On Connes’ Godbillon-Vey Theorem

Nigel Higson

Department of MathematicsPennsylvania State University

Victoria, July 2009

Nigel Higson

Introduction

The following quite spectacular theorem relates the differentialtopology of a foliation (V ,F ), in the form of the Godbillon-Veyclass, to the measure theory of (V ,F ), in the form on Connes’foliation von Neumann algebra.

Theorem (Connes, 1984)Let (V ,F ) be an oriented and transversally oriented,codimension-one foliation.

If the Godbillon-Vey class of (V ,F ) is nonzero, then the vonNeumann algebra of the foliation admits a nonzero invariantmeasure on its flow of weights.

In particular, the von Neumann algebra has a Type IIIcomponent.

Nigel Higson

Contents

The goal is to prove the theorem using de Rham theory.

Preliminaries on differential formsThe von Neumann algebra of a foliationGeometry: jet spaces and characteristic classesConstruction of invariant measures

Homotopy invariance of cohomology will imply the measuresare invariant.

Poincaré duality will guarantee that at least one of themeasures is nonzero.

Nigel Higson

Differential Forms

We shall need to work extensively with differential forms sincethe Godbillon-Vey class is defined using differential forms on V .

Differential forms are used to generalize the fundamentaltheorem of calculus to higher dimensions, as Stokes theorem.

Ω∗ = Graded-commutative algebra.Ω0 = C∞(V ).Ω1 is dual to the C∞(V )-module of vector fields on V .There exists d : Ωp → Ωp+1, obeying Leibniz, such that

〈X ,df 〉 = X (f )

on functions.Ω∗ is contravariantly functorial.

Nigel Higson

Differential Forms and Cohomology

One has d2 = 0 and the de Rham cohomology groups of V are

HpdR(V ) = kernel(d : Ωp → Ωp+1

)/image

(d : Ωp−1 → Ωp

)= closed forms

/exact forms.

We shall use the Poincaré Lemma and Poincaré Duality:

TheoremIf ν is a closed differential form on an oriented manifold V , andif the cohomology class of ν is nonzero, then there is a closed,compactly supported differential form ω such that∫

Vν ∧ ω 6= 0.

Nigel Higson

Foliations and Differential Forms

V = Smooth manifoldF = Foliation of V

The distribution F ⊆ TV is integrable, that is, [F ,F ] ⊆ F , and byFrobenius F determines the leaves of the foliation uniquely.

The foliation is also determined by the ideal of differential formson V generated by the one-forms that vanish on F . Integrabilityof F implies that it is a differential ideal.

Nigel Higson

Flat Bundle Construction

One source of foliations:

V = (T × L)/Γ

Here:Γ acts freely and properly on LΓ acts on T in any mannerF consists of the vectors tangent to L.

For example, the Kroneckerfoliation is of this type:

L = real lineT = circleΓ = integers

↓

Nigel Higson

Codimension-One Foliations

We shall be concerned with codimension-one foliations of V ,whose leaves have dimension one less than the dimension ofV . Thus

rank(F ) + 1 = rank(TV ).

In the flat bundle examples, this means the transversal space Thas dimension one.

The foliation is transversally oriented if there is a one-form ω onV such that

F = kernel(ω).

The foliation is oriented if the leaves are consistently oriented,meaning that F is oriented.

Oriented + transversally oriented⇒ V is oriented.

Nigel Higson

The Godbillon-Vey Class

(V ,F ) = Codimension-one, transversally oriented, oriented.ω = An associated one-form on V , as above.

LemmaThere is a one-form α on V such that dω = ω ∧ α.

This is the differential forms version of the fact that F isintegrable.

Theorem (Godbillon-Vey)The differential form α ∧ dα is closed. Its cohomology class

GV = [α ∧ dα] ∈ H3dR(V )

is independent of ω and α.

Nigel Higson

The Roussarie Example

G = PSL(2,R)K = PSO(2)P = Upper triangular matrices in G

Iwasawa decomposition:

G = K × P = G/P ×G/K = RP1 ×H2.

If Γ = surface group, then we can carry out the flat bundleconstruction

V = G/Γ = (RP1 ×H2)/Γ.

This is a closed, oriented, foliated 3-manifold.

Proposition (Roussarie)In this example, GV 6= 0 in H3dR(V )

∼= R.

Nigel Higson

The von Neumann Algebra of a Foliation

(V ,F ) = Foliated manifoldvN(V/F ) = Connes’ foliation von Neumann algebra

vN(V/F ) = algebra of measurable families of boundedoperators Tv , parametrized by v ∈ V , where:

Tv acts on the L2-space of the leaf through v .Tv1 = Tv2 if v1 and v2 lie on the same leaf.

PropositionIf V = (T × L)/Γ (and if Γ acts analytically), then vN(V/F ) isequivalent to the crossed product (group measure spaceconstruction) vN(T/Γ) associated to the action of Γ on T .

Nigel Higson

KMS and the Flow of Weights

M = von Neumann algebra

Theorem (Tomita, Takesaki, Connes)The process

Faithful normal state 7→ Modular automorphism group7→ Crossed product by modular group7→ Center with dual R-action

constructs an invariant of M.

The invariant, the flow of weights, is a flow on a measurablespace.

The theorem is difficult to prove, but the invariant is easy tocompute.

Nigel Higson

Crossed Product von Neumann Algebras

Γ× T −→ T Action on a measurable space.µ Fully supported measure on T .

The measure µ gives a faithful normal state on vN(T/Γ).

Define an action Γ on T × R+ by

g : (t , x) 7→(g · t , dg∗µ

dµ(t) · x

).

It commutes with the obvious multiplication action of R+.

PropositionThe flow of weights for the crossed product vN(T/Γ) is themultiplication action of R+ on L∞(T × R+)Γ.

We identify R and R+ via the exponential.

Nigel Higson

Measures from the Godbillon-Vey Class

The theorem we are trying to prove boils down to this:

Theorem (Connes)Assume that Γ acts on a circle or line T by orientationpreserving diffeomorphisms.

Assume also that for some oriented V =(T × L

)/Γ the

Godbillon-Vey class GV ∈ H3dR(V ) is nonzero.

Then there is a nonzero flow-invariant finite measure on themeasurable space of Γ-invariant subsets of T × R+.

Nigel Higson

Jets and Secondary Classes

T = Smooth oriented manifold of dimension n

DefinitionA k-jet at t ∈ T is an equivalence class of localdiffeomorphisms Rn → T at t (mapping 0 to t).

Two local diffeos are equivalent if they agree to k th order at 0.

DefinitionDenote Tk be the smooth bundle over T with fiber over t ∈ Tthe k -jets at t .

The group of orientation-preserving diffeomorphisms of Tacts on Tk on the right, by composition.The group Gk of k -jets at 0 ∈ Rn acts on the right, also bycomposition. This is a finite-dimensional Lie group,coordinatized by Taylor coefficients.

Nigel Higson

Thickenings and Gelfand-Fuks

Let V =(T × L

)/Γ, as usual.

Define Vk =(Tk × L

)/Γ.

There is Vk −→ V . Fibers are copies of Gk .When dim(T ) = 1 the map is a homotopy equivalence.

Gelfand-Fuks: The process

Closed Diff+(T )-invariant forms on Tk7→ Closed Γ-invariant forms on Tk × L7→ Closed forms on Vk =

(Tk × L

)/Γ

7→ Classes in H∗dR(Vk )7→ Classes in H∗dR(V )

creates characteristic classes for foliations.

Nigel Higson

Codimension One

Assume dim(T ) = 1 and fix a (local) coordinate on T .

ThenT1 = T ×G1 = T × R+

and Diff+(T ) acts by φ · (t , x) = (φ(t), φ′(t)x). In addition

T2 = T ×G2,

where

G2 ={(x y

0 x2

): x > o

}and Diff+(T ) acts by

φ ·(t ,(

x y0 x2

))=(φ(t),

(φ′(t) φ′′(t)

0 φ′(t)2

)(x y0 x2

)).

Nigel Higson

Godbillon-Vey in the Gelfand-Fuks Framework

LemmaThe differential form

η =1x3

dt ∧ dx ∧ dy

on T2 is closed and Diff+(T )-invariant.

The associated characteristic class is the Godbillon-Vey class

The Essential Point:The Godbillon-Vey class corresponds to an invariant 3-form onthe manifold T2, and this manifold happens to be 3-dimensional.

As a result GV corresponds to an invariant measure.

Nigel Higson

Ruelle-Sullivan Current

V =(T × L)

/Γ Oriented foliation (flat bundle construction).

ννν = Γ-invariant measure on T .

Ruelle-Sullivan current: Cννν : Ωdim(L)c (V ) −−−−−−−→ R

Projection

V

U

W

Lift ω, supported on thechart in V , to ω̃supported on the chartabove.

If ω is supported in a smallchart in V , then

Cννν(ω) =∫T

(∫{t}×L

ω̃

)dν(t).

In general, use partitions ofunity.

Nigel Higson

Cohomological Properties

LemmaThe Ruelle-Sullivan current

Cννν : Ωdim(L)c (V ) −→ R

is closed: Cννν(dη) = 0 for all η.

As a result, Cννν defines a map

Cννν : Hdim(L)dR,c (V ) −→ R.

ExampleIf the invariant measure ννν is associated to a volume form ν onT , then

Cννν(ω) =∫

Vν ∧ ω.

Nigel Higson

Some Simple Invariance Properties

Functoriality

ννν Γ-invariant measure on T , as before.

f : T → T Γ-equivariant diffeomorphism.

Cf∗ννν(ω) = Cννν(f∗ω)

Restriction to an Invariant Set

E Γ-invariant subset of T .

νννE (S) := ννν(E ∩ S).

If f∗ννν = ννν, then

Cννν f∗E (ω) = Cf∗νννE (ω) = CνννE (f∗ω).

Nigel Higson

Construction of Finite Invariant Measures

Ingredients

The manifold T2.The Diff+(T )-invariant volume form ν.A closed compactly supported form on V2 =

(T2 × L

)/Γ.

Recipe

We want a measure on the Γ-invariant subsets of T × R+.

Recall that T × R+ = T1.

Do this:

Γ-invariant set in T1 7→ Inverse image E in T2 7→ CνννE (ω)

This is flow-invariant, by the homotopy invariance ofcohomology.

Nigel Higson

Non-Vanishing and Poincaré Duality

SummaryFor each closed, compactly supported ω on V2, there is aninvariant measure on the flow of weights T1.

But are any of them non-zero?

The total measure of the space T1 is Cννν(ω), and as we noted,

Cννν(ω) =∫

V2ν ∧ ω

But the cohomology class of ν is nonzero, so, according toPoincaré duality, the integral is nonzero for some ω.

This completes the proof of Connes’ theorem.

Nigel Higson