hi+1)⊗ . . .⊗ hn+1 ;

Cyclic operator τn : Cn(H)→ Cn(H)

τn(h1⊗. . .⊗hn) = (∆n−1S(h1))·h2⊗. . .⊗hn⊗1 .

38

Normalized bicomplex (CC∗,∗(H), b, B)

The Hopf cyclic cohomology is computed from

the bicomplex CCp,q, where:

CCp,q(H) = Cq−p(H), q ≥ p,CCp,q(H) = 0, q < p ;

with

Cn(H) = ∩ Ker σi , ∀n ≥ 1, C0(H) = C;

b : Cn−1(H)→ Cn(H), b =n∑

i=0

(−1)i∂i

B = A B0 , n ≥ 0 ,

where

A = 1 + (−1)nτn + . . .+ (−1)n2τnn .

B0(h1⊗. . .⊗hn+1) = (∆n−1S(h1))·h2⊗. . .⊗hn+1

B0(h) = ν(h), h ∈ H

39

Characteristic map HC∗Hopf (H1) → HC∗ (A)

One lets A ⊂ A be the smooth subalgebra.

The map

χτ(h1⊗. . .⊗hn)(a0, . . . , an) = τ(a0 h1(a1) . . . hn(an))

where h1, . . . , hn ∈ H1 and a0, a1, . . . , an ∈ A ,

induces a characteristic homomorphism

χ∗τ : HC∗Hopf (H1) → HC∗ (A) .

40

Characteristic map and Gelfand Fuchs co-

homology

κ∗1 : H∗(a1,C)'−→ PHC∗Hopf (H1) ,

The element δ1 ∈ H1 is a Hopf cyclic cocycle,

which gives a nontrivial class

[δ1] ∈ HC1Hopf (H1) .

Moreover, [δ1] is a generator for PHCoddHopf (H1)

and corresponds to the Godbillon-Vey class in

the isomorphism κ∗1 with the Gelfand-Fuchs co-

homology.

41

Schwarzian derivative

Schwarzian derivative

y ; x :=d2

dx2

(log

dy

dx

)− 1

2

(d

dx

(log

dy

dx

))2

.

The element δ′2 := δ2 − 12δ

21 ∈ H1 is a Hopf

cyclic cocycle, whose action on the crossed

product algebra A = C∞c (J1+(S1)) o Γ is given

by the Schwarzian derivative

δ′2(fU∗ϕ) = y21 ϕ(y) ; y fU∗ϕ

and whose class

[δ′2] ∈ HC1Hopf (H1)

is equal to B(c), where c is the following Hochschild

2-cocycle,

c := δ1 ⊗X +1

2δ21 ⊗ Y .

42

Transverse fundamental class

The generator of PHCevenHopf(H1) is the class of

the cyclic 2-cocycle

F := X ⊗ Y − Y ⊗X − δ1 Y ⊗ Y ,

which in the foliation context represents the

‘transverse fundamental class’.

It is “integral” and is the Chern character in

K-homology of the spectral triple given by the

hypoelliptic transverse signature operator:

Q =

(0 Y 2 +X

Y 2 −X 0

)

43

Local index formula (ac + hm)

• The equality∫−P = Resz=0 Trace (P |D|−z)

defines a trace on the algebra generated by

A, [D,A] and |D|z, where z ∈ C.

• Cocycle in the bicomplex (b, B) of A,

ϕn(a0, . . . , an) =

∑

k

cn,k

∫−a0[D, a1](k1) . . . [D, an](kn) |D|−n−2|k|

• The pairing of the cyclic cohomology class

(ϕn) ∈ HC∗(A) with K1(A) gives the Fred-

holm index of D with coefficients in K1(A).

44

Modular Hecke algebra (ac + hm)

α · z =az + b

cz+ d, j(α, z) = cz + d

f |k α (z) = det(α)k/2 f(α · z) j(α, z)−k

Let Γ be a congruence subgroup. By a Hecke

operator form of level Γ we mean a map

F : Γ\GL+2 (Q)→M , Γα 7→ Fα ∈ M ,

with finite support and satisfying the covari-

ance condition

Fαγ = Fα|γ , ∀α ∈ GL+2 (Q), γ ∈ Γ .

(F1 ∗ F2)α :=∑

β∈Γ\GL+2 (Q)

F1β · F2

αβ−1|β

turns the vector space A(Γ) of all Hecke opera-

tor forms of level Γ into an associative algebra.

45

Hopf action of H1 on A(Γ)

The Hopf algebra H1 admits a canonical action

on A(Γ).

Y (f) =k

2· f , ∀ f ∈ Mk .

X :=1

2πi

d

dz− 1

12πi

d

dz(log∆) · Y

(where ∆(z) = η24 = q∏∞n=1(1 − qn)24 with

q = e2πiz)

Y (F )α := Y (Fα) , ∀F ∈ A(Γ) , α ∈ G+(Q) ,

X(F )α := X(Fα) ,

δn(F )α := µn,α · Fα ,where

µn, α := Xn−1(µα) , ∀n ∈ N .

µγ (z) =1

2πi

d

dzlog

η4|γη4

.

46

Z(z) :=2πi

3

∫ z

i∞η4dz ,

dZ :=1

3η4dq

q=

2πi

3η4 dz .

47

J+1 (S1) L−2

S1 Γ′(1)\H∗

θ ∈ R/2π Z Z ∈ C/Λ

ei θ = cosθ+ i sinθ (℘Λ(Z), ℘′Λ(Z)) =

( 3√j, −2 E6

η12 )

φ(θ) Z| γ

φ′(θ) J(γ) =dZ| γdZ

( ddθ)nLog (φ′(θ)) ( d

dZ)nLog (J(γ))

48

Hopf Symmetry of Modular Hecke Alge-

bras

Let Γ be any congruence subgroup.

10. The above define a Hopf action of the Hopf

algebra H1 on the algebra A(Γ).

20. The Schwarzian derivation δ′2 = δ2− 12δ

21 is

inner and is implemented by ω4 = −E472 ∈

A(Γ) , E4(q) := 1 + 240∑∞

1 n3 qn

1−qn .

30. The image of the tranverse fundamental

class [F ] ∈ HC2Hopf(H1) under the canoni-

cal map from the Hopf cyclic cohomology

of H1 to the Hochschild cohomology of

A(Γ), gives the natural extension of the

first Rankin-Cohen bracket · , ·1 to the

algebra A(Γ).

49

Some available papers

Here are relevant papers that can be down-

loaded on the site

http://www.alainconnes.org/downloads.html

10. A survey of foliations and operator alge-

bras.

20. Cyclic cohomology and the transverse fun-

damental class of a foliation.

30. Hopf algebras, cyclic cohomology and the

transverse index theorem (with Henri Moscovici).

40. Modular Hecke Algebras and their Hopf Sym-

metry (with Henri Moscovici).

50. L2 Homology for von-Neumann Algebras

(with D. Shlyakhtenko).

50