AF-algebras and topology of 3-manifolds

0.31 MB·English

by  Igor Nikolaev

9
0 AF-algebras and topology of 3-manifolds
0
2

r Igor Nikolaev
a ∗
M

1
1 Abstract

We consider 3-dimensional manifolds, which are surface bundles over
]
T the circle. It is shown that to every infinite order (pseudo-Anosov) mon-
G odromy φ of the bundle, one can assign an AF-algebra Aφ (an operator
algebra). It is proved that the assignment is functorial, i.e. each mon-
.
h odromy φ′, conjugate to φ, maps to an AF-algebra Aφ′, which is stably
t isomorphic to A . This approach gives new topological invariants of the
a φ
m 3-dimensional manifolds, coming from the known invariants of the stable
isomorphism classes of the AF-algebras. Namely, the main invariant is
[
a triple (Λ,[I],K), where Λ is an integral order in the (real) algebraic

6 numberfieldK and[I]anequivalenceclassoftheidealsinΛ. Asacorol-
v lary, one gets the numerical invariants of the 3-dimensional manifolds:
7 determinant ∆ and signature Σ, which we compute for the case of torus
2 bundles.
2
0
1 Key words and phrases: AF-algebras, 3-dimensional manifolds
1
0
/ AMS (MOS) Subj. Class.: 19K, 46L, 57M.
h
t
a
1 Introduction
m
:
v 1.1 Background and motivation
i
X
A.Algebraictopology,inlarge,isafunctorwhichtakesthe categoryconsisting
r of topological spaces and continuous maps between the spaces into a category
a
consisting of the algebraic objects and morphisms between the objects. The
algebraiccategoryisusuallyacategoryofthegroups,ringsormodulesoverthe
rings with appropriate homomorphisms between the groups, rings or modules.
As a rule, the functor is not injective. Given such a functor, it is possible to
detect the topological invariants in terms of (normally easier) algebraic invari-
ants.
∗PartiallysupportedbyNSERC.
1
B. In the 1930’s Murray and von Neumann introduced the rings of bounded
operators on Hilbert space, which are now called von Neumann algebras. The
von Neumann algebras is a subcategory of a more general category of the C -
∗
algebras. The operator algebras are relatively new in the context of algebraic
topology. Inthe1970’sNovikovconjecturedahomotopyinvarianceofthehigher
signatures of smooth manifolds. It was first shown by Mischenko [19], that
the Novikov conjecture can be settled (in a special case) in terms of the C -
∗
algebras. BuildingontheideasofAtiyahandSinger[1],Kasparov[15]invented
anequivariantKK-theoryfortheC -algebras. Suchatheoryallowstoprovethe
∗
homotopyinvarianaceofthehighersignaturesforabroadclassofmanifolds. In
an independent development, V. F. R. Jones [14] discovered a new polynomial
invariant of knots and links appearing in the theory of subfactors of the von
Neumannalgebras. Formoreonthe interactionsbetweenthe operatoralgebras
and topology, we refer the reader to the last chapter of book [5].
C. Let X be anorientablesurface ofgenus g. We shalldenote by Mod (X)the
mapping class group, i.e. a group of the orientation-preserving automorphisms
of X modulo the normal subgroupof trivial automorphisms. Let φ Mod (X)
∈
and consider a mapping torus:
M = X [0,1] (x,0) (φ(x),1), x X .
φ
{ × | 7→ ∈ }
Note that Mφ is a 3-dimensionalmanifold and Mφ ∼=Mφ′ are homotopy equiv-
alent if and only if φ = ψ φ ψ 1 are conjugate by a ψ Mod (X) [12].
′ −
◦ ◦ ∈
Equivalently, M is calleda surface bundle overthe circle with the monodromy
φ
φ. The surface bundles make by far the most interesting, the most complex
and the most useful part of the 3-dimensional topology [22], p.358. Recall that
the automorphisms of X fall into three disjoint classes [23]: (i) the periodic
(φn = Id) automorphisms, (ii) the infinite order (pseudo-Anosov) automor-
phisms or (iii) a mixture of the types (i) and (ii). In the rest of this paper, we
shallbeinterestedintopologyofthemanifoldsM ,whereφisanautomorphism
φ
of type (ii).
D. It is known that any simple finite-dimensional C -algebra is isomorphic to
∗
the algebra M (C) of complex n n matrices. A natural completion of the
n
×
finite-dimensional semi-simple C -algebras (as n ) is known as an AF-
∗
→ ∞
algebra. The AF-algebra is most naturally given by an infinite graph, which
recordstheinclusionoffinite-dimensionalsubalgebrasintotheAF-algebra. The
graph is known as a Bratteli diagram. In an important special case when the
diagramisperiodic,theAF-algebraiscalledstationary. Finally,intheaddition
to a regular isomorphism ∼=, the C∗-algebras A,A′ are called stably isomorphic
wheneverA =A ,where istheC -algebraofcompactoperators. The
⊗K∼ ′⊗K K ∗
intrinsic invariants of both the isomorphism and stable isomorphisms classes of
the AF-algebras have been introduced by Elliott [9] and studied by Effros [8],
Handelman [11] and others.
2
E. Let φ Mod (X) be a pseudo-Anosovautomorphism. The mainidea of the
∈
presentpaperistoassigntoφanAF-algebra,A ,sothatforeveryψ Mod(X)
φ
∈
the following diagram commutes:
conjugacy
φ - φ =ψ φ ψ 1
′ −
◦ ◦
? ?
isomorphism
-
Aφ Aφ′
⊗K ⊗K
(In other words, if φ,φ are conjugate pseudo-Anosov automorphisms, then the
′
AF-algebras Aφ,Aφ′ are stably isomorphic.) For the sake of clarity, we shall
consider an example illustrating the idea in the case X =T2.
1.2 Model example
A. Let T2 be a two-dimensional torus. It is known that Mod (T2) = SL (Z),
∼ 2
where SL (Z) is the modular group. In the case of tori, an infinite order auto-
2
morphismsarecalledAnosov’s. Wheneverφissuchanautomorphism,itisgiven
by a hyperbolic matrix A SL (Z), i.e. matrix A such that tr (A ) > 2.
φ 2 φ φ
∈ | |
For the sake of simplicity, let tr (A ) > 2. (The case tr (A ) < 2 is treated
φ φ
−
likewise.) Denote by [A ]= TA T 1 : T SL (Z) a conjugacy class of A .
φ φ − 2 φ
Theconjugacyclasscontainsa{matrixA+ ∈[A ],who}seentriesarenon-negative
φ ∈ φ
integers. In view of the fact, we further restrict to the Anosov automorphisms
given by such matrices.
B. Let φ Mod (T2) be an Anosov automorphism. Consider a (stationary)
∈
AF-algebra, A , given by the following (periodic) Bratteli diagram:
φ
a a a
11 11 11
b b b b ...
(cid:0)@ (cid:0)@ (cid:0)@ (cid:0)
(cid:0)b a12@(cid:0) a12@(cid:0) a12@(cid:0) A = a11 a12 ,
@ a (cid:0)@ a (cid:0)@ a (cid:0)@ φ (cid:18)a21 a22(cid:19)
21 21 21
@b(cid:0) @b(cid:0) @b(cid:0) @b ...
a a a
22 22 22
Figure 1: The AF-algebra A .
φ
where a indicate the multiplicity of the respective edges of the graph. We
ij
encourage the reader to verify that F : φ A is a correctly defined function
φ
7→
3
on the set of Anosov automorphisms given by the hyperbolic matrices with the
non-negative entries.
C. Let us show that if φ,φ Mod (T2) are the conjugate pseudo-Anosov
′
∈
automorphisms, then Aφ,Aφ′ are stably isomorphic AF-algebras. Indeed, let
φ′ =ψφψ−1foraψ Mod(X). ThenAφ′ =TAφT−1foramatrixT SL2(Z).
∈ ∈
Note that (A )n = (TA T 1)n = TAnT 1, where n N. We shall use the
′φ φ − φ − ∈
following criterion of stable isomorphism ([8], Theorem 2.3): the AF-algebras
A,A are stably isomorphic if and only if their Bratteli diagrams contain a
′
common block of the arbitrary length. Consider the following sequences of
matrices:
A A ...A
φ φ φ

n
 T A|φAφ.{.z.AφT}−1,
n
which mimic the Bratteli diagram|s of{Azφ an}d Aφ′: the upper (lower) row cor-
responds to the Bratteli diagram of the AF-algebra Aφ (Aφ′). Since n can be
made arbitrary large, we conclude that Aφ⊗K∼=Aφ′ ⊗K.
D.The conjugacyproblemcannowbe recastintermsofthe AF-algebras: find
the intrinsic invariants of the stable isomorphism classes of the stationary AF-
algebras. One such invariant is due to Handelman [11]. Consider an eigenvalue
problem for the hyperbolic matrix A SL (Z): A v = λ v , where λ >
φ 2 φ A A A A
∈
(1) (2)
1 is the Perron-Frobenius eigenvalue and v = (v ,v ) the corresponding
A A A
eigenvector with the positive entries normalized so that v(i) K = Q(λ ).
A ∈ A
Denote by m = Zv(1) +Zv(2) a Z-module in the number field K. Recall that
A A
the coefficient ring, Λ, of module m consists of the elements α K such that
αm m. It is known that Λ is an order in K (i.e. a subring o∈f the algebraic
num⊆bers containing 1) and, with no restriction, one can assume that m Λ. It
followsfromthe definition, thatmcoincideswithanideal,I,whoseequi⊆valence
class in Λ we shall denote by [I]. It has been proved by Handelman, that the
triple(Λ,[I],K)isanarithmeticinvariantofthestableisomorphismclassofA :
φ
the Aφ,Aφ′ are stably isomorphic AF-algebras if and only if Λ = Λ′,[I] = [I′]
and K =K .
′
E. It is interesting, that the same set of arithmetic invariants has been used
to classify the conjugacy classes of the hyperbolic matrices, see e.g. [25]. The
phenomenon is due to a fortunate fact that the Anosov automorphisms of tori
are bijective with the hyperbolic matrices – an advantage which is no longer
(immediately) available for the surfaces. In this sense, the AF-algebras can be
regarded as a proper substitute for the hyperbolic matrices in the case g 2.
≥
1.3 Objectives
A. Let be a category, whose objects Ob ( ) are the pseudo-Anosov au-
M M
4
tomorphisms φ Mod (X) and morphisms Mor ( ) are the conjugacies
∈ M
φ = ψ φ ψ 1 between the automorphisms. Denote by a category, whose
′ −
◦ ◦ A
objects Ob ( ) are the AF-algebras and morphisms Mor ( ) are the stable
A A
isomorphisms of the AF-algberas. In view of our motivating example, the ob-
jectives of the present paper can be formulated as follows.
Mainproblem. ConstructafunctorF : (ifany),whichmapsthecon-
M→A
jugate pseudo-Anosov automorphisms into the stably isomorphic AF-algebras.
Find the invariants (numerical or other) of the stable isomorphism classes of
the AF-algebras.
B. In the present paper a functor F : is constructed. The functor
M → A
is non-injective with Ker F consisting of the commensurable pseudo-Anosov
automorphisms 1 . As for the second part of the main problem, two families of
the invariants are constructed: (i) the arithmetic invariants (the integer orders
in the algebraic number fields and equivalence classes of ideals in the orders);
(ii) the numerical invariants (the determinant and signature of module in the
number field). Since F is a functor, both (i) and (ii) are the topologicalinvari-
ants of the surface bundles M . Let us pass to the exact formulation of our
φ
results.
1.4 AF-algebras attached to measured foliations
A. It is useful to rephrase our model example as follows. Let A SL (Z)
φ 2
∈
be a hyperbolic matrix and let A v = λ v be the corresponding eigenvalue
φ A A A
problem. Denote θ = (v(2)/v(1)) K = Q(λ ) and consider a foliation, ,
A A A ∈ A FθA
on the torus given by the parallel lines of slope θ (see Fig. 2).
A
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
(cid:8) (cid:8)
Figure 2: The foliation on T2 =R2/Z2.
FθA
The foliation is preserved by the automorphism φ in the sense that the
FθA
line y = θ x is invariant under the action of φ, while the remaining lines map
A
to each other and tend to the invariant line. We refer to the foliation as
FθA
Anosov’s.
m,1nTheNa.utomorphisms φ1,φ2 ∈Mod (X) are said to be commensurable if φm1 = φn2 for a
∈
5
B. The matrix A can be recovered from the Anosov foliation. Indeed, let
φ
θ =[a ,a ,a ,...]be the regularcontinuedfractionofthe realnumber θ. Since
0 1 2
θ K,thecontinuedfractionθ =[a ,a ,...,a ,a ,...,a ]iseventually
A A 0 1 k 1 k k+p
∈ −
periodic, where a ,...,a is the minimal period of continuedfraction. It can
k k+p
be verified that
a 1 a 1
A = 0 ... n+k .
φ (cid:18) 1 0(cid:19) (cid:18) 1 0(cid:19)
C.Ameasured foliation onthesurfaceX isapartitionofX intothesingular
F
points x ,...,x of the order (multiplicity) k ,...,k and the regular leaves
1 m 1 m
(1-dimensionalsubmanifolds). OneachopencoverU ofX x ,...,x there
i 1 m
−{ }
exists a non-vanishing real-valued closed 1-form ω such that: (i) ω = ω on
i i j
±
U U ; (ii) at each x there exists a local chart (u,v) : V R2 such that for
i j i
∩ →
z = u+iv, it holds ωi = Im (zk2idz) on V Ui for some branch of zk2i. The
∩
pair (U ,φ ) is called an atlas for the measured foliation . Finally, a measure
i i
F
µisassignedtoeachsegment(t ,t) U ,whichistransversetotheleavesof ,
0 i
t ∈ F
via the integral µ(t ,t) = ω . The measure is invariant along the leaves of
0 t0 i
, hence the name. The mReasured foliation given by trajectories of a globally
F
defined closed 1-form, is called oriented.
D. For brevity, let be an oriented measured foliation given by the closed 1-
F
form ω. (In case is a general measured foliation, it is possible to cover by
F F
anorientedfoliation, ,onasurfaceX,whichisthedoublecoverofX ramified
F
over the singular points of odd order.) Let Sing = x ,...,x be the set
e e F { 1 m}
of zeroes of ω. Then the relative homology H (X,Sing ( );Z) = Zn, where
1
F
n = 2g+m 1. The integration of ω along a basis in the relative homology
−
yieldsv(i) = ω. Thevector(v(1),...,v(n))isknowntobeacoordinatevector
γi
of in the sRpace of all measured foliations on X with the same set of singular
F
points [7].
E. Denote by θ =(θ ,...,θ ) a vector with coordinates θ =v(i+1)/v(1). To
1 n 1 i
assign an AF-algebra, A , to−the measured foliation, consider the convergent 2
θ
Jacobi-Perroncontinued fraction of the vector θ:
1 0 1 0 1 0
= lim ... ,
(cid:18)θ(cid:19) k (cid:18)I b1(cid:19) (cid:18)I bk(cid:19)(cid:18)I(cid:19)
→∞
where b = (b(i),...,b(i))T is a vector of the non-negative integers, I the unit
i 1 n
matrix and I = (0,...,0,1)T. The AF-algebra in question is given by the
Bratteli diagram in Fig. 3. (The diagram shows the case n = 6. The numbers
b(i) indicate the multiplicity of edges of the graph.)
j
2This property is known to be generic, i.e. true for almost all vectors θ ∈ Rn−1. In
particular,theperiodicfractionsarealwaysconvergent.
6
b b b ...
(cid:8) (cid:8)
(cid:0)L (cid:8) L (cid:8)
(cid:8) (cid:8)
(cid:0) L (cid:8) L (cid:8)
(cid:8) (cid:8)
(cid:0) (cid:8)b L b(cid:8)L b ...
(cid:8) (cid:8) (cid:8)
(cid:0) (cid:8) AA L (cid:8) AA L (cid:8)
(cid:0)(cid:8)(cid:8) b(1)AA L(cid:8)(cid:8) b(2)AA L(cid:8)(cid:8)
(cid:8) 1 (cid:8) 1 (cid:8)
b(cid:0)(cid:8) (cid:8)b AA L b(cid:8)AA L b ...
H (cid:8) (cid:8)
@J H JJ AA L (cid:8) JJ AA L (cid:8)
J@HH b(1)JJ A(cid:8)A L(cid:8) b(2)JJ (cid:8)AA L(cid:8)
H 2 (cid:8) 2 (cid:8)
J@ H(cid:8)b JJ AAL b(cid:8) JJ AAL b ...
(cid:8) (cid:8)
J @ @@ JJ AAL(cid:8) @@ JJ AAL(cid:8)
(cid:8) (cid:8)
J @b(1)@@(cid:8)JJAAL b(2)@@(cid:8)JJAAL
3 (cid:8) 3 (cid:8)
JJJ@JJ(cid:8)HHbbb(cid:8)(41HH(cid:8)) HH@@(cid:8)HH@@(cid:8)JJHAHAJ@@J(cid:8)LAAJJLHAAH@@(cid:8)LJJJJAALLbb(cid:8)(cid:8)bHH(42(cid:8)HH) (cid:8)HH@@(cid:8)HH@@JJ(cid:8)HH@@AAJJLAAJJL(cid:8)HH@AA@LJJJJAALLbb ......
b(1) b(2)
5 5
Figure 3: The AF-algebra A (case n=6).
θ
1.5 Pseudo-Anosov foliations
There exists a special countable set of measured foliations, which will play an
important rˆole in the sequel. These are the invariant foliations of the pseudo-
Anosov automorphisms of surfaces. The topology of the invariant foliation en-
codesagooddealofinformationaboutthepseudo-Anosovautomorphism,hence
ourinterest. The formaldefinitionis asfollows. Letφ Mod(X)be apseudo-
∈
Anosov automorphism of the surface. By the definition of φ [23], there exist a
stable and unstable mutually orthogonalmeasuredfoliations on X, such
s u
F F
thatφ( )= 1 andφ( )=λ ,whereλ >1iscalledadilatationofφ.
Fs λφFs Fu φFu φ
The foliations , are minimal, uniquely ergodic and describe the automor-
s u
F F
phism φ up to a power. Since the foliations and determine each other,
s u
F F
we shall denote by one of them and call it a pseudo-Anosov foliation.
φ
F
1.6 Main results
We retain the notation of the preceding sections. By A we understand the
φ
AF-algebra of a pseudo-Anosov foliation and by F a mapping acting by
φ
F
the formula φ A . For a matrix A GL (Z) with the positive entries, we
φ n
7→ ∈
letλ bethePerron-Frobeniuseigenvalueand(v(1),...,v(n))thecorresponding
A A A
normalizedeigenvectorwithv(i) K =Q(λ ). Thecoefficient(endomorphism)
A ∈ A
ringofthemodulem=Zv(1)+...+Zv(n) willbedenotedbyΛ. Theequivalence
A A
class of ideals in the ring Λ generated by the ideal m, we shall write as [I].
7
Finally, we denote by ∆ = det (a ) and Σ the determinant and signature of
ij
the symmetric bilinear form q(x,y) = n a x x , where a = Tr (v(i)v(j))
i,j ij i j ij A A
are the traces of the algebraic numbers.POur main results can be expressed as
follows.
Theorem 1 A is a stationary AF-algebra.
φ
Theorem 2 The mapping F : has the following properties:
M→A
(i) F is a functor, i.e. F maps conjugate pseudo-Anosov automorphisms to
the stably isomorphic AF-algebras;
(ii) Ker F = [φ], where [φ] = φ Mod (X) (φ)m = φn, m,n N is
′ ′
{ ∈ | ∈ }
the commensurability class of the pseudo-Anoov automorphism φ.
Corollary 1 The following are invariants of the stable isomorphism class of
the AF-algebra A :
φ
(i) triples (Λ,[I],K);
(ii) integer numbers ∆ and Σ.
Corollary 2 Let M be surface bundle over the circle with a pseudo-Anosov
φ
monodromy φ. Then:
(i) (Λ,[I],K) is an arithmetic invariant of the homotopy type of M ;
φ
(ii) ∆ and Σ are numerical invariants of the homotopy type of M .
φ
1.7 Structure of the paper
The paper is organized as follows. In section 2 the notion of a Jacobian of
measured foliation is introduced and the basic properties of the Jacobian are
established. The Jacobians will be used to prove our main results. In section 3
it is shown that the topologically conjugate foliations correspond to the stably
isomorphic AF-algebras. This fact is critical for the proof of our results. The-
orems 1, 2 and corollaries 1, 2 are proved throughout section 4. In section 5,
the automorphisms of tori are considered. It is shown that the determinant ∆
discerns the conjugacy classes of such automorphisms better than the Alexan-
der polynomial. Some open problems and conjectures, related our results, are
formulated in section 6. Finally, section 7 is designed to help the interested
reader to briefly recall the main concepts and notions involved in the paper.
2 The Jacobian of a measured foliation
A.Recallthatthe JacobianJac(R)ofaRiemannsurfaceR isthe factorspace
Cg/Λ,whereΛisalatticegivenbytheperiods ω oftheholomorphic1-forms
γi
ω on R. The lattice Λ is an invariantly attachRed object, which records a good
8
deal of data on the Riemann surface R. In this section, we shall introduce an
analogofthelatticeΛforthemeasuredfoliationsonasurfaceX. Theresulting
object (a generalized pseudo-lattice [17]), we shall refer to as a Jacobian of the
foliation.
B. Let be a measured foliation on a compact surface X [23]. For the sake
F
of brevity, we shall always assume that is an oriented foliation, i.e. given
F
by the trajectories of a closed 1-form ω on X. (The assumption is no restric-
tion – each measured foliation is oriented on a surface X, which is a double
cover of X ramified at the singular points of the half-integer index of the non-
e
oriented foliation [13].) Let γ ,...,γ be a basis in the relative homology
1 n
{ }
group H (X,Sing ;Z), where Sing is the set of singular points of the foli-
1
F F
ation . It is well known that n = 2g+m 1, where g is the genus of X and
F −
m= Sing ( ). The periods of ω in the above basis we shall write as:
| F |
λ = ω. (1)
i
Z
γi
The λ are the coordinates of in the space of all measured foliations on X
i
F
(with a fixed set of the singular points) [7].
Definition 1 By a Jacobian Jac ( ) of the measured foliation , we under-
stand a Z-module m = Zλ +...+FZλ regarded as a subset ofFthe real line
1 n
R.
C. The Jacobiansare usefulin view of the following observation. Note that de-
spitetheperiodsλ dependonthebasisinthehomologygroupH (X,Sing ;Z),
i 1
F
the Jacobian does not. Moreover, up to a scalar multiple, the Jacobian is an
invariantofthe topologicalconjugacyofthe foliation . We shallformalizethe
F
observations in the following two lemmas.
Lemma 1 TheZ-modulemisindependentofthechoiceofabasisinH (X,Sing ;Z)
1
F
and depends solely on the foliation .
F
Proof. Indeed, let A=(a ) GL (Z) and let
ij n
∈
n
γ = a γ (2)
i′ ij j
Xj=1
be a new basis in H (X,Sing ;Z). Then using the integration rules:
1
F
λ = ω = ω =
′i Rγi′ Z nj=1aijγj
nP
n
= ω = a λ . (3)
j=1 γj ij j
P R Xj=1
9
To prove that m=m, consider the following equations:
′
n n
m = n Zλ = Z a λ =
′ i=1 ′i ij j
P Xi=1 Xj=1
= n ( n a Z)λ m. (4)
j=1 i=1 ij j ⊆
P P
LetA 1 =(b ) GL (Z)beaninversetothematrixA. Thenλ = n b λ
− ij ∈ n i j=1 ij ′j
and P
n n
m = n Zλ = Z b λ =
i=1 i ij ′j
P Xi=1 Xj=1
= nj=1( ni=1bijZ)λ′j ⊆m′. (5)
P P
Since both m m and m m, we conclude that m =m. Lemma 1 follows. (cid:3)
′ ′ ′
⊆ ⊆
Recallthatthefoliations and aresaidtobetopologically conjugate,ifthere
′
F F
existsanautomorphismh Mod(X),whichsendsthe leavesofthe foliation
∈ F
to the leaves of the foliation , see e.g [16], p.388. Note that the equivalence
′
F
deals with the topological foliations (i.e. the projective classes of measured
foliations [22]) and does not preserve the transversalmeasure of the leaves.
Lemma 2 Let , be the topologically conjugate measured foliations on a
′
F F
surface X. Then
Jac ( )=µ Jac ( ), (6)
′
F F
where µ>0 is a real number.
Proof. Let h:X X be an automorphism of the surface X. Denote by h its
actiononH (X,S→ing( );Z)andbyh onH1(X;R)connectedbytheform∗ula:
1 ∗
F
ω = h (ω), γ H (X,Sing ( );Z), ω H1(X;R). (7)
∗ 1
Z Z ∀ ∈ F ∀ ∈
h∗(γ) γ
Let ω,ω H1(X;R) be the closed 1-forms whose trajectories define the folia-
′
∈
tions and , respectively. Since , are topologically conjugate measured
′ ′
F F F F
foliations,
ω =µ h (ω) (8)
′ ∗
for a µ>0.
Let Jac ( )=Zλ +...+Zλ and Jac ( )=Zλ +...+Zλ . Then:
F 1 n F′ ′1 ′n
λ′i =Z ω′ =µ Z h∗(ω)=µ Z ω, 1≤i≤n. (9)
γi γi h∗(γi)
By lemma 1, it holds:
n n
Jac ( )= Z ω = Z ω. (10)
F Z Z
Xi=1 γi Xi=1 h∗(γi)
10