Joint International Meeting

American Mathematical Society

Real Sociedad Matematica Espanola

Sevilla, June 20, 2003

Characteristic classes ofRiemannian foliations

Rationality properties of the secondary classes of Riemannian

foliations and some relations between the values of the classes

and the geometry of Riemannian foliations are discussed.

Steven HurderUniversity of Illinois at Chicago

http://www.math.uic.edu/∼hurder

1

In memorial ofConnor Lazarov, 1937 – 2003

Born – September 26, 1937

Died – February 27, 2003

Connor was the only child of Julian and Mildred Lazarov,who were immigrants from Poland. Many of Connor’srelatives, who remained in Poland, subsequently died inthe Holocaust.

Connor married Sharon Rosenthal in 1970.

Connor was an avid skier and sailor.

B.A. University of Michigan 1959

Ph.D. Harvard University 1966 (advisor was Raoul Bott)

Member Institute for Advanced Studies, 1974-75

Professor of mathematics and computer science at LehmanCollege of CUNY 1968–2003

26 papers in major mathematics research journals (in-cluding the ”Annals” and ”Inventiones”).

Coauthors were James Heitsch, Joel Pasternack, HerbShulman, Al Vasquez, Arthur Wasserman

2

Publications by Connor Lazarov

[1] Connor Lazarov. Secondary characteristic classes in K-theory.Trans. Amer. Math. Soc., 136:391–412, 1969.

[2] Connor Lazarov and Arthur Wasserman. Cobordism of regular0(n)-manifolds. Bull. Amer. Math. Soc., 75:1365–1368, 1969.

[3] C. Lazarov and A. Vasquez. The difference construction inK-theory. Proc. Amer. Math. Soc., 21:315–317, 1969.

[4] Connor Lazarov and Arthur Wasserman. Cobordism of regularO(n)-manifolds. Ann. of Math. (2), 93:229–251, 1971.

[5] Connor Lazarov. Actions of groups of order pq. Trans. Amer.Math. Soc., 173:215–230, 1972.

[6] Connor Lazarov. Quillen’s theorem for MU∗. In Proceedings ofthe Second Conference on Compact Transformation Groups (Univ.Massachusetts, Amherst, Mass., 1971), Part I, pages 183–190.Lecture Notes in Math., Vol. 298, Berlin, 1972. Springer.

[7] Connor Lazarov and Arthur Wasserman. Cobordism of U(n)-actions. Bull. Amer. Math. Soc., 78:1035–1038, 1972.

[8] Connor Lazarov and Arthur Wasserman. Complex actions of Liegroups. American Mathematical Society, Providence, R.I., 1973.Memoirs of the American Mathematical Society, No. 137.

[9] Connor Lazarov and Arthur G. Wasserman. Free actions andcomplex cobordism. Proc. Amer. Math. Soc., 47:215–217, 1975.

[10] Connor Lazarov and Joel Pasternack. Residues and charac-teristic classes for Riemannian foliations. J. Differential Geometry,11(4):599–612, 1976.

[11] Connor Lazarov and Joel Pasternack. Secondary character-istic classes for Riemannian foliations. J. Differential Geometry,11(3):365–385, 1976.

[12] Connor Lazarov and Herbert Shulman. Obstructions to folia-tion preserving Lie group actions. Topology, 18(3):255–256, 1979.

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[13] Connor Lazarov. A permanence theorem for exotic classes. J.Differential Geom., 14(3):475–486 (1980), 1979.

[14] Connor Lazarov. A vanishing theorem for certain rigid classes.Michigan Math. J., 28(1):89–95, 1981.

[15] Connor Lazarov and Herbert Shulman. Obstructions to foliation-preserving vector fields. J. Pure Appl. Algebra, 24(2):171–178,1982.

[16] Connor Lazarov. An index theorem for foliations. Illinois J.Math., 30(1):101–121, 1986.

[17] Connor Lazarov. Spectral invariants of foliations. MichiganMath. J., 33(2):231–243, 1986.

[18] Connor Lazarov. A relation between index and exotic classes.In Index theory of elliptic operators, foliations, and operator al-gebras (New Orleans, LA/Indianapolis, IN, 1986), volume 70 ofContemp. Math., pages 125–143. Amer. Math. Soc., Providence,RI, 1988.

[19] James L. Heitsch and Connor Lazarov. A Lefschetz theoremfor foliated manifolds. Topology, 29(2):127–162, 1990.

[20] James L. Heitsch and Connor Lazarov. A Lefschetz theorem onopen manifolds. In Geometric and topological invariants of ellipticoperators (Brunswick, ME, 1988), volume 105 of Contemp. Math.,pages 33–45. Amer. Math. Soc., Providence, RI, 1990.

[21] James L. Heitsch and Connor Lazarov. Homotopy invarianceof foliation Betti numbers. Invent. Math., 104(2):321–347, 1991.

[22] James L. Heitsch and Connor Lazarov. Rigidity theorems forfoliations by surfaces and spin manifolds. Michigan Math. J.,38(2):285–297, 1991.

[23] James L. Heitsch and Connor Lazarov. Spectral asymptoticsof foliated manifolds. Illinois J. Math., 38(4):653–678, 1994.

[24] James L. Heitsch and Connor Lazarov. A general families indextheorem. K-Theory, 18(2):181–202, 1999.

[25] C. Lazarov. Transverse index and periodic orbits. Geom.Funct. Anal., 10(1):124–159, 2000.

[26] James L. Heitsch and Connor Lazarov. Riemann-Roch-Grothendieckand torsion for foliations. J. Geom. Anal., 12(3):437–468, 2002.

4

Characteristic Classes

Let F be a Riemannian foliation of smooth manifold M with tangentbundle TF of dimension p. Choose a Riemannian metric g on TMwhich is projectable when restricted to the normal bundle Q = TF⊥.Let ∇g be the basic connection on Q associated to this metric.

For each 1 ≤ i ≤ q/2 the closed Pontrjagin form pi(∇g) ∈ Ω4i(M)determines the Pontrjagin class pi ∈ H4i(M).

If q = 2n and Q is oriented, there is also the Euler class χq ∈ Hq(M)whose square χ2

q = pn.

The weight of an index I = (i1 < i2 < · · · < ik) is

|I| = 4(i1 + 2i2 + · · · + kik)

The degree of a monomial term pI = pi1 · · · pik is the weight of |I|.In the case q is even, the degree of the monomial χq · pI is |I| + q.We let P denote a monomial of either pI or χq · pI type.

THEOREM: (Pasternack) If degP > q then

P (∇g) = pi1(∇g) ∧ · · · ∧ pik(∇g) = 0

DEFINITION: If q = 2n, set

R[χq, p1, p2, . . . , pn−1]q ≡ R[χq, p1, p2, . . . , pn−1]/P | degP > qand for q = 2n + 1, set

R[p1, p2, . . . , pn]q ≡ R[p1, p2, . . . , pn]/P | degP > q

COROLLARY: (Pasternack) There is a well-defined map

q = 2n, ∆∗:R[χq, p1, p2, . . . , pn−1]q → H∗(M)

q = 2n + 1, ∆∗:R[p1, p2, . . . , pn]q → H∗(M)

5

Transgression Classes

Assume there is a trivialization s: εq = M × Rq ∼= Q.

Let ∇s be the associated flat connection for which s is parallel.

Set ∇t = (1 − t)∇g + t∇s with pi(∇t) ∈ Ω4i(M × R).

hi(∇g, s) =

∫ 1

0

pi(∇t) ∈ Ω4i−1(M)

dhi(∇g, s) = pi(∇g)

REMARK: i > q/4 =⇒ dhi(∇g, s) = 0.

If q = 2n, then also set

hχq(∇g, s) =

∫ 1

0

χq(∇t) ∈ Ωq−1(M)

DEFINITION: (q = 2n)

RWq = R[χq, p1, p2, . . . , pn−1]q ⊗ Λ(hχq, h1, . . . , hn−1

)d(1 ⊗ hχq) = χq ⊗ 1, d(1 ⊗ pi) = pi ⊗ 1

DEFINITION: (q = 2n + 1)

RWq = R[p1, p2, . . . , pn]q ⊗ Λ(h1, . . . , hn)

d(1 ⊗ pi) = pi ⊗ 1

DEFINITION: The basic connection ∇g and framing s define themap of DGA’s

∆(∇g, s):RWq → Ω(M)

pi → pi(∇g, s)

χq → χq(∇g, s)

hi → hi(∇g, s)

hχq → hχq(∇g, s)

6

Secondary Classes

THEOREM: (Lazarov) Assume that Q has framing s. The mapon cohomology

∆∗(s):H∗(RWq) → H∗(M)

is independent of the choice of basic connection ∇g, and dependsonly on the homotopy class of the framing s.

THEOREM: (Lazarov) Suppose that two sections s, s′ of Q arerelated by a gauge transformation ϕ:M → SO(q),

s′(x) = s(x) · ϕ(x), x ∈ M

Then on the level of forms,

∆(∇g, s′)(hi) = ∆(∇g, s)(hi) + ϕ∗(τi)

where τi ∈ Ω2i−1(SO(q)) is the transgressive class formed from theMaurer-Cartan form. In particular, for i > q/4,

∆∗(s′)(hi) = ∆∗(s)(hi) + ϕ∗[τi] ∈ H2i−1(M)

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Some Questions

QUESTION 1: Given a class z ∈ H∗(RWq), is there a Riemannianfoliation of codimension q with framed normal bundle such that∆∗(s)(z) = 0?

QUESTION 1’: Given a class z ∈ H∗(RWq), is there a Riemannianfoliation of codimension q with framed normal bundle on a compactmanifold M such that ∆∗(s)(z) = 0?

QUESTION 2: Given a class z ∈ H∗(RWq), how does the valueof ∆∗(s)(z) ∈ H∗(M) depend upon “the geometry and topology”of F?

QUESTION 3: (Molino, Tokyo 1993) How do the values of theclasses z ∈ H∗(RWq) relate to the Molino structure theory of F?

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Universal Classifying Spaces

An RΓq–structure on M is an open covering U = Uα | α ∈ A andfor each α ∈ A, there is given

(i) a smooth map fα:U → Rq

(ii) a Riemannian metric gα on Rq

such that the pull-backs f−1α (TRq) → Uα define a vector bundle

Q → M with Riemannian metric g | Q = gα = f∗αgα

THEOREM: (Haefliger) A Riemannian foliation F on M definesan RΓq–structure on M . The homotopy class of the composition

hF :M ∼= BU → BRΓq

depends only on the integrable homotopy class of F.

The normal bundle to the universal RΓq–structure on BRΓq admitsa classifying map ν:BRΓq → BO(q). The homotopy fiber of ν is

the space BRΓq This classifies RΓq–structures with a (homotopyclass of) framing for Q.

M

P

O(q)

↓↑s

↓hF ,s−→

=

hF−→ BRΓq

ν↓BO(q)

BRΓq

BO(q)

=

=

ν↓

FRΓq

↓BRΓq

↓

O(q)

↓ΩBO(q)

↓

QUESTION 4: What is the homotopy type of BRΓq? of BRΓq?

9

Homotopy Type of BRΓq

THEOREM: (Pasternack) BRΓ1 BRδ

THEOREM: (Lazarov) If q = 4k − 2,4k − 1, then

π4k−1(BRΓq) → R → 0, > 0

THEOREM: (Hurder) ν:BRΓq → BO(q) is a q-connected map.

That is, BRΓq is q − 1-connected.

Proof: Milnor’s remark =⇒ by the Phillips immersion theorem, anRΓq–structure on S for 0 < < q corresponds to a Riemannianmetric defined on an open neighborhood retract S ⊂ U ⊂ Rq.

An RΓq–structure on an open set in Rq is explicitly homotopic tothe Euclidean metric on an open neighborhood retract S ⊂ V ⊂ Rq,which is the “trivial” RΓq–structure.

Thus, the given RΓq–structure on S can be homotoped to a trivial

structure, hence represents the trivial RΓq–structure on S.

QUESTION 5: Is BRΓq q-connected for q = 4k − 1?

QUESTION 5’: What are the integrable homotopy invariants ofa 1–dimensional Riemannian foliation on an open manifold?

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Injectivity of the Pontrjagin Classes

THEOREM: (Thom) There is a compact, orientable Riemannianmanifold B of dimension q such that all of the Pontrjagin and Eulerclasses up to degree q are independent in H∗(B).

If q is odd, then B can be chosen to be a connected manifold.

Proof: For q even, B is the disjoint union of all products of theform

CPi1 × · · · × CPik × S1 × · · · × S1

with dimension q.

For q odd, B is the connected sum of all products of the form

CPi1 × · · · × CPik × S1 × · · · × S1

with dimension q.

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Injectivity of the Secondary Classes

THEOREM: (Hurder) There exists a compact manifold M and aRiemannian foliation F on M with trivial normal bundle, such thatF is defined by a fibration over a compact manifold of dimensionq, and the secondary characteristic map injects:

∆∗(s):H∗(RWq) → H∗(M)

Moreover, if q is odd, then M can be chosen to be connected.

Proof: Let B be the compact, orientable Riemannian manifolddefined in the proof of Thom’s Theorem.

Let M be the bundle of oriented orthonormal frames for TB.

The basepoint map π:M → B defines a fibration

SO(q) → M → B

with fiber over x ∈ B the group SO(q) of orthonormal frames inTxB.

F is the foliation defined by the fibration. The Riemannian metricon B lifts to the transverse metric on the normal bundle Q = π−1TB.s is the canonical framing of Q.

The basic connection ∇g and framing s define the map of DGA’s

∆(∇g, s):RWq → Ω(M)

Consider a fiber Lx = π−1(x) – the normal bundle restricted to Lx

is trivial, as it is just the constant lift of TxB. Thus, the basicconnection ∇g is the connection associated to the product bundle.

The connection ∇s for which the canonical framing is parallel,equals the connection defined by the Maurer-Cartan form on SO(q).

12

By Chern-Weil theory, the forms hi(∇g, s) and hχq(∇g, s) restrictedto Lx are closed, and their classes in cohomology define free exteriorgenerators for the cohomology H∗(SO(q)).

The characteristic map on forms

∆(∇g, s):RWq → Ω(M)

induces a map of the basic spectral sequence,

∆∗,∗p :E∗,∗

2 (RWq) → E∗,∗p (M)

At the E2 stage, this is

∆∗,∗2 :RWq → H∗(SO(q)) ⊗ H∗(B)

which is injective by construction.

Pass to the E∞–limit to obtain that

∆∗(s):H∗(RWq) → H∗(M)

induces an injective map of associated graded algebras, hence it isalso injective.

Note that in this proof:

• All leaves are compact.

• The “geometry” of F (seems) to make no difference.

• The image of the “basis” classes are integral

∆∗(s)(hJpI) = ∆∗(s)(hj1∧· · ·∧hi·pi1 · · · pik) ∈ H∗(M,R) ← H∗(M,Z)

QUESTION 1”: Does there exists a compact connected manifoldM and a Riemannian foliation F with even codimension and trivialnormal bundle, such that the secondary characteristic map ∆∗(s)injects?

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Variable and Rigid Classes

The secondary classes of a foliation are divided into two types,variable and rigid.

The variable classes are most sensitive to geometry and dynamics.

The rigid classes seem to be topological in nature.

A basis element

hJpI = hj1 ∧ · · · ∧ hi · pi1 · · · pik

is a rigid class if it lies in the image of

H∗(RWq+1) → H∗(RWq)

The variable classes are the cokernel of this map.

A rigid class is invariant for a deformation Ft, 0 ≤ t ≤ 1 throughRiemannian foliations: The family Ft defines a Riemannian foliationF × [0,1] on M × [0,1] of codimension q + 1, and the rigid classesare defined for F × [0,1] so restrict to the same class at the ends.

The variable classes do not have this property, so a priori may varynon-trivially under deformation.

The variable classes exist only when q = 4k − 2 or q = 4k − 1.

A spanning set for the variable classes is obtained by consideringall terms

hj1 · pi1 · · · pik ∈ RWq

of degree 4k − 1, where j1 ≤ i for all . We then extend this set toinclude all terms hj1 ∧ · · · ∧ hi · pi1 · · · pik where j > ji for > 1.

14

Integrality

The integrality of the secondary classes in the proof of the non-triviality theorem illustrates a general property of the secondaryclasses:

THEOREM: (Hurder) Let F be a Riemannian foliation of codi-mension q on M with trivial normal bundle. Suppose there existsa manifold B of dimension q and a submersion π:M → B whosefibers define F.

If q = 2n is even, then the image of the basis classes hJpI arerational

∆∗(s)(hJpI) ∈ Image H∗(M,Q) → H∗(M,R)When deg pI = q, the image of ∆∗(s)(hJpI) lies in the image of theintegral cohomology.

If q = 2n + 1 is odd, then the image of the rigid basis classes arerational

∆∗(s)(hJpI) ∈ Image H∗(M,Q) → H∗(M,R)When deg pI = 2n + 2, the image of ∆∗(s)(hJpI) lies in the imageof the integral cohomology.

Proof: Use standard rational homotopy techniques and functorial-ity of the characteristic maps to show that an integral (or rational)model of RWq factors through the rational de Rham complex of M .

The restriction on the types of classes for q odd is essential. Theproof uses “rigidity” properties of the classes, but applied within thelevel of DGA’s and not to a geometric deformation of the foliation.

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Compact Hausdorff Foliations

DEFINITION: A foliation of a manifold M is compact Hausdorffif every leaf of F is a compact manifold, and the leaf space M/Fis a Hausdorff space.

THEOREM: (Epstein, Millet) A compact Hausdorff foliation F isRiemannian – there exists a transverse holonomy invariant Rieman-nian metric on the normal bundle Q for F.

THEOREM: (Hurder) Let F be a compact Hausdorff foliation ofcodimension q on M with trivial normal bundle. If hJpI is a rigidclass, or if q is even and hJpI is any basis element, then

∆∗(s)(hJpI) ∈ Image H∗(M,Q) → H∗(M,R)

16

Super-rigidity and Riemannian Foliations

THEOREM: Suppose that M is a compact manifold with funda-mental group Γ. Assume that Γ is an irreducible lattice of real-rankr. (e.g., Γ = SL(r + 1,Z)) Then every Riemannian foliation of Mof codimension q < r is compact Hausdorff.

When the codimension q > r where r is the real-rank of Γ, thenDupont and Kamber have obtained rationality results for certain ofthe secondary classes, when F is a point foliation. See “Cheeger-Chern-Simons classes of transversally symmetric foliations,” Math.Ann. 295 (1993)

17

Continuous Variation

Consider S3 as the Lie group SU(2) with Lie algebra spanned by

X =[

i 00 −i

], Y =

[0 ii 0

], Z =

[0 −11 0

].

THEOREM: (Chern-Simons) Let gu be the Riemannian metric onS3 where the parallel Lie vector fields u ·X, Y, Z is an orthonormalbasis. Then for Fu the point-foliation on S3 with transverse metricgu, ∆∗(s)(h1) ∈ H3(M) is a non-constant function of u.

THEOREM: (Lazarov) Let q = 4k − 2. Then for every non-zero z ∈ H4k−1(RWq), there is a family of Riemannian foliationsFt with trivial normal bundle on a compact manifold M such thatthe images of the class ∆(s)(z) ∈ H4k−1(M) is a non-constantcontinuous function of t.

Lazarov proved much more – he showed the complete independentvariation of all classes in these degrees.

THEOREM: (Lazarov) Let q = 4k−2. The universal characteristicmap

∆∗:Hq+1(RWq) → Hq+1(BRΓq)is injective, and all classes in the image “vary independently”.

The geometry of deformation examples involves either:

• For q = 4k−1, a deformation of the transverse Riemannian metricon Q. This is analyzed via Cheeger-Chern-Simons’ theory.

• For q = 4k − 2, a geometric deformation of the foliation Ft in aneighborhood of a singular set. This is analyzed via residue theory.

18

Molino Theory

Let F be a Riemannian foliation of codimension q on a compactmanifold M , with oriented normal bundle Q.

Let π:P → M be the bundle of oriented orthonormal frames in Q,with fiber group SO(q).

The foliation F is covered by a Riemannian foliation F on P of thesame dimension.

THEOREM: (Molino) The closures of the leaves of F define a

Riemannian foliation G = F of P with all leaves compact and noholonomy. Thus, G is defined by a fibration P → W .

As F is SO(q)-invariant, the foliation G and is also SO(q)-invariant,W is a SO(q)-space, and the quotient map P → W is SO(q)-equivariant.

For a leaf L of F, the restriction of F to the closure L in P is a LieG-foliation with all leaves dense. That is, for each x ∈ P there is aconnected Lie group Gx, a complete manifold Zx with fundamental

group Γx = π(Zx, x) and universal cover Zx, and representation

ρx: Γx → Gx so that the quotient space (Zx ×Gx)/Γx∼= L as foliated

manifolds. The identification is given by a natural map induced

from the Lie algebra of Gx realized as basic vector fields for F.

The for a leaf L ⊂ P which covers a leaf L ⊂ M , π(L) = L.

Let Fx denote the fiber of π:M → B containing x. We identifyFx with SO(q) where x corresponds to the identity element. Then

Hx = L ∩ Fx is a closed subspace of Fx, and is identified with asubgroup of SO(q).

The set of leaf closures L for F define a singular foliation F on M .

Plus more. . .

19

Regular Riemannian Foliations

DEFINITION: A Riemannian foliation of a compact manifold Mis regular if the set of leaf closures form a non-singular foliation F.

EXAMPLE: A Riemannian foliation with every leaf compact isregular. These are the compact Hausdorff foliations.

Assume that M is compact and the normal bundle Q to F is framed.

Let L be a leaf of F with closure L.

Let L a leaf of F which covers L.

The Molino structure theory gives L ∼= (Zx × Gx)/Γx so

L ∼= (Zx × Gx/Hx)/Γx

where Hx ⊂ Gx is a closed subgroup.

The normal bundle of F restricted to L is identified with the bundleassociated to the adjoint representation of Γx on m = Tx(Gx/Hx)

The restriction of Q to L decomposes into Q = Q1 ⊕ Q2 whereQ1 is tangent to L and Q2 is orthogonal to L. Let q2 denote thecodimension of F, hence equal to the dimension of Q2.

LEMMA: If F is regular, then Q2 is finitely covered by a trivialbundle.

Proof: F is a Riemannian foliation with all leaves compact, henceis compact Hausdorff. The holonomy of each leaf L of F is finite,so there is a finite covering of L on which the restriction of Q2 istrivial.

COROLLARY: The Pontrjagin forms of Q vanish in degrees greater

than the codimension q2 of F.

20

Rationality and Vanishing for Regular Foliations

THEOREM: Let F be a regular Riemannian foliation of codi-mension q on M with trivial normal bundle. Suppose that theleaf closures F define a foliation of codimension q2 < q. Then∆∗(s)(hJpI) = 0 ∈ H∗(M,R) if deg pI > q2.

THEOREM: Let F be a regular Riemannian foliation of codimen-sion q on M with trivial normal bundle. If hJpI is a rigid class, or ifq is even and hJpI is any basis element, then

∆∗(s)(hJpI) ∈ Image H∗(M,Q) → H∗(M,R)

The requirement on dimension and rigid class is required, as thefollowing example shows.

EXAMPLE: Let S3 have the point foliation Fu with transversemetric gu such that ∆∗(s)(h1) ∈ H3(M) is a non-constant functionof u.

Let B be a compact oriented Riemannian manifold of dimension 8such that p2(TB) = 0. Let P → B be the bundle of oriented framesfor TB. Then P has a codimension 8 foliation F′ with trivial normalbundle for which ∆∗(h2 · p2) = 0 ∈ H15(P ).

The product manifold V = S3 × P has a family of Riemannianfoliations F′′

u = Fu × F′ of codimension 11, whose normal bundleis framed by the sum of the framings of the normal bundles oneach factor. Then the class ∆∗(s)(h1 ∧ h2 · p2) ∈ H18(V ) and is anon-constant function of u.

Note that h1 · p2 is a variable class, so also is h1 ∧ h2 · p2.

21

Vanishing for Regular Foliations

The last result is a vanishing theorem for all of the secondary classesif the topology of M is restricted.

THEOREM: Let F be a regular Riemannian foliation of codimen-sion q on M with trivial normal bundle. Suppose that π1(M) isfinite, and the leaf closures F define a foliation of codimensionq2 < q. Then ∆∗(s)(hJpI) = 0 ∈ Hm(M,R) where M has dimen-sion m. That is, all of the secondary characteristic numbers for Fvanish.

22