Transversality and Lefschetz numbers for foliation maps

0.35 MB·English

by  Jesús A. Álvarez López

TRANSVERSALITY AND LEFSCHETZ NUMBERS FOR
FOLIATION MAPS
8
0 JESU´SA.A´LVAREZLO´PEZANDYURIA.KORDYUKOV
0
2
Abstract. LetF beasmoothfoliationonaclosedRiemannianmanifoldM,

n and let Λ be a transverse invariant measure of F. Suppose that Λ is abso-
a lutely continuous with respect to the Lebesgue measure on smooth transver-
J sals. Thenatopological definitionoftheΛ-Lefschetznumber ofanyleafpre-
0 servingdiffeomorphism(M,F)→(M,F)isgiven. Forthispurpose,standard
3 results about smooth approximation and transversality are extended to the
case of foliation maps. It is asked whether this topological Λ-Lefschetz num-

] berisequaltotheanalyticΛ-LefschetznumberdefinedbyHeitschandLazarov
T whichwouldbeaversionoftheLefschetztraceformula. HeitschandLazarov
G haveshownsuchatraceformulawhenthefixedpointsetistransversetoF.
.
h
t
a
m Contents
[
1. Introduction 1

1 2. Preliminaries 3
v
3. Spaces of foliation maps 5
8
4. Leafwise approximations 11
2
6 5. Transversality for foliation maps 13
4 6. Local saturation and clean intersection 15
. 7. Transversality in good local saturations 16
1
0 8. Leafwise transversality 18
8 9. Approximation and transversality on foliated manifolds with boundary 19
0 10. Leafwise homotopies 20
:
v 11. Coincidence points 22
i 12. Fixed points 24
X
13. Λ-coincidence and Λ-Lefschetz number 25
r
a References 29
1. Introduction
Let F be a smooth foliation of codimension q on a closed Riemannian manifold
M, and let Λ be a transverse invariant measure of F. Continuing the ideas of
Connes’ index theory for foliations [8, 9], Heitsch and Lazarov use Λ to define
an analytic version of Lefschetz number for any leaf preserving diffeomorphism
1991 Mathematics Subject Classification. 57R30,58J20.
Key words and phrases. foliation, foliation map, transversality, integrable homotopy, Λ-
Lefschetznumber,Lefschetztraceformula.
ThefirstauthorispartiallysupportedbytheMECgrantMTM2004-08214.
Thesecondauthor ispartiallysupportedbytheRFBRgrant06-01-00208.
1
2 J.A.A´LVAREZLO´PEZANDY.A.KORDYUKOV
φ : (M,F) → (M,F) which is geometric with respect to a leafwise Dirac complex
[12]. For the sake of simplicity, consider only the case of the leafwise de Rham
complex. Then, by using the Hodge isomorphism on the leaves, the “local super-
trace” of the induced map on the reduced L2 cohomology of the leaves can be
defined as a density on the leaves. The pairing of this leafwise density with Λ is
a measure on M, whose integral is the analytic Λ-Lefschetz number, which will be
denoted by L (φ).
Λ,an
LetFix(φ)denotethefixedpointsetofφ. In[12],HeitschandLazarovhavealso
proved a Lefschetz trace formula when Fix(φ) is a regular submanifold transverse
toF. AssumethatdimFix(φ)=q forthesakeofsimplicity;thusFix(φ)isaclosed
transversalof F. Then the Lefschetz trace formula is
(1) L (φ)= ǫ Λ,
Λ,an φ
ZFix(φ)
where ǫ :Fix(φ)→{±1} is the locally constant function defined by
φ
ǫ (x)=signdet(id−φ :T F →T F).
φ ∗ x x
The right hand side of (1) can be called the topological Λ-Lefschetz number of φ,
and denoted by L (φ). Thus (1) becomes
Λ,top
(2) L (φ)=L (φ).
Λ,an Λ,top
In this paper, we give some steps towards a generalization of (2) by relaxing
the transversalityconditiononFix(φ). The motivationis thatexistenceofaclosed
transversalisastrongrestrictiononF. ButwehavetoassumethatΛisabsolutely
continuous with respect to the Lebesgue measure on smooth transversals;i.e., it is
given by a continuous density on smooth transversals.
For that purpose, we extend some standard results about maps to the case
of foliation maps between foliated manifolds (those which map leaves to leaves).
This kind of study was begun in [3, 4], where the set of smooth foliation maps is
endowed with certain topologies, called the strong and weak plaquewise topologies.
That study is recalled and pursued further here; in particular, we study certain
version of transversality for foliation maps, called ls-transversality.
It is shown that any leaf preserving diffeomorphism φ : (M,F) → (M,F) is
“homotopic along the leaves” to a leaf preserving map ξ :(M,F)→(M,F) whose
graph is ls-transverse to the diagonal of M ×M. With this condition, it turns out
that Fix(ξ) is a closed regular submanifold of M of dimension q. Let Fix (ξ) ⊂
0
Fix(ξ)denotetheopensubsetwhereFix(ξ)istransversetoF,whichisatransversal
of F. Then we prove that the topological Λ-Lefschetz number of φ is well defined
by the formula
(3) L (φ)= ǫ Λ;
Λ,top ξ
ZFix0(ξ)
i.e., we show that this integral is defined and is independent of ξ.
With more generality, the coincidence point set of two foliation maps φ,ψ :
(M,F)→(M,F) is
Coin(φ,ψ)={x∈M | φ(x)=ψ(x)}.
If φ and ψ are “transversely smooth” and induce the same map on the leaf space
M/F,then,withtheobviousgeneralizationofthenotationof (3),itisprovedthat
TRANSVERSALITY AND LEFSCHETZ NUMBERS FOR FOLIATION MAPS 3
their topological Λ-coincidence is well defined by a formula
Coin (φ,ψ)= ǫ Λ.
Λ,top ξ,ζ
ZCoin0(ξ,ζ)
Therefore, for any “transversely smooth” leaf preserving map φ : M → M, the
topological Λ-Lefschetz number of φ is well defined by
(4) L (φ)=Coin (id,φ).
Λ,top Λ,top
This work raises the following question.
Question 1.1. With the definition of L (φ) given by (3), is the Lefschetz
Λ,top
trace formula (2) valid without the assumption that the fixed point set of φ is a
transversal?
Besides its own interest, an affirmativeanswer to this questionwould be helpful
to describe the Lefschetz distribution of a Lie foliation on a closed manifold [2].
There are some other works related with measurable Lefschetz numbers for foli-
ation maps. Note that the analytic Λ-Lefschetz number of the identity map coin-
cideswiththe Λ-Eulercharacteristicofthe foliationintroducedbyConnesin[8,9].
Connes has also proven the corresponding Lefschetz trace formula, which is just a
measurableindextheoremforthe leafwisede Rhamcomplexonacompactfoliated
manifold. ThetopologicalΛ-EulercharacteristicwasstudiedrecentlybyBermu´dez
[6] in a more general context of measurable foliations.
In [1], the authors showed that non-vanishing of the Λ-Euler characteristic of a
transitive codimension one foliation implies that the reduced leafwise cohomology
of this foliation is infinite-dimensional. This result was extended to the case of Lie
foliations in [2] and to the case of Riemannian foliations in [15].
Finally,letusmentionthatBenameurin[5]hasprovedaK-theoreticalversionof
theLefschetzformulaforleafpreservingisometries,extendingConnesandSkandalis
K-theoreticalindex theorem for foliations [10].
Intherestofthepaper,wewillonlyconsiderthetopologicalΛ-Lefschetznumber
and the topological Λ-coincidence. So the word “topological” and the subindex
“top” will be removed from this terminology and notation.
2. Preliminaries
Recall the following terminology and notation about foliations (see e.g. [7]).
Let M be a manifold of dimension n = p+q. A foliation F of dimension p and
codimension q on M can be defined as a maximal atlas {U ,θ } of M such that
i i
θ (U ) = T × B for some open subset T ⊂ Rq and some convex open subset
i i i i i
B ⊂Rp, and the corresponding changes of coordinates are of the form
i
(5) θ ◦θ−1(x,y)=(h (x),g (x,y)).
j i i,j i,j
These charts are called foliation charts of F. The pair (M,F) is called a foliated
manifold. Usually, it is assumed that each B is the standard ball of Rp, but, with
i
the above freedom to choose them, it will be easier to give foliation charts. Each
U iscalledadistinguished open set. Thecompositeofθ withthefactorprojection
i i
T ×B →T is called the local projection of θ or U , and denoted by p :U →T .
i i i i i i i i
The fibers of each p are called plaques. Each T can be identified to the quotient
i i
space of plaques of U , which is called the local quotient of U . The plaques form a
i i
base of a topology, called the leaf topology. The connected components of the leaf
4 J.A.A´LVAREZLO´PEZANDY.A.KORDYUKOV
topology are p-dimensional immersed submanifolds of M called leaves. The leaves
determineF. LetM/F denotethequotientspaceofleaves. AsubsetofM iscalled
saturated if it is a union of leaves.
For each y ∈ B , the q-dimensional regular submanifold θ−1(T ×{y}) ⊂ M is
i i i
calledalocal transversal of F. Atransversal ofF is any regularsubmanifold ofM
which has an open covering by local transversals. A transversal is called complete
if it meets all leaves.
The restriction of F to any open subset U ⊂ M is the foliation F| whose
U
foliation charts are the foliation charts of F defined on subsets of U. The leaves of
F| are the connected components of the intersections of the leaves of F with U.
U
AcollectionoffoliationchartswhosedomainscoverM iscalledafoliation atlas,
anddetermines F. Afoliationatlas {U ,θ } is calledregular if U ∪U is contained
i i i j
insomedistinguishedopensetwheneverU meetsU . Therealwaysexistsalocally
i j
finite regular atlas.
Forr∈N∪{∞},aCr structure onF isamaximalfoliationatlaswhosechanges
of coordinates are Cr maps. When F is endowed with a Cr structure, then F is
called a Cr foliation, or it is said that F is of class Cr.
If F is of class Cr with r ≥1, then the vectors tangent to the leaves form a Cr
vector subbundle TF ⊂TM, whose fiber at each x∈M is denoted by T F. Then
x
ν =TM/TF is called the normal bundle of F.
Let N be another Cr manifold, with r ≥1. A Cr map φ:N →M is said to be
transverse to F if it is transverse to all leaves of F; i.e., when
φ (T N)+T F =T M
∗ x φ(x) φ(x)
for all x ∈ N. In this case, the inverse image by φ of the leaves of F are the
leavesof a foliationφ∗F onN, calledthe pull-back of F by φ. If N is a regularCr
submanifoldofM andφistheinclusionmap,thenφ∗F iscalledtherestriction ofF
toN,anddenotedbyF| . ObservethattheCr transversalsaretheq-dimensional
N
regular submanifolds transverse to F.
Let {U ,θ } be any regular foliation atlas of F with θ (U ) = T ×B . Then
i i i i i i
the maps h of (5) are local homeomorphisms of T = T , and thus generate
i,j i i
a pseudogroup H. The pseudogroups generated by different foliation atlases are
F
equivalent inasenseintroducedin[11]. Theequivalenceclassofthesepseudogroups
is called the holonomy pseudogroup of F. If {U } is locally finite, then T can be
i
identified to a transversal of F, and the maps h can be considered as “slidings”
i,j
of local transversals along the leaves. For any transversal of F, the “slidings” of
its small enough open sets along the leaves, called holonomy transformations, also
generate a representative of the holonomy pseudogroup.
AnyH-invariantmeasureΛonT iscalledantransverse invariant measure ofF.
ThisdefinitionisindependentoftherepresentativeHoftheholonomypseudogroup
because invariant measures can be pushed forward by pseudogroup equivalences.
Thus Λ can be considered as a measure on transversals invariant by holonomy
transformations. There is a maximal saturated open subset U ⊂ M such that Λ
vanishes on transversals contained in U; then M \U is called the support of Λ.
Suppose that F is C1. Then the class of the Lebesgue measure is well defined
onC1 transversals. Thus it makes sense to assume that Λ is absolutely continuous
with respect to the Lebesgue measure on C1 transversals. This means that Λ can
be considered as a continuous density on C1 transversals.
TRANSVERSALITY AND LEFSCHETZ NUMBERS FOR FOLIATION MAPS 5
Orientations and transverse orientations of F can be defined as orientations of
TF and ν, respectively; thus they can be representedby non-vanishing continuous
sections, χ of pTF∗ and ω of qν∗. Since there is a canonical injection ν∗ →
TM∗, ω can be considered as a differential q-form on M. With the choice of a
splitting TM V∼= ν ⊕TF, we canValso consider χ as a differential p-form on M.
Moreoverω∧χ is independent of the splitting and does not vanish; thus it defines
an induced orientation of M. Of course, orientations and transverse orientations
could also be defined without using differentiability; in particular, a transverse
orientation can be given as an H-invariant orientation on T.
A continuous differential form α on M is called horizontal if i α = 0 for all
X
vector field X tangent to the leaves; i.e., it can be considered as a continuous
section of ν∗. If α is horizontal and invariant by the flow of any C1 vector
field X tangent to the leaves, then it is said to be basic. With respect to local
V
foliation coordinates, the coefficients of the expression of a continuous basic form
are constant on the plaques. On the other hand, if α is a C1 horizontal q-form,
then the local expressions of dα involve only leafwise derivatives. Therefore, if α
is a continuous basic q-form, then there is a sequence of C1 horizontal q-forms α
i
convergingto αsuchthat dα convergesto zero,where the compact-opentopology
i
is considered. Since the basic forms on each U are the pull-back of differential
i
forms on T by the corresponding local projection, it follows that basic forms can
i
be also considered as H-invariant differential forms on T.
Suppose that F is transversely oriented, and thus T has a corresponding H-
invariant orientation. This orientation can be used to identified H-invariant con-
tinuous densities on T with H-invariant continuous differential forms of degree q,
which in turn can be considered as continuous basic forms of degree q. There-
fore any transverse invariant measure Λ, absolutely continuous with respect to the
Lebesgue measure, can be identified to a continuous basic q-form α. It turns out
that, for any C1 transversal Σ and any compactly supported continuous function
f :Σ→R, we have
fΛ= fα,
ZΣ ZΣ
where the restriction of α to Σ is also denoted by α. With respect to the opposite
orientation on T, Λ is identified to −α.
3. Spaces of foliation maps
In this section, we adapt to foliations the usual strong and weak topologies on
the sets of mapsbetween manifolds,andwe shallshowsome elementaryproperties
of these spaces.
For r ∈ N∪{∞}, let F and F′ be Cr foliations on Cr manifolds M and M′
respectively. A foliation map φ : (M,F) → (M′,F′) (or simply φ : F → F′) is a
map M →M′ that maps leaves of F to leaves of F′. If r ≥1, for any Cr foliation
mapφ:F →F′,itstangentmapφ :TM →TM′restrictstoaCr homomorphism
∗
φ :TF →TF′.
∗
Let Cr(F,F′) denote the set of Cr foliation maps F → F′. The notation
C(F,F′) will be also used in the case r = 0. Assume first that r is finite, and fix
the following data:
• Any φ∈Cr(F,F′).
6 J.A.A´LVAREZLO´PEZANDY.A.KORDYUKOV
• Any locally finite collection of foliation charts of F, Θ = {U ,θ } with
i i
θ (U )=T ×B.
i i i
• A collectionoffoliationchartsofF′ with the sameindex set,Θ′ ={U′,θ′}
i i
withθ′(U′)=T′×B′. Letπ′ :U′ →T′ denotethe localprojectiondefined
i i i i i i
by each θ′.
i
• AfamilyofcompactsubsetsofM withthesameindexset,K={K },such
i
that K ⊂U and φ(K )⊂U′ for all i.
i i i i
• A family of positive numbers with the same index set, E ={ǫ }.
i
Then let Nr(φ,Θ,Θ′,K,E) be the set of foliation maps ψ ∈Cr(F,F′) such that
(6) ψ(K )⊂U′ ,
i i
(7) π′◦ψ(x)=π′◦φ(x),
i i
(8) Dk θ′◦ψ◦θ−1 (θ (x))−Dk θ′◦φ◦θ−1 (θ (x)) <ǫ
i i i i i i i
for each i, an(cid:13)y x(cid:0)∈ Ki and ev(cid:1)ery k ∈ {0,...(cid:0),r}. Observ(cid:1)e that ((cid:13)7) means that, for
(cid:13) (cid:13)
eachplaque P of θ , the images of P ∩K by φ and ψ lie in the same plaque of θ′.
i i i i
For each finite r, the family of such sets Nr(φ,Θ,Θ′,K,E) is a base of a topology
on Cr(F,F′), which will be called the strong plaquewise topology, or simply the
sp-topology. When φ is fixed, all possible sets Nr(φ,Θ,Θ′,K,E) form a base of
neighbourhoods of φ in this space.
On the other hand, the sets Nr(φ,Θ,Θ′,K,E), with i running in a finite index
set, form a base of a coarser topology on Cr(F,F′), which will be called the weak
plaquewise topology, or simply the wp-topology. For families Θ, Θ′, K and E with
just one element, say θ, θ′, K and ǫ, the corresponding sets Nr(φ,θ,θ′,K,ǫ) =
Nr(φ,Θ,Θ′,K,E) form a subbase of this topology. When φ is fixed, all possible
sets Nr(φ,θ,θ′,K,ǫ) form a subbase of neighbourhoods of φ in this space.
In the case r = 0, consider also the set N(φ,Θ,Θ′,K) of foliation maps ψ ∈
C(F,F′) satisfying (1) and (2). Observe that all of these sets form a base of the
sp-topology of C(F,F′). A base of its wp-topology is similarly given with finite
families Θ, Θ′ and K.
Now let us consider the case r = ∞. With the above notation, for arbitrary
s∈N and φ, Θ, Θ′, K and E as above, all sets
C∞(F,F′)∩Ns(φ,Θ,Θ′,K,E)
formabaseofatopologyonC∞(F,F′), whichwillbecalledthestrong plaquewise
topology,orsimplythesp-topology. Finally,theweakplaquewisetopology,orsimply
thewp-topology,onC∞(F,F′)canbesimilarlydefinedbyconsideringfamilieswith
i running in a finite index set.
Forr =0,abaseofthesp-topologyofC(F,F′)isgivenbythesetsN(φ,Θ,Θ′,K)
offoliation maps ψ :(M,F)→(M′,F′) satisfying (6) and (7), whereφ, Θ, Θ′ and
K vary as above. A base of the wp-topology of C(F,F′) can be given in the same
way with i running in a finite index set.
If M is compact, then both of the above topologies on each Cr(F,F′) are obvi-
ously equal. In this case, it will be called the plaquewise topology on Cr(F,F′).
Thesubindex“SP”or“WP”willmeanthattherestrictionofthestrongorweak
plaquewisetopologyisconsideredonanysetofCr foliationmaps;thesubindex“P”
willbeusedwhenthemanifoldsarecompact. Moreoverexpressionslike“sp-open,”
“sp-continuous,” etc. mean that these topological properties hold with respect
to the sp-topology on sets of Cr foliation maps, or for the induced topologies on
TRANSVERSALITY AND LEFSCHETZ NUMBERS FOR FOLIATION MAPS 7
quotients, products, etc. A similar notation will be used for the wp-topology. The
subindex“S”or“W”willmeanthattheusualstrongorweaktopologyisconsidered
on any set of Cr maps.
The following extreme examples may help to clarify the above topologies.
Example 3.1. If the leaves of F′ are points (dimF′ = 0), then Cr (F,F′) is
SP
discrete.
Example 3.2. If the leaves of F are points (dimF = 0) and the leaves of F′ are
openinM′ (dimF′ =dimM′),thenCr (F,F′)=Cr(M,M′)andCr (F,F′)=
SP S WP
Cr (M,M′).
W
Recall that a subset of a topological space is called residual if it contains a
countable intersection of dense open subsets; the complement of each residual set
is called a meager set. Also, recall that a topological space is called a Baire space
when every residual subset is dense.
Theorem 3.3. Cr (F,F′) and Cr (F,F′) are Baire spaces for 0≤r ≤∞.
WP SP
Proof. Thisresultcouldbeshownbyadaptingtheproofofthecorrespondingresult
fortheusualweakandstrongtopologies[13,Chapter2,Theorem4.4]. Butashorter
proofisgivenbyrelatingtheweakandstrongplaquewisetopologieswiththe usual
weak and strong topologies.
For φ,Θ,Θ′,K as above, let Nr(Θ′,K) denote the set of foliation maps ψ ∈
Cr(F,F′) satisfying (6) for all i, and let Nr(φ,Θ,Θ′,K) denote the set of folia-
tion maps ψ ∈ Cr(F,F′) satisfying (6) and (7) for all i and x ∈ K . The set
i
Nr(φ,Θ,Θ′,K) is easily seen to be closed in Nr (Θ′,K). Hence there is a closed
W
subspace A of Cr (F,F′) such that
W
Nr(φ,Θ,Θ′,K)=A∩Nr(Θ′,K).
It is easy to check that Cr(F,F′) is closed in Cr (M,M′). Then A is closed in
W
Cr (M,M′) as well, and thus A is a Baire space [13, Chapter 2, Theorem 4.4].
W S
On the other hand, since Cr (M,M′) is completely metrizable [13, Chapter 2,
W
Theorem 4.4], its closed subspace A is completely metrizable too, and thus A
W W
is a Baire space by the Baire Category Theorem.
Since Nr(Θ′,K) is open in Cr(F,F′), it follows that Nr(φ,Θ,Θ′,K) is open
S
in the Baire space A . Therefore Nr(φ,Θ,Θ′,K) is a Baire space too [14, Propo-
S S
sition 8.3]. Then it easily follows that Cr (F,F′) is a Baire space because all
SP
possible sets Nr(φ,Θ,Θ′,K) form an open covering of Cr (F,F′).
SP
Now assume that i runs in a finite index set. Then Nr(Θ′,K) is also open in
Cr (F,F′), and thus Nr(φ,Θ,Θ′,K) is open in the Baire space A . Again, it
W W
follows that Nr (φ,Θ,Θ′,K) is a Baire space, and thus so is Cr (F,F′) because
W WP
all possible sets Nr(φ,Θ,Θ′,K), with i running in a finite index set, form an open
covering of Cr (F,F′). (cid:3)
WP
Lemma 3.4. For 0≤r ≤∞, any open subset of Cr (F,F′) is the intersection of
SP
a countable family of open subsets of Cr (F,F′).
WP
Proof. The statement is obvious for the above basic open sets of Cr (F,F′).
SP
AnarbitraryopensetMofCr (F,F′)isaunionofbasicopensetsN asabove
SP α
with α running in an arbitrary index set. In turn, each N is the intersection of
α
8 J.A.A´LVAREZLO´PEZANDY.A.KORDYUKOV
open subsets N of Cr (F,F′) with i running in N. Then M= A , where
α,i WP i∈N i
A = (N ∩···∩N ) T
i α,0 α,i
α
[
is open in Cr (F,F′). (cid:3)
WP
Corollary 3.5. For 0≤r≤∞, any residual subset of Cr (F,F′) is also residual
SP
in Cr (F,F′).
WP
Proof. Let A be a residual subset of Cr (F,F′). Thus A contains a countable
SP
intersectionofopendensesubsetsA ofCr (F,F′),k ∈N. Inturn,byLemma3.4,
k SP
each A is a countable intersection of open subsets A of Cr (F,F′), ℓ ∈ N.
k k,ℓ WP
MoreovereachA is dense in Cr (F,F′) because it contains A , which is dense
k,ℓ WP k
in Cr (F,F′) since the sp-topologyis finer than the wp-topology. Therefore A is
WP
residual in Cr (F,F′) because it contains the intersection of the sets A . (cid:3)
WP k,ℓ
The proofs of the following two lemmas are easy exercises, similar to the proofs
ofthe correspondingresults for the usualweak andstrongtopologies [13, page41].
Lemma 3.6. Let U be an open subset of M, F = F| , and ι : U → M the
U U
inclusion map. Then the restriction map
ι∗ :Cr(F,F′)→Cr(F ,F′), φ7→φ| =φ◦ι,
U U
is sp-open for 0≤r ≤∞.
Lemma 3.7. Let {V } be a locally finite family of open subsets of M. For each i,
i
let F = F| , let ι : V → M denote the inclusion map, and let A be a wp-open
i Vi i i i
subset of Cr(F ,F′) for 0≤r ≤∞. Then (ι∗)−1(A ) is sp-open in Cr(F,F′).
i i i i
LetF′′ beanotherCr foliationonaCr mTanifoldM′′. Theproofofthefollowing
lemma is an easy exercise, similar to the proof of the corresponding result for the
usual weak and strong topologies [13, page 64–65].
Lemma 3.8. For 0≤r ≤∞, the following properties hold:
(1) The composition defines a wp-continuous map
Cr(F′,F′′)×Cr(F,F′)→Cr(F;F′′).
(2) If σ :F′ →F′′ is Cr, then the map
σ :Cr(F,F′)→Cr(F,F′′), ψ 7→σ◦ψ ,
∗
is sp-continuous.
(3) If τ :F →F′ is proper and Cr, then the map
τ∗ :Cr(F′,F′′)→Cr(F,F′′), φ7→φ◦τ ,
is sp-continuous.
Lemma 3.9. Let U′ be an open subset of M′, FU′′ =F′|U′, and ι′ :U′ →M′ the
inclusion map. Then the map
ι′∗ :Cr(F,FU′′)→Cr(F,F′), φ7→ι′◦φ,
is an sp-open embedding for 0≤r ≤∞.
TRANSVERSALITY AND LEFSCHETZ NUMBERS FOR FOLIATION MAPS 9
Proof. Themapι′ isclearlyinjective,andmoreoveritissp-continuousbyLemma3.8-
∗
(2).
Toshowthatι′ issp-open,assumefirstthatr <∞. ThenletN denotethebasic
∗
open subset of Cr (F,F′ ) defined with some data φ, Θ, Θ′, K and E. Since the
SP U′
elements of Θ′ are also foliation charts of F′, we can also consider the basic open
subset M of Cr (F,F′) defined with the data ι′◦φ, Θ, Θ′, K and E. Obviously,
SP
ι′(N) is the set of maps in M whose image is contained in U′. But the image of
∗
any map of M is contained in U′ when K covers M, and thus ι′(N) = M in this
∗
case. So ι′ is open as desired.
∗
When r = ∞, the result follows similarly by adding an arbitrary s ∈ N to the
data needed to define the above basic open sets. (cid:3)
Let F′ and F′ be Cr foliations on manifolds M′ and M′, respectively, and let
1 2 1 2
F′ be the product foliation F′ ×F′ on M′ = M′ ×M′, whose leaves are the
× 1 2 × 1 2
products of leaves of F′ with leaves of F′.
1 2
Lemma 3.10. For 0≤r ≤∞, the canonical map
Cr(F,F′)×Cr(F,F′)→Cr(F,F′ ),
1 2 ×
which assigns the mapping x 7→ (φ (x),φ (x)) to each pair (φ ,φ ), is a homeo-
1 2 1 2
morphism with the weak and strong plaquewise topologies.
Proof. The canonical map of the statement is obviously bijective. Moreover it is
easilycheckedto be continuouswith the weakand strongplaquewise topologiesby
taking products of foliation charts of F′ and F′ as foliation charts of F′ to define
1 2 ×
basic open sets as above.
ThefactorprojectionsoftheproductM′×M′ arefoliationmapspr′ :F′ →F′
1 2 1 × 1
and pr′ : F′ → F′. Then the inverse of the canonical map of the statement is
2 × 2
given by
σ 7→(pr′ ◦σ,pr′ ◦σ),
1 2
whichiscontinuouswiththeweakandstrongplaquewisetopologiesbyLemma3.8-
(1)–(2). (cid:3)
The canonical map of Lemma 3.10 will be considered as an identity.
Let M′ and F′ be as before. Moreoverlet F and F be other Cr foliations on
× × 1 2
manifolds M and M , respectively, and let M =M ×M and F =F ×F .
1 2 × 1 2 × 1 2
Lemma 3.11. For 0≤r ≤∞, the product map
Cr(F ,F′)×Cr(F ,F′)→Cr(F ,F′ ), (φ ,φ )7→φ ×φ ,
1 1 2 2 × × 1 2 1 2
is a wp-embedding.
Proof. Thismapisobviouslyinjective. Moreoveritiswp-continuousbyLemma3.8-
(1) because, accordingto Lemma 3.10, it canbe identified with the mappr∗×pr∗,
1 2
where
pr∗ :Cr(F ,F′)→Cr(F ,F′)
i i i × i
is induced by the factor projection pr :F →F for i=1,2.
i × i
To finish the proof, assume first that r is finite. For each i = 1,2, let N be a
i
subbasic neighbourhood of some φ in Cr (F ,F′) determined by some data θ ,
i WP i i i
θ′, K and ǫ , as explained in the definition of the wp-topology. Let M be the
i i i
subbasic neighbourhood of φ ×φ in Cr (M ,F ;M′ ,F′ ) defined by the data
1 2 WP × × × ×
θ ×θ , θ′ ×θ′ , K ×K , min{ǫ ,ǫ }.
1 2 1 2 1 2 1 2
10 J.A.A´LVAREZLO´PEZANDY.A.KORDYUKOV
It is easy to check that
ψ ×ψ ∈M =⇒ (ψ ,ψ )∈N ×N
1 2 1 2 1 2
for any ψ ∈Cr(F ,F′), i=1,2. Hence the image of N ×N by the product map
i i i 1 2
is a wp-neighbourhood of φ ×φ in the image of the product map, which finishes
1 2
the proof in this case.
If r =∞, the resultfollows ina similar wayby adding anarbitrarys∈N to the
data needed to define the above subbasic neighbourhoods. (cid:3)
Lemma 3.12. For 1 ≤ r < ∞, let F and F′ be a Cr foliations on manifolds M
and M′, respectively. Let U be an open subset of M. For each Cr foliation map
φ : F → F′, there is an sp-neighborhood M in Cr(F| ,F′) of the restriction φ|
U U
such that, for each ψ ∈M, the combination H(ψ) of ψ and φ| is Cr foliation
M\U
map F → F′, and moreover the assignment ψ 7→ H(ψ) defines a continuous map
H :M →Cr (F,F′).
SP SP
Proof. This is analogous to the proof of Lemma 2.8 of Chapter 2 in [13]. Let
{Vk}k∈N be anopencoveringofU suchthat Vk ⊂Vk+1. Considersome data Θ, Θ′
and K, like in the definition of the sp-topology,such that {U } is an open covering
i
of U. For each i, let
1
ǫ = ,
i min{k∈N | K ⊂V }
i k
and set E ={ǫ }. Then the result follows with M=Nr(φ| ,Θ,Θ′,K,E). (cid:3)
i U
When a foliationmap φ:F →F preserveseachleaf, thenit willbe calledaleaf
preserving map. For 0≤r ≤∞, the family ofCr leafpreservingmaps F →F will
be denoted by Cr (F,F). The proof of the following result is elementary.
leaf
Proposition 3.13. The set Cr (F;F) is open in Cr (F,F).
leaf SP
Corollary 3.14. Cr (F;F) is a Baire space.
leaf,SP
Proof. By [14, Proposition 8.3], this is a direct consequence of Theorem 3.3 and
Proposition 3.13. (cid:3)
Let Diffr(M) be the set of Cr diffeomorphisms M → M; here, a C0 diffeomor-
phism is just a homeomorphism. Then let
Diffr(M,F)=Diffr(M)∩Cr(F,F),
Diffr(F)=Diffr(M)∩Cr (F;F).
leaf
With the operation of composition, Diffr(M,F) is a group, and Diffr(F) is a nor-
mal subgroup of Diffr(M,F). Assuming r = ∞, this notation is compatible with
the usual notation X(M,F) for the Lie algebra of infinitesimal transformations of
(M,F), and X(F) ⊂ X(M,F) for the normal Lie subalgebra of vector fields tan-
gent to the leaves. The flow of any vector field in X(M,F) (respectively, X(F)) is
a uniparametric subgroup of Diff∞(M,F) (respectively, Diff∞(F)).
Corollary 3.15. For 1 ≤ r ≤ ∞, the sets Diffr(M,F) and Diffr(F) are open in
Cr (F,F).
SP
Proof. ThisisaconsequenceofProposition3.13sinceDiffr(M)isopeninCr(M,M)
S
[13, page 38]. (cid:3)
Corollary 3.16. Diffr (M,F) and Diffr (F) are Baire spaces for 1≤r≤∞.
SP SP