Table Of ContentThe Atiyah Patodi Singer signature formula for
measured foliations
Paolo Antonini
antoninimat.uniroma1.it
paolo.antongmail.
om
9
0 January 6, 2009
0
2
n Abstra
t
a (X0,F0) F0
J Let be a
ompa
t manifold with boundary endowed with a foliatΛion whi
h
isassumedtobemeasuredandtransversetotheboundary. Wedenoteby aholonomy
6 (X ,F ) R
0 0 0
invariant transverse measure on and by the equivalen
e relation of the
(X,F)
] foliation. Let be the
orresRponding manifold with
ylindri
al end and extended
G foliation with equivalen
e relation . L2 Λ
D In the (cid:28)rst part of tDhiFs worXk we prove a formula for the - index of a longitudinal
Dira
-typeoperator on inthespiritofAlainConnes' non
ommutativegeometry
.
h [26℄ ind (DF,+)=hAˆ(TF)Ch(E/S),C i+1/2[η (DF∂)−h++h−].
t L2,Λ Λ Λ Λ Λ
a
m In the se
ond part we spe
ialize to the signature operator. We de(cid:28)ne three types
[ of signσature(Xfo,r∂Xthe) pair (foLl2iaΛtion, boundary foliation): the analyti
signature, de-
noted Λ,an σ0 is th(Xe,∂X- )-index of the signature operator on the
ylinRder; the
2 Λ,Hodge 0
Hodgesignature ,de(cid:28)nedusingthenatural representationof onthe
v
(cid:28)eld of square integrable harmoni
forms on the leaves and the de Rham signature,
3 σΛ,dR(X,∂X0) R0
,de(cid:28)nedusingthenatural representationof onthe(cid:28)eldof relative de
4
Rhamspa
es of theleaves. We prove that thesethree signatures
oin
ide
1
0 σΛ,an(X0,∂X0)=σΛ,Hodge(X,∂X0)=σΛ,dR(X,∂X0).
.
1
0 Asa
onsequen
eoftheseequalitiesandoftheindexformulawe(cid:28)nallyobtainthemain
9 result of this work, theAtiyah-Patodi-Singer signature formula for measuredfoliations:
0 F
σ (X,∂X )=hL(TF ),C i+1/2[η (D ∂)]
: Λ,dR 0 0 Λ Λ
v
i
X
Contents
r
a
1 Introdu
tion 2
2 Geometri
Setting 13
2.1 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Longitudinal Dira
operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Atiyah Patodi Singer index theorem 16
4 Von Neumann algebras, foliations and index theory 18
4.1 Non(cid:21)
ommutativeintegration theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Holonomyinvarianttransverse measures . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.1 Measures and
urrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Paolo Antonini
4.2.2 Tangential
ohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.3 Transverse measures and non
ommutativeintegration theory . . . . . . . . . . 32
4.3 VonNeumannalgebras and Breuer Fredholmtheoryfor foliations . . . . . . . . . . . . 33
4.3.1 Thesplitting prin
iple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Analysis of the Dira
operator 39
5.1 Finite dimensionalityof theindexproblem. . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Breuer(cid:21)Fredholm perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Cylindri
al (cid:28)nite propagation speed and Cheeger Gromov Taylor type estimates. 49
6.1 The standard
ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 The
ylindri
al
ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 The eta invariant 58
7.1 The
lassi
al eta invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2 The foliation
ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.3 Eta invariantfor perturbationsof theDira
operator . . . . . . . . . . . . . . . . . . . 62
8 The index formula 63
9 Comparison with Rama
handran index formula 72
9.1 The Rama
handranindex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
10 The signature of a foliated manifold with boundary 76
10.1 The Hirzebru
hformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
10.2 Computations with theleafwise signature operator . . . . . . . . . . . . . . . . . . . . 78
10.3 The Analyti
signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
10.4 The Hodge signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.5 Analyti
al signature=Hodge signature . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
L2
10.6 The (cid:21)deRhamsignature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.6.1 manifolds with boundarywith bounded geometry. . . . . . . . . . . . . . . . . 84
10.6.2 RandomHilbert
omplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10.6.3 De(cid:28)nitionof thede Rhamsignature . . . . . . . . . . . . . . . . . . . . . . . . 88
L2
10.7 de Rhamsignature=Harmoni
signature . . . . . . . . . . . . . . . . . . . . . . . . 88
R
0
10.7.1 Theboundaryfoliation and . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.7.2 Weaklyexa
tsequen
es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.7.3 Spe
traldensityfun
tions and Fredholm
omplexes. . . . . . . . . . . . . . . . 94
10.7.4 Theproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A Analysis on Manifolds with bounded geometry 105
A.1 Spe
tral fun
tions of ellipti
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Some
omputations onCli(cid:27)ord algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 109
1 Introdu
tion
X 4k
0
Let bea (cid:21)dimensionalorientedmanifoldwithout boundary. One
angivefourdi(cid:27)erent
de(cid:28)nitions of the signature.
σ(X )
0
• The topologi
al signature is de(cid:28)ned as the signature of the interse
tion form in
(x,y):= x y,[X ] , x,y H2k(X ,R).
0 0
the middle degree
ohomology; h ∪ i ∈
σ (X )
dR 0
• The de Rham signature is the signature of the Poin
aré interse
tion form in
([ω],[φ]):= ω φ; ω,φ H2k(X ).
the middle de Rham
ohomology; X0 ∧ ∈ dR 0
R
3
σ (X )
Hodge 0
• The Hodge signature, is the signature of the Poin
aré interse
tion form
2k
de(cid:28)ned in the spa
e of Harmoni
forms with respe
t to some
hoosen Riemannian
(ω,φ):= ω φ; ω,φ 2k(X ).
stru
ture X0 ∧ ∈H 0
R
1
• The analyti
al signature is the index of the
hiral signature operator
σ (X ):=ind(Dsign,+).
an 0
One
an provethat all these numbers
oin
ide,
σ(X )=σ (X )=σ (X )=σ (X ).
0 dR 0 Hodge 0 an 0
(1)
The Hirzebru
h formula
σ(X )= L(X )
0 0
ZX0
an be proven using
obordism arguments as in the original work of Hirzebru
h or
an be
seen asa
onsequen
eofthe Atiyah(cid:21)Singerindex formula[10℄ and the
hainofequalities (1).
X X Γ X
0 0 0
If −→ is a Galois
overing with de
k group with as above Atiyah [2℄ used the
Γ
Von Neumann algebra asso
iated to the regular right representation of to normalize the
f L2 σ (X )
Γ 0
signature on middle degree harmoni
forms on the total spa
e. This signature
again enters in a Hirzebru
h type formula
σ (X )= L(X ).
Γ 0 0
ZX0
L2
This is the
elebrated Atiyah (cid:21)signature theorem.
L2
The Atiyah (cid:21)signature theorem was extended by Alain Connes [26℄ to the situation in
X
0
whi
h the total spa
e is foliated by an even dimensional foliation. This is the realm of
non(cid:21)
ommutative geometry. We shall have the o
asion to des
ribe extensively this kind of
result.
X
0
What
an one say if has non empty boundary ?
X
0
So let now be an oriented
ompa
t manifold with boundary and suppose the metri
is
produ
t typenear the boundary. Atta
h an in(cid:28)nite
ylinder a
rossthe boundaryto form the
manifold with
ylindri
al ends,
X =X ∂X [0, )
0 0
× ∞
∂[X0h i
IntheseminalpaperbyAtiyahPatodiandSinger[5℄isshowenthattheFredholmindexofthe
generalized boundary value problem with the pseudodi(cid:27)erential APS boundary
ondition on
X L2
0
for the signature operator (or a
hiral Dira
type operator) is
onne
ted to the (cid:21)index
X L2 X
oftheextendedoperatoron . IndeedthisFredholmindexisthe indexon plusadefe
t
dependingonthespa
eofextendedsolutionsonthe
ylinder. Morepre
iselytheoperatoron
L2
the
ylinder a
ting on the natural spa
e of (cid:21)se
tions is no more Fredholm (in the general
d+d∗
hir1atlhigsraisditnhgeτdi:(cid:27)=er(e−n1ti)akl∗op(−er1a)t|o·|r(|·2|−1),Da
sitginng=o„ntDhesi
0gonm,+plexDosif0gnd,i−(cid:27)er«ential forms,odd w.r.t. the natural
4 Paolo Antonini
ase in whi
h the boundary operator is not invertible) but its kernel and the kernel of its
2
formal adjoint are (cid:28)nite dimensional and this di(cid:27)eren
e is given by the formula
η(0) h (D ) h (D+)
indL2(D+)= Aˆ(X,∇)Ch(E)+ 2 + ∞ − −2 ∞ ;
ZX0
h (D ) L2
±
where ∞ are the dimensions of the limiting values of the extended solutions and
η(0)
is the eta invariant of the boundary operator.
4k
Then, in the
ase of the signature operator, in dimension the authors investigate the
X (X ,∂X )
0 0 0
relationshipbetweentheAPSindexoftheoperatoron ,thesignatureofthepair ,
L2 X X
the indexon andthespa
eofharmoni
formson . The
on
lusionisthatthesignature
σ(X ) L2 h
0 ±
is exa
tly the (cid:21)index on the
ylinder i.e. the di(cid:27)eren
e of the dimensions of
X
3
positive/negativesquare integrable harmoni
forms on ,
σ(X )=h+ h =ind (Dsign,+)
0 − L2
−
h (Dsign, )=h (Dsign,+)
−
while ∞ ∞ by spe
i(cid:28)
simmetries of the signature operator. In parti
-
ular the APS signature formula be
omes
σ(X )= L(X , )+η Dsign .
0 ZX0 0 ∇ |∂X0
(cid:0) (cid:1)
Γ
In the
ase of (cid:21)Galois
overings of a manifold with boundary with a
ylinder atta
hed,
X X
−→ this program is partially
arried on by Vaillant [93℄ in his Master thesis. More
spe
i(cid:28)
ally he estabilishes a Von Neumann index formula in the sense of Atiyah [2℄ for a
e Γ
Dira
type operator and relates this index with the (cid:21)dimensions of the harmoni
forms on
thetotalspa
e. Theremainingpartofthestoryi.e. therelationwiththetopologi
allyde(cid:28)ned
L2
(cid:21)signatureis
arried out by Lü
k and S
hi
k [60℄. Call the index of Vaillant the analyti
al
L2 X σ (X ) σ (X )
0 an,(2) 0 Ho 0
(cid:21)signatureofthe
ompa
tpie
e , insymbols whileHarmoni
isthe
L2 X˜
signature de(cid:28)ned using harmoni
forms on . Then Vaillant proves that
σ (X )= L(X , )+η Dsign =σ (X ).
an,(2) 0 ZX0 0 ∇ Γ |∂X˜ Hodge 0
(cid:0) (cid:1)
L2 σ (X )
dR,(2) 0
Lu
k and S
hi
k de(cid:28)ne other di(cid:27)erent types of signatures, de Rham and
σ (X )
top,(2) 0
simpli
ial and prove that they are all the same and
oin
ide with the signatures
of Vaillant. To be more pre
ise they prove
σ (X )=σ (X )=σ (X ).
Hodge 0 dR,(2) 0 top,(2) 0
None of these steps are easy adaptations of the
losed
ase sin
e in the
lassi
al proof a
fundamental role is played by the existen
e of a gap around the zero in the spe
trum of the
boundaryoperator. Thissituationfailstobetrueinnon
ompa
t(also
o
ompa
t)ambients.
I this thesis I
arry out this program for a foliated manifold with
ylindri
al ends endowed
Λ
with a holonomy invariant measure [26℄. The framework is that explained by Connes in
his seminal paper on non
ommutative integration theory [26℄ in parti
ular I use in a
ru
ial
way the semi(cid:28)nite Von Neumann algebras asso
iated to a measured foliation. Working with
the Borel groupoid de(cid:28)ned by the equivalen
e relation R I (cid:28)rst extend the index formula of
Vaillant.
2oppositeorientation w.r.t. APS
3indeed theinterse
tion formispassestobenon(cid:21)degenerate totheimagLe2oftherelative
ohomXologyinto
theabsoluteone. Thisve
tor spa
eisnaturally isomorphi
tothespa
e of (cid:21)harmoni
formson .
5
L2 Λ
Theorem 1.0 (cid:22) The Dira
operator has (cid:28)nite dimensional − (cid:21)index and the following
formula holds
indL2,Λ(D+)=hAˆ(X)Ch(E/S),[CΛ]i+1/2[ηΛ(DF∂)−h+Λ +h−Λ]. (2)
The dimensions of the spa
es of extended solutions, h±Λ are suitably de(cid:28)ned using Von Neu-
mannalgebrasasso
iatedtosquareintegrablerepresentationsofR,thefoliationetainvariant
is de(cid:28)ned by Rama
handran [76℄ and the usual integral in the APS formula is
hanged into
the distributional pairing with a tangential distributional form with the Ruelle(cid:21)Sullivan
ur-
rent [65℄. In the proof of (81) a signi(cid:28)
ative role is played by the introdu
tion of a notion of
Λ Λ
(cid:21)essential spe
trum of an operator, relative to the tra
e de(cid:28)ned by in the leafwise Von
Λ
Neumann algebra of the foliation. This is stable by (cid:21)
ompa
t perturbations (in the sense
of Breuer [17℄) and translate to the foliation
ontest the general philosophy of Melrose [62℄
stating that the operator is Fredholm i(cid:27) is invertible at the boundary. Then we de(cid:28)ne a two
parameter perturbation with invertible family at the boundary. This is a Breuer(cid:21)Fredholm
perturbation. The strategy is to prove (cid:28)rst the formula for the perturbation then let the
parameters go to zero.
In the se
ond part, inspired by the de(cid:28)nitions of Lü
k and S
hi
k [60℄ I pass to the study of
0
three di(cid:27)erent representationsofR (the equivalen
erelation ofthe foliation on the
ompa
t
X σ (X ,∂X )
0 Λ,an 0 0
pie
e )inordertode(cid:28)netheAnalyti
al Signature, (i.e. themeasuredindex
σ (X ,∂X )
Λ,dR 0 0
ofthesignatureoperatoronthe
ylinder),thede Rham signature (i.etheone
indu
ed by the representation whi
h is valued in the relative de Rham spa
es of the leaves)
σ (X ,∂X )
Λ,Hodge 0 0 0
and the Hodge signature, (de(cid:28)ned in terms of the representationof R in
X
the harmoni
forms on the leaves of the foliation on ).
L2
Combining a generalization of the notion of the long exa
t sequen
e of the pair (folia-
tion,boundary foliation), in the sense of sequen
es of Random Hilbert
omplexes (the analog
L2 Γ
ofthehomology longsequen
eofHilbert (cid:21)modulesinCheegerandGromov[22℄)together
with the analysisof boundary value problems of [87℄, one shows that the methods in [60℄
an
be generalized and give the following
Λ X
0
Theorem 1.0 (cid:22) The above three notions of (cid:21)signature for the foliation on
oin
ide,
σ (X,X )=σ (X,X )=σ (X,X )
Λ,dR 0 Λ,Hodge 0 Λ,an 0
and the followingAPS signature formula holds true
σΛ,an(X0,∂X0)= L(X),[CΛ] +1/2[ηΛ(DF∂)]
h i
A more detailed des
ription of the various se
tions follows.
Geometri
setting
In this se
tionthe whole geometri
stru
ture is introdu
ed. We speak about
ylindri
alfolia-
tionsandallthedataneededtode(cid:28)nethelongitudinalDira
operatorasso
iatedtoaCli(cid:27)ord
bundle. Every
ylindri
alfoliation arisesfrom a gluingpro
ess, startingfrom afoliated man-
ifold with boundary and foliation transverseto the boundary. The (cid:28)rst geometri
alinvariant
of a foliation is its holonomy. It enters into index theory in an essentialway providing a nat-
ural desingularization of the leaf spa
e. The various holonomy
overs glue all together into a
G
manifold, the holonomy groupoid where one
an speak about smooth fun
tions and apply
6 Paolo Antonini
theusualanalyti
alte
hniques. FollowingRama
handranweworkatlevelofthe equivalen
e
relation R of being on the same leaf. This is the most elementary level of desingularization
where boundary value problems
an be set up without ambiguity.
Von Neumann algebras, foliations and index theory
Von Neumann algebras and Breuer Fredholm theory with tra
es. In this se
tion
generalitiesaboutVonNeumannalgebrasaregiven. Theseareparti
ular∗(cid:21)subalgebrasofall
bounded operators a
ting on an Hilbert spa
e. We spe
ialize to Von Neumann algebras that
an be equipped with a semi(cid:21)(cid:28)nite normal faithful tra
e like Von Neumann algebras arising
from foliations admitting a holonomy invariant transverse measure. Indeed Connes [27℄ has
shownthatholonomyinvarianttransversemeasure
orrespondonetoonetosemi(cid:28)nitenormal
faithful tra
es on the Von Neumann algebra of the foliation. More generally a transverse
measuregivesaweight. Thisweightisinvariantunderinteriorautomorphisms,i.e. isatra
e,
i(cid:27) the measure is invariant under holonomy. Then in some sense holonomy is represented in
the Von Neumann algebra by the interior automorphisms. One
an also use the language of
foliated
urrent. A transverse measure sele
ts a transverse
urrent. This
urrent is
losed i(cid:27)
the transversemeasure is hol. invariant.
M τ : M+ [0, ]
So let be a Von Neumann agebra with a tra
e −→ ∞ one has a natural
M V
notion of dimension of a
losed subspa
e a(cid:30)liated to , i.e. a subspa
e whose proje
tion
Pr M τ(Pr )
V V
belongs to . This is by de(cid:28)nition the relative dimension . Relative dimension
M.
is the
ornerstone of a theory of Fredholm operators inside This story goes ba
k to the
seminal work of Breuer [16, 17℄. For this reason relatively Fredholm operators are
alled
Breuer(cid:21)Fredholm. A Breuer(cid:21)Fredholm operator has a (cid:28)nite real index with some stability
properties as in the
lassi
al theory.
Transverse measures and Von Neumann algebras.
In the spirit of Alain Connes non
ommutative geometry Von Neumann algebras stand for
C
∗
measurespa
eswhile (cid:21)algebrasdes
ribestopologi
alspa
es. Intheseminalwork[26℄hehas
shown that a foliation with a given transversemeasure gives rise to a Von Neumann algebra
whose properties re(cid:29)e
t the properties of the measure. First we de(cid:28)ne transverse measures
as measures on the sigma ring of all Borel transversals. This is a
ted by the holonomy
pseudogroup. When the measure is invariantw.r.t. this a
tion one has a holonomy invariant
measures.
W I II
∗
If a holonomy invariant measure exists then the asso
iated (cid:21) algebrais type or type
(the (cid:28)rst type appears only in the ergodi
ase). In parti
ular there's a natural tra
e whose
de(cid:28)nition is expli
itly given as an integral of suitable obje
ts living along leaves against the
transversemeasure.
Then transverse measures
an be
onsidered as some kind of measure on the spa
e of the
leaves.
In this se
tion we de(cid:28)ne the Von Neumann algebraasso
iatedto the transversemeasureand
x y x y
a square representation of the Borel equivalen
e relation R i(cid:27) and are in the same
E
leave. For a ve
tor bundle this is the algebra of uniformly bounded (cid:28)elds of operators
x A :L2(L ;E) L2(L ;E) L x
x x x x
7−→ −→ ( is the leaveof ) a
tingon the Borel (cid:28)eld of Hilbert
x L2(X;E)
spa
es 7−→ suitably identi(cid:28)ed using the transverse measure. Thinking of an
operator as a family of leafwise operators the tra
e has a natural meaning, it is the integral
against the transverse measure of a family of leafwise measures
alled lo
al tra
es.
M
ForselfadjointintertwiningoΛperators,usingthespe
traltheoreRmandthetra
eon (
oming
from a transverse measure ) one
an de(cid:28)ne a measure on
alled the spe
tral measure
(depending on the tra
e). Breuer(cid:21)Fredholmproperties ofthe operatorareeasilydes
ribed in
7
terms of this spe
tral measure. In parti
ular one
an de(cid:28)ne some kind of essential spe
trum
Λ
alled the (cid:21)essentialspe
trum. Belongingof zero to the essential spe
trum is equivalent for
the operator to be Breuer(cid:21)Fredholm. We show also that for ellipti
operators the essential
spe
trumisgovernedbythebehavioroftheoperatoroutside
ompa
tsubsetsontheambient
K X K
manifold. Noti
e that if one (cid:28)x a
ompa
t set on every leaf
an interse
t in(cid:28)nitely
K
many times then our notion of "lying outside " must be explained with
are. We
all this
result the Splitting prin
iple. It will be useful in the study of the Dira
operator.
X
Analysis of the Dira
operator. Consider the leafwise Dira
operator on asso
iated
to the geometri
datas of the (cid:28)rst se
tion. This is obtained from the
olle
tion of all Dira
D L
x x x
oZperators {D =}Do+ne fDor ea
h leave . If the foliation is assumed even dimEensional this is
2 −
(cid:21)graded ⊕ withrespe
ttoanaturalinvolutiononthebundle . Thisis
alled
the Chiral Dira
operator.
M
This leafwise family of operatorsgivesan operatora(cid:30)liated to the Von Neumann algebra
(the transverse measure gives the glue to join all the operators together). In parti
ular ea
h
D M
spe
tralproje
tionof de(cid:28)nesaproje
tionin . Ifthefoliatedmanifoldis
ompa
tConnes
han shown that this is a Breuer(cid:21)Fredholm operator and the index, the relative dimension of
KernelminusCoKernelisrelatedtotopologi
alinvariantsofthefoliationbytheConnesindex
formula. ind (D+)= Ch(D+)Td(X),[C ]
Λ Λ
h i
Onthe righthandside one(cid:28)nds the distributional
ouplingbetweena longitudinal
hara
ter-
C
Λ
isti
lassandthehomology
lassofa
losed
urrent asso
iatedtothetransversemeasure
by the Ruelle(cid:21)Sullivan method.
Finite dimensionality of the index problem.
In our
ylindri
al
ase, the operator is in general non Breuer(cid:21)Fredholm. As a general philo-
sophi
al prin
iple for manifolds with
ylindri
al ends and produ
t(cid:21)stru
tureoperators, Fred-
L2
holm properties of the operator on the natural spa
e are essentially
aptured by the
spe
trum at zero of the operator on the
ross se
tion (the base of the
ylinder). Thanks to
the splitting prin
iple the Philosophy
invertibility at boundary ⇐⇒ Freholm property
Λ
arriesontothefoliated
aseifonelooksatthe (cid:21)essentialspe
trumoftheleafwiseoperator
on the foliation indu
ed on the transverse se
tion of the
ylinder (this is to be thought of as
the foliation at in(cid:28)nity).
Now it's a well known fa
t that lots of Dira
type operatorsof
apital importan
e in Physi
s
and Geometry are not invertible at the boundary (in(cid:28)nity). One example for all is the
Signature operator, our main appli
ation here.
Howeversome work on ellipti
regularity and the use of the generalized eigenfun
tion expan-
Λ L2
sion of Browderand Gårding shows that the (cid:21)dimension of the proje
tion on the kernel
D+ D M
−
of and are (cid:28)nite proje
tions of the V.N. algebra . In parti
ular we
an de(cid:28)ne the
L2 D+
hiral index of as
ind (D+)=dim Ker (D+) dim Ker (D ).
L2,Λ Λ L2 Λ L2 −
−
Ona
ompa
tfoliatedmanifold,ifafamilyofoperatorsisimplementedbyafamilyofleafwise
uniformly smoothing s
hwartz kernels the (cid:28)nite tra
e property follows immediately from the
remarkable fa
t that integrating a longitudinal Radon measure against a transversemeasure
gives a (cid:28)nite mass measure on the ambient. Now the ambient is a manifold with a
ylinder,
hen
e Radon longitudinal measures do not give (cid:28)nite measures in general. Our strategy
L2 L2
to prove the (cid:28)nite dimensionality of the index problem is to show that the (cid:28)eld of
8 Paolo Antonini
D+
proje
tions on the kernel of x enjoys the additional property to be lo
ally tra
eable with
respe
ttoabiggerfamilyofBorelsets. Tobemorepre
iseweprovethatforevery
ompa
tset
K on theL2b(oLun)dary of the
ylinder of a leave the operator χK×R+ΠKerL2(D+)χK×R+ is tra
e
x
lasson . Thisis su(cid:30)
ient(by the integrationpro
ess)to assure(cid:28)nite dimensionality.
Breuer(cid:21)Fredholm perturbation.
On
e(cid:28)nitedimensionalityofkernelsisprovenweperformaperturbationargumentto
hange
the Dira
operator into a Breuer(cid:21)Fredholm one. This is done following very
losely Boris
Vaillant paper [?℄ where the same problem is studied for Galois
overings of manifolds with
ylindri
al ends. Sin
e we are working with Von Neumann algebras the possibilty to use
Borel fun
tional
al
ulus gives a great help in a way that we
an de(cid:28)ne our two parameters
perturbation essentially by boundary operator minus the boundary operator restri
ted to
some small spe
tral interval near zero
D ;D , D :=D
ǫ,u ǫ,0 ǫ
D
ǫ,u
Next we prove (through the splitting prin
iple) that is Breuer(cid:21)Fredholm for small pa-
rametersanditsindexapproximatesthe
hiralindex. A
tuallywehaveto
onsiderseparately
the two parameters limits.
L2
The analysis of the relation between the perturbed Fredholm index and the
hiral index
L2 euθL2 u > 0 r
requires the introdu
tion of weighted spa
es along the leaves, for ( is
the
ylindri
al
oordinate). Smooth solutions belonging to ea
h weighted spa
e are
alled
Ext(D )
±
Extended Solutions, . They enternaturallyinto the A.P.S index formulabut do not
L2 Λ
form a
losed subspa
e in . Some
are is needed in de(cid:28)ning their (cid:21)dimension and prove
that this is (cid:28)nite.
The remaining part of the se
tion is devoted to the proof of the fundamental asymptoti
relations
limind (D+)=ind (D+), limdim Ext(D )=Ext(D ).
L2,Λ ǫ L2,Λ Λ ǫ± ±
ǫ 0 ǫ 0
→ →
Cylindri
al (cid:28)nite propagation speed and Cheeger Gromov Taylor type estimates.
To prove the index formula we need some pointwise estimates on the S
hwartz kernels of
fun
tions of the leafwise Dira
operator. Our perturbation on the
ylinder has the shape
D+Q Q
where is some selfadjoint order zero pseudodi(cid:27)erential operator on the base of the
Q uId
ylinder (a
tually is just a sum of a uniformly smoothing operator and ) in parti
ular
one
an repeat the proof of energy estimates as in the Book by John Roe for example [79℄
∂L (a,b) ∂L
x x
for the wave equation no more on a small geodesi
ball but on a strip × ( is
ξ
0
the base of the
ylinder) (cid:28)nding out that unitary
ylindri
al di(cid:27)usion speed holds i.e. if
∂L (a,b) eiQξ
x 0
is supported in × then the solution of the wave equation is supported in
∂L (a t,b+ t).
x
× −| | | | Thisissu(cid:30)
ienttoextimatekernelsof
lasss
hwartzspe
tralfun
tions
D Q
of and following the method of Cheeger, Gromov and Taylor [23℄ obtaining de
aying
estimates for the heat kernel
|∇lz1∇kz2[Tme−tT2](z1,z2)|≤C(k,l,m,T)e(|s1−s2|−r1)2/6t. (3)
[] r
1
With the notation · one denotes the S
hwartz kernel, and is some positive number while
z =(x ,s ) s s >2r
i i i 1 2 1
aretwopointsonthe
ylinderwith | − | . Itis
learwhy oneisbrought
to
all these the Chegeer Gromov Taylor estimates in the
ylindri
al dire
tion. There is also
an extremely useful relative version of estimate (3) where one
an estimate the di(cid:27)eren
e of
U
the kernels of spe
tral fun
tions of two operators that agree on some open subset of the
ylinder. This is an estimate of the form
2n¯+l+k
l k([f(P )] [f(P )]) (P ,k,l,r ) fˆ(j)(s)ds,
|∇x∇y 1 − 2 (x,y)|≤C 1 2 Xj=0 ZR−J(x,y)| |
9
l r >0 x,y U Q(x,y):=max min d(x,∂U);d(y,∂U) r ;0
2 2
where is a leaf, , ∈ , { { }− },
Q(x,y) Q(x,y)
J(x,y):= − , f (R)
c c and ∈S isaS
hwartzfun
tiontoapplytotheoperator
(cid:16) (cid:17)
using the spe
tral theorem.
In pra
ti
e we shall
olle
t all these estimates, one for ea
h leaf. Thanks to the uniformly
bounded geometry of the leaves that run trough a
ompa
t manifold (with boundary) the
onstants
an be made independend. This is an extremely important fa
t.
The foliated eta invariant.
Sin
e its (cid:28)rst apparitionin [5℄ the eta invariantof a Dira
operator asthe di(cid:27)eren
e between
the lo
al and global term on the Atiyah Patodi Singer index formula
1/2η(D )= ω ind(D+)+1/2dimKer(D )
0 D 0
−{ }
ZX
or the spe
tral asimmetry de(cid:28)ned as the regular value at zero of the meromorphi
fun
tion
(summation overeigenvalues)
signλ
η (s):= , Re(s)>dim∂X
D0 λs (4)
λ=0 | |
X6
has be
ome a key
on
ept a in Spe
tral geometry and modern Physi
s.
The foliation eta invariant on a
ompa
t manifold (when a transverse invariant measure is
(cid:28)xed)wasde(cid:28)nedindependentlyandessentiallyinthesamewaybyPeri
[70℄andRama
han-
dran [76℄ and enters into our A.P.S index formula exa
tly in the way it enters
lassi
ally. It
should be remarked that Peri
and Rama
handran numbers are not the same. The reason
is simple. Peri
uses the holonomy groupoid to desingularize the spa
e of the leaves while
Rama
handran works dire
tly on the Borel equivalen
e relation. Due to their global nature
the eta invariants obtained are not the same. As a striking
onsequen
e one gets that on a
ylindri
alfoliatedmanifoldevery
hoi
eofdesingularizationfromtheequivalen
erelationto
the holonomy (or the monodromy groupoid ) leads to di(cid:27)erent index formulas with di(cid:27)erent
eta invariants. This is a genuine new feature of the boundary (
ylindri
al)
ase.
Sin
eweworkwiththeBorelequivalen
erelationoureta(cid:21)invariantisthatofRama
handran.
DF∂ 4 DF∂ Re(s) 0
So
onsiderthebaseDira
operator the etafun
tion of isde(cid:28)ned for ≤ by
η(DF∂,s):= 1 ∞ts−21 trΛ(DF∂e−DF∂)dt, λ >0, s> 1.
Γ((s+1)/2) | | −
Z0
Re(λ) 0 (dim
∂
It
an be shown that this is meromorphi
for ≤ with simple poles at F −
k)/2, k =0,1,...
and a regular value at zero.
In this se
tion we des
ribe this result and we extend it to some
lasses of perturbations of
theoperatorneeded in the proofoftheindexformula. Weshall
onsiderperturbationsofthe
Q = DF∂ +K K K = f(DF∂) f :
f(oram,a) R f =witχh( ǫ,ǫ)some uniformly smoothing spe
traQl fuu:n=
tDionF∂+DF∂f(DF∂,)+u
− −→ . For − more
anbesaidaboutthefamily
in fa
t we
an de(cid:28)ne
ηΛ(Qu)=LIMδ 0 k t−1/2 trΛ(Que−tQ2u)dt+ ∞ t−1/2 trΛ(Que−tQ2u)dt
→ Zδ Γ(1/2) Zk Γ(1/2)
LIM δ
where is the
oknstant term in the asymptoti
development in powers of near zero of
δ
the fun
tion 7−→ δ . Moreovertwo important formulas hold true
4therelationwith(R4)
omesfromtheidentitysign(λ)|λ|−1=Γ s+21 −1 0∞ts−21λe−tλ2dt
` ´ R
10 Paolo Antonini
ηΛ(Qu) ηΛ(Q0)=sign(u)trΛ(f(DF∂)
• −
•
η (Q )=1/2(η (Q )+η (Q )).
Λ 0 Λ u Λ u
− (5)
This only requires a minimal modi(cid:28)
ation of the proof given by Vaillant [93℄ for Galois
ov-
erings. The point is that the relevant Von Neumann algebras are
losed under the Borel
fun
tional
al
ulus.
The index formula.
Finally we prove the index formula
indL2,Λ(D+)=hAˆ(X)Ch(E/S),[CΛ]i+1/2[ηΛ(D0F)−h+Λ +h+Λ]
whereh±Λ :=dimΛ(Ext(D±)−dimΛ(KerL2(D±).Ourproofisamodi(cid:28)
ationofVaillantproof
L2
that in turnis inspiredby Müller proofofthe (cid:21)index formulaon manifolds with
ornersof
odimension two [66℄. This is a (of
ourse) a proof based on the heat equation.
The starting point is the identity
indL2,Λ(Dǫ+)=luim01/2{indΛ(Dǫ+,u)+indΛ(Dǫ+,−u)+h−Λ,ǫ−h+Λ,ǫ} (6)
↓
where
h±Λ,ǫ :=dimΛExt(Dǫ±)−dimΛKerL2(Dǫ±)
ǫ=0
de(cid:28)nition also valid for .
Next we prove
indΛ(Dǫ+,u)=hAˆ(X)Ch(E/S),[CΛ]i+1/2ηΛ(DǫF,u∂)+g(u) (7)
g(u) 0
u
with −→ .
u
Equation (7)
ombined with (6) and (5) be
omes, after the (cid:21)limit
indL2,Λ(Dǫ+)=hAˆ(X)Ch(E/S),[CΛ]i+1/2ηΛ(DF0)+h−ǫ −h−ǫ .
ǫ 0 ǫ
The last step is to assure that under → ea
h (cid:21)depending obje
t in the above equation
ǫ=0
goes to the
orresponding value for .
Some words about the proof of (7). This is inspired from the work of Müller [66℄. We
[exp( tD2 )]
start from the
onvergen
e into the spa
e of leafwise smoothing kernels of − ǫ,u to
[Ker (D )] φ X X
L2 ǫ,u k k+1 m
. The
hoi
e of
ut o(cid:27) fun
tions supported in ( is the manifold
r =m X
trun
ated at ) gives an exaustion of into
ompa
t pie
es. Consider the equation
indΛ(Dǫ+,u)=strΛχ 0 (Dǫ,u)= lim lim strΛ(φke−tDǫ2,uφk)=
{ } k + t +
→ ∞ → ∞
kl→im∞strΛ(φke−sDǫ2,uφk)−Zs∞strΛ(φkDǫ2,ue−tDǫ2,uφk)dt. (8)
√k
t + ∞
The (cid:21)integral is splitted into s √k the se
ond one going to zero thanks to the Breuer(cid:21)
D
ǫ,u
Fredholmpropertyof . MoRreworRkis needed in the studyofthe(cid:28)rst one, theresponsible
for the presen
e of the eta invariant in the formula. Using heavily the relative version of the
Cheeger(cid:21)Gromov(cid:21)Taylorestimate (3) one shows that
√k
lim LIMs 0 =1/2ηΛ(DǫF,u0).
k→∞ → Zs
