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M', both of which make the diagram homotopy commutative, we prove that 'P c::= 'P'. First the above Proposition 1.45 implies that p' 0 'P c::= p' 0 'P'. Hence we have a homotopy H. : MI ---> A connecting them. If we set No = M I8i M here, we can write MI = N o(i5",) because of the definition of MI. Consider the following commutative diagram:
No
A is a minimal model, the above group is trivial. Hence if exists and this gives the desired homotopy between 'P and 'P'. Finally the fact that 'P : M ---> M' becomes an isomorphism follows from the following proposition. PROPOSITION 1.46. Let M, M' be generalized nilpotent minimal d.g.a. 's wh'ich are generated by elements of degrees smaller- than or equal to i where i = 1,2"" ,00. If there exists a d.g.a. map
'P: M
--->
M'
such that the induced homomorphism 'P' : H*(M) -) H*(M') 'is bijective up to degree i and injective for degree i + 1, then 'P is an isomorphism, PROOF. To begin with, let 171 be the set of all d-closed elements of
M whose degree is positive and the smallest (which we denote by nd, Then we see that there exists a natural isomorphism 171 ~ Hnl (M). We apply similar construction to M' and let VI ~ Hn~ (M') be the corresponding isomorphism, Then from the assumption, we can deduce that n~ = nl and 'P sends 171 bijectively to VI. If we set Ml = A(Vd and M~ = AW{), then they become sub d.g.a.'s of M and M' respectively. Next we set M/icleal MT F1 and choose an isomorphism M ~ M I 0F1 . Regarding M', we can also construct a similar isomorphism M' ~ M~ 0 F{ such that 'P(Fd C F(, Let 172 be the set of all homogeneous elements x E F1 with the smallest positive degree (which we clenote by n2) such that dx E MI' Let V{ be the corresponding set for M' and let n~ be its degree. Then (-dx, x) becomes a relative co cycle with respect to the inclusion MI -) M so that it defines an element of the cohomology group Hn 2 +I(M 1 , M). It can be shown that this correspondence induces an isomorphism V2 ~ Hn2+I(MI,M). By Proposition 1.43, we obtain an exact sequence
0---> H n2(M I ) ---> Hn2(M) --->
172
--->
H n 2+ I (M 1 )
--->
H n 2+ 1 (M)
---> ... ,
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
33
II' we compare this exact sequence with the corresponding one for .M' and use two isomorphisms Ml 9'. M~, <po : H*(M) 9'. H*(M'), Lhen we can conclude that n~ = n2 and
B be a d.g. a. map inducing an isomorphism H*(A) 9'. H*(B) in cohomology. Then for any generalized nilpotent, minimal d.g.a. M, the natural map [M,A]---> [M,B] is a b~jection. Here [M, A] denotes the set of homotopy clas.ses of d.g.a. maps from M to A.
L DE RHAM HOMOTOPY THEORY
34
COROLLARY 1.48. Let A, B be cohomologically connected d.g.a. 's and let f : A -> B be a d.g.a. map. Also let PA : MA -> A and Pl3 : Ml3 -> B be minimal models of A and B re8pectively. Then there exists a d.g.a. map j: MA -> Ml3 which makes the diagram
MA~A
M6~B P13
homotopy commutative. Moreover, such a map is unique up to homotopy. PROOF. It is enough to set A in the statement of Theorem 1.47.
= Ml3,f = Pl3,B = B,M
co
MA 0
1.3. The main theorem 1.3.1. Differential forms on simplicial complexes. The de Rham complex A*(M) of a Coo manifold M is a. d.g.a. over ~ so that in principle we cannot deduce information on the rational homotopy type of M from it. If there a.re given enough cydes of Mover Z, then by investiga.ting values of integrals over them, we can decide whether a given closed form represents a rational cohomology class or not. However, this procedure cannot be considered as an intrinsic structure of the de Rham complex. Then there appeared the de Rham theory for simplicial complexes which turn out to be utilized to obtain information about their structure over Q. To define the de Rham complex of simplicial complexes, first consider the k-dimensional standard simplex
!J.k = {(to,tI'''' ,tk)
E JRk+l;ti
2
O,'L)i = l}.
1.49. (i) The restriction of any Coo differential form on JRHI to 6,k is called a Coo form on it. (ii) The restriction of a form DEFINITION
j 'I(to , ... ,tk)dt· t , 1 1\···l\dt·t..,whose coefficients h(to,'" , tk) E Q[to, tl,'" , tk] are polynomials with rational coefficients to !J.k is called a Q polynomial form.
1.3. THE MAIN THEOREM
35
We denote by A * (6, k) and AQ (6, k) the sets of all Coo forms and Q polynomial forms on 6, k respectively. For any face 6,r c D. k, the restriction of forms induces a natural homomorphism A*(6,k) ----> A*(6,T) and similarly for the Q polynomial forms. DEFINITION 1.50. Let K be a simplicial complex. A Coo q-form on K is a collection
(w,,- ),,-EK of q-forms w,,- E Aq((J) on each simplex (J of K such that if T is a face of (J, w,,-IT = w T . We denote by A*(K) the set of all Coo forms on K and call it the de Rham complex of K. If L c K is a subcomplex, the relative de Rham complex of the pair (K, L) is defined by
A*(K,L) = {w
E
A*(K);w,,- = 0 for any
0'
E
L}.
Similarly the de Rham complex AQ(K) of all Q polynomial forms on K as well as its relative version AQ(K, L) are defined. The exterior differential and the exterior product are defined for
A*(K,L), AQ(K,L) exactly the same way as the case of manifolds. Namely we set
d(w,,-) = (dw,,-),
(w,,-)
1\
(T,,-) - (w"
1\
T,,-).
Thus A*(K, L) and AQ(K, L) become d.g.a.'s over lR and Q respectively. The integration on each simplex induces natural maps
I: A*(K,L)
(w,,-)
f------>
I: A;Q(K,L):OJ (w,,-)
f------>
=:1
1 ~1
{O' ~
w,,}
E
C*(K,L;lR)
{O'
w,,}
E
C*(K,L;Q)
which commute with the differential operators by virtue of the theorem of Stokes. For the second map, we use the fact that the integral of any Q polynomial form on a standard simplex is a rational number. It follows that I induces a homomorphism in cohomology. THEOREM 1.51 (de Rham theorem for simplicial complexes). Let K be a simplicial complex. Then the map I induced by the integration gives isomorphisms
H*(A*(K)) H*(AQ(K))
~ ~
H*(K;lR) H*(K; Q)
of algebras. The same is true for' the relative cohomology groups of any subcomplex L c K.
1. DE RHAM HOMOTOPY THEORY
36
We first show that the claim holds for t:J. k . We assume that the case A*(t:J.k) is already known and consider the case AQ(t:J.k). Since the unique relation between the coordinates and I-forms on t:J.k is to + ... + tk = 1, dto + ... + dtk = 0, PROOF.
we can conclude that there exists a natural isomorphism
A;Q(t:J.k)
=:!
Q[tl,··· ,tkJ 0 E A*(K,K(n-I)) - - - 4 A*(K)
11 o ---->
C*(K, K(n-I)) -
C*(K)
---4
---->
A*(K(n-I))
---4
C*(K(n-I)) - -
---->
0
... 0
where K(n-I) denotes the (n -I)-skeleton of K. Now A*(K, K(n-I)) is isomorphic to the direct sum of d copies of A * (L\ n, 8L\n) , where d is the number of n-simplices of K. Similarly C*(K, K(n-I)) is isomorphic to the direct sum of d copies of C*(L\",8L\rt). Hence by (1.7), their cohomology groups are isomorphic. By the same argument as above, applying the five lemma to the induced long exact sequence in cohomology, we obtain an isomorphism H*(A*(K))
~
H*(K).
The relative version can be proved similarly. This completes the proof.
o
For the product structure, we refer the reader to [GlVI). 1.3.2. Homotopy groups of minimal d.g.a. 'so
1.52. Let M be a d.g.a. over the ground field K. (i) Let MI C M be the sub d.g.a. generated by elements of degree 1. We define sub d.g.a.'s Ni (i = 0,1,2,· .. ) of MI as follows. DEFINITION
No=K
= sub d.g.a. N2 = sub d.g.a.
Nl
of Ml generated by {x E M\ dx
= O}
of j\1/1 generated by {x E M\ dx ENd
N3 = sub d.g.a. of MI generated by {x E MI; dx E N 2 }
Then each N, C Ni+l is a Hirsch extension and MI If we take the dual of these, we obtain a series . . . ----+
£i
----t . . . ----t
.c 3 ----t £2 ----t £ 1 ----t
°
= UiNi.
1.3. THE MAIN THEOREM
39
of nilpotent Lie algebras £i over K. Here each surjective homomorphism £i+l --; £i is a central extension of Lie algebras. If we set Li to be the Lie group corresponding to £;, the Baker-Campbell-Hausdorff formula implies that the exponential map exp : £i --; L; is bijective and the product in L; can be expressed by explicit polynomial mappings. We define the fundamental group 7rl (M) of M to be the tower . . . ----->
Li
------> . . . ------>
L3
------>
L2
------>
L1
------>
0
of nilpotent Lie groups. These matters will be further explained in §1.4. (ii) Let M+ = EBi>oM(i) be as before and set
I(M) = M+ /M+ M+
= EBi>O I(M)C'i). In other words, I(M)(i) is the set of all indecomposable elements of M. We set
7ri(M) = Hom(I(M)(;), K) and call it the i-th homotopy group of M. The differential d induces a bilinear mapping
7ri(M) ®7rJ (M)
------>
7r·i+J-l(M).
It can be deduced from the fact dod = 0 that 7r. (M) EBi7ri(M) becomes a graded Lie algebra. DEFINITION 1.53. Let (X, xo) be a topological space with a base point. A mapping
7ri(X,XO) x 7rj(X,Xo)
3
(0:,,6)
f------+
[a,,6]
E
7ri+j-l(X,XO),
called t.he Whitehead product, is defined as follows. If two elements a, ,6 are represented by continuous mappings f: (r',ar') --; (X,xo),
g: (Ij,[)Jj) --; (X,xo)
respectively, then [a,,6] is the element represented by
ar'+J
=
Ii
X
uP
U
ali x Ij ~ X.
If i = j = 1, [a,,6] is the usual commutator in the fundamental group. In the cases i > 1 or j > 1, the Whitehead product is distributive with respect to the corresponding factor and also
[,6, 0:]
=
(-1) ij [a, ,6].
40
L DE RHAM HOMOTOPY THEORY
Moreover it is known that the Jacobi identity
(_I)ik[[a,/JJ,"/]
+ (-I)ij[[/J,,,/J,a] + (-1)1k[b,a],!3]
holds where "/ E 7rk(X,XO). With these structures, 7r*(X) = EBk7rk(X) becomes a graded Lie algebra over Z. The next theorem is the main theorem of the de Rham homotopy theory. THEOREM 1.54 (Sullivan [Su2][Su3]). (i) Let K be a simplicial complex which is nilpotent and of finite type; that is the act'ion of Jrl (K) on 7ri (K) is nilpotent and Jri(K) is .finitely generated for any i. Let AQ(K) be the d.g.a. consisting of all Q polynom'ial forms on K and let M f( be its minimal model. Then M J( and the rational homotopy type of K are related as duals to each other 80 that each of them determines the other completely. In particulaT there e:risi;s a natural isomorphism
as graded Lie algebras over Q. (ii) Let f : K -7 L be a simpl'icial map between nilpotent simplicial complexes and let f* : AQ (L) -) AQ (K) be the induced homomorphism. Denote by .1* : M L -7 M f( the d. g. a. map between the m'inimal models induced by f*. Then it is the d'ual of the localization map fa : Ko -7 Lo of f so that each of.f* and fo deteTm'ines the other completely. In particular, under the isomorphism of (i), the mop .1* : Jr * (M K ) -7 Jr. (Md coincides wil;h f. : 1f* (K) ® Q - 7 Jr * (L) 0Q. THEOREM 1.55 (Sullivan [Su3]). Let M be a Coo manifold which is nilpotent and of finite type and let K be a Coo triangnlation of d. If we denote by Ai /1'/ the minimal model of the de Rham comple:r A * (Ai) of Ai, then theTe exists a natural isomorphism MM~MJ(0R
In particular- we have 1f*(Ml'vd ~ 1f.(M) @ JR. A similaT statement as that of the pTev'ioll,s theorem holds fOT Coo mappings between two manifolds.
Sketch of Proof of Theorem 1.54.. The key to the proof is the following fact which exhibits certain bijective correspondence between principal fibrations with fiber an Eilenberg-MacLane space and Hirsch extensions. Let X be a nilpotent simplicial complex and suppose
41
1.3. THE MAIN THEOREM
that there is given a d.g.a. map M --> AQ(X) which induces an isomorphism in cohomology. Let V be a finite dimensional vector space over Q and let
K(V,n)
--4
E
--4
X
be a principal fibration over X with fiber K(V, n). Let
o E Hn+l(x; V) ~ Hom(Hn+ 1 (XiQ), V) be its characteristic cohomology class. Then the dual element can be expressed as 0* E
0*
of
0
Hom(V*,H"+l(X;Q)) ~ Hom(V*,Hn+l(M)).
On the other hand, the isomorphism class of a Hirsch extension
is classified by its characteristic cohomology class 0' E
Hom(V*,HM1(M))
(see Proposition 1.32). Then, as is expected by the fact that the two elements 0*, 0' belong to the same set, we can conclude as follows. Namely, there exists a d.g.a. map
M ®d A(V*)"
--4
AQ(E),
which induces an isomorphism in cohomology and coincides with the given map M -, AQ(X) on M, if and only if the equality 0*
= 0'
holds. This fact is natural and should be easy to understand conceptually. However, we need many technical preparations for a rigorous proof of it. For example, in the above, E is not a simplicial complex as it stands so that we have to construct a model of E --> X in terms of simplicial complex ami simplicial maps. We refer the reader to the book [GM] for details. Assuming the above fact, let us examine the ingredients of the theorem on simply connected simplicial complex K for simplicity. If we set 1f2(K)®Q = V2 , the first stage in the Postnikov decomposition of the rational homotopy type K 0 of K is given by
Ko
--4
K(V2, 2).
On the other hand, there exists a natural isomorphism H2(K; Q) ~ V2* so that we can write MJ{ = A(V;*). Then J(MK )(2) = V2* = Hom(7T'2(K), Q).
1. DE RHAM HOMOTOPY THEORY
42
Next, if we set 11"3 (K) Q9 Ql = V3 , the second stage in the Postnikov decomposition is given by the principal fibration
K(Va,3)
K O(3)
---->
---->
K(V2, 2).
If we denote its characteristic cohomology class, namely the k-invariant of Ko, by
k 4 (Ko) E H4(K(V2' 2); Va) ~ Hom(H4 (K(V2, 2», V3), its dual element can be written as
k4 (Ko)* E Hom(V3*' H 4(K(V2, 2»). Then the claim is that the Hirsch extension
A(Vn Q9d A(Vn which corresponds to the above element is nothing but M 3 , that is the 3-minimal model of AQ(K). Moreover, if we take the dual of the differential where we can identify
A 2 (Vn = {homogeneous polynomial of degree 2 generated by H2 (K; Ql) },
we obtain the dual mapping {homogeneous polynomial of degree 2 generated by H 2 (K; Ql)} ---->
V3 = 7r3(K)
Q9
Ql.
Now the claim is that this corresponds to the Whitehead product
1f2(K) Q9 1I"2(K) ....... 1f3(K). Explicitly, if we choose a basis Xb··· ,Xe of H 2 (K;Ql), we can write M7( = A(X1,··· ,xe). Let xi,··· ,xi be the dual basis of 1I"2(K) Q9 Ql. Let Y1,··· , Ym, Z1>··· , Zn be new generators of MK of degree 3. Here Yl,··· , Ym is a basis of H:3 (K; Ql) and Zl,··· , Zn is a basis of Ker(H 4 (M}) ....... H 4 (K;Ql»). Hence we can write dYi = 0,
Then yi, ... , y:n, zi, and we have
... , z~
dZ k = LatxiXj.
can serve as the dual basis of 1f3(K) Q9 Ql,
* Xj*] [Xi'
k * = "" L...- aijzk·
The situation is similar for higher degrees. Namely, the k-invariant of K 0 at each stage corresponds to the characteristic cohomology class
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 43
of the Hirsch extension appearing in the corresponding stage of MK, and quadratic terms of the differential d of the Hirsch extensions correspond to the Whitehead products on 1f*(K) 0 rQ. 1.4. Fundamental groups and de Rham homotopy theory 1.4.1. Lower central series and nilpotent groups. Let r be a group. We put ro = r, and for general k ::::: 0 we inductively define rk+l = [rk, r]. r 1 = [r, r] is the commutator subgroup of r. rk is a normal subgroup of r for any k. The series
r :J r 1 :J ... :J r k :J ... of these normal subgroups is called the lower central series of r. If r k is trivial for some k, r is said to be nilpotent. Any abelian group is nilpotent. The quotient group Nk = r / r k is called the k-th nilpotent quotient of r. Nl = r/[r,r] is the abelianization of r, and Nk is nilpotent for any k. A short exact sequence 1 ~ A ~ G ~ Q ~ 1 consisting of homomorphisms of groups is called an extension of the group Q by A. In the case where A is an abelian group contained in the center of G, it is called a central extension and denoted by
o ----> A
---->
G
---->
Q
---->
1.
If we set Ak = rk-drk, it is easily seen to be an abelian group contained in the center of Nk = r / r k. Hence we obtain a series
(1.8)
0
-----.c,
Ak
---->
Nk
---->
N k- 1 ----> 1 (k
= 1,2,· .. )
of central extensions. By collecting them, we obtain homomorphisms from the given group r to a tower
(1.9)
II
1
r
N3
1
II r
N2
.)
1
II r
----->
N1
=
r/[r,r]
1. DE RHAM HOMOTOPY THEORY
44
of nilpotent groups such that each step tension.
Nk -)
Nk -
1
is a central ex-
1.4.2. Central extensions of groups and the Euler class. Given a group G and an abelian group A, let us consider the problem of constructing central extensions of the form
(1.10)
0
----->
A ~G~Q~1
and also the classification of them. To investigate the structure of such an extension, choose a map s : Q -> G such that 7r 0 s ,idQ. If we can take s to be a homomorphism, then it is easy to sec that G ~ A x Q. Therefore we consider the correspondence
Qx Q
3
(0:,{3) ........ c s (a,{3) = s(a)s({3)8(af3)-1.
E
A.
A simple computation shows that the equality
cs ({3, 'Y) - cs (a{3, 'Y) + cs(a, (3'Y) - cs(o:, (3) = 0 holds for any a, (3, 'Y E Q. It is called the co cycle condition. We denote by Z2(Q; A) the set of all mappings c : Q x Q -> A satis{ying the co cycle condition, and any element of it is called a 2-cocyde of Q with values in A. The above mapping Cs is called the 2-cocycle associated to s. Let C s ' be the 2-cocyde associated to another map 8' : Q -> G. In this case, it can be shown that if we define a map d: Q -> A by d(a) = s'(a)8(a)-1 E A, we have (1.11)
c'(o:, (3) - c(a,f3) = d({3) - d(a[3)
+ d(a).
In general, for any given map d : Q -> A, if we define od : Q x Q -) A by setting lSd(a, (3) to be the right hand side of (1.11), then it becomes a 2-cocycle. Such a 2-cocycle is called a 2-coboundary, and the set of all 2-coboundaries is denoted by B2(Q; A) c Z2(Q; A). The equation (1.11) can now be written as c' - (' = od. Given any element c E Z2(Q; A), define a product on the set A x Q by
(a, a)(b, (3) = (a
+ b + c(a, (3), 0:(3)
(a, bE A, 0:, (3 E Q).
Then it can be shown that A x Q becomes a group which is a central extension of Q by A. We denote this group by A XC Q. In this case, if we define s : Q -> A Xc Q by s(a) - (0, a), then clearly Cs = c. Moreover, for another 2-cocycle c' E Z"2(Q; A) if there exists a map d : Q -> A such that c' - c = lSd, then it can be shown that we can construct a group isomorphism A xc' Q ~ A xcQ which is the identity on A. We then consider the quotient group
H2(Q; A) = Z2(Q; A)jB 2 (Q; A)
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 45
;1I1d call it the two dimensional cohomology group of the group Q with coefficients in A. It follows from the definition that the element hl E JI2(Q;A) is well defined independent of the choice of s. We (all it the Euler class of the central extension (1.8). Summing up the above argument, we obtain the following theorem. THEOREM 1.56. Let Q be a group and let A be an abelian group. Then the set of isomorphism classes of central extensions of Q by A can be naturally identified with H 2(Q;A) = Z2(Q;A)/B 2 (Q;A) by associating the Eule'r class.
We identify Sl with the Lie group SO(2). As is well known, the set of isomorphism classes of Sl-bundles Sl ---4 E ---4 X over a topological space X can be identified with H2(X; Z) by associating the Euler class to each Sl-bundle. The above theorem is an analogue of this fact in the context of group theory. In fact, the classification of central extensions of a group Q by Z corresponds exactly to the geometrical classification of Sl-buncHes over K(Q,l). Here the 2dimensional cohomology of a group appeared. We can also consider the cohomology group of any dimension. DEFINITION 1.57. Let r be a group. The cohomology group H*(K(T, 1)) of the Eilenberg-MacLane space K(T, 1) is called the cohomology group of r and is denoted by H* (r).
This is a. geometrical definition of cohomology of groups. There is also a purely algebraic definition (see §3.6.2). The above description of central extemiions in terms of 2-dimensional cohomology of groups is one such example. Also we can consider any r-module for cocHicients of cohomology groups. For a general theory of cohomology of groups, we refer the reader to [Brl. Let M be a manifold, or more generally a topological space, a.nd let M ---4 K (7r1 (M), 1) be the first stage in the Postnikov decomposition of it. This map induces a hOlIlomorphism H*(7rl(M))
--->
H*(M).
This homomorphism already played a key role in the formulation of the Novikov conjecture, which has been one of the fundamental problems in topology of manifolds. It seems that the importance of this kind of homomorphism will increase in the future in not merely geometry but also in algebraic geometry and number theory.
1. DE RHAM HOMOTOPY THEORY
46
r
1.4.3. Malcev completion. Let be a finitely generated group. In this subsection, we define r ® Q by using the results of the preceding subsections §1.4.1 and §1.4.2 (see [M] for details). It is a certain projective system consisting of nilpotent Lie groups over Q and homomorphisms between them. It contains all the nilpotent information over Q of the group r. r ® Q is called the Malcev completion or the rational nilpotent completion of r. Roughly speaking, r ® Q is the diagram
1
II r
(1.12)
N3®Q
-->
1
II r
N2®Q
----?
1
II r
----?
NI®Q=Hl(r;Q)
which is obtained by replacing each nilpotent group Nk by Nk ® Q in the tower (1.9) of nilpotent groups. Nk ®Q can be defined inductively as follows. If k = 1, NI is an abelian group so that NI ® Q is defined and we have a natural homomorphism Nl ---t Nl ® Q. In this case, H*(NI ®Q; Q) ---t H*(NI; Q) is an isomorphism. We now assume the existence of N k - I ®Q and a homomorphism N k - I ---t N k - 1 ®Q such that the induced homomorphism H*(N~'_I ® Q;Q) 2:' H*(Nk-I;Q) in Q cohomology is an isomorphism. Consider the homomorphism H 2 (Nk_ 1 ; A k )
--7
H 2 (Nk _ 1 ; Ak ® Q)
2:'
]-J2(Nk_l ® Q; Ak ® Q).
The Euler class of the central extension (1.8) is an element of the group H 2 (Nk _ 1 ;A k ). We then consider its image, under the above homomorphism, in H2(Nk_1 ® Q; Ak ® Q). If we consider the associated central extension, we obtain a commutative diagram
o -----+
Ak
1
----t
1
1
1
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 47
By this, Nk Q9 Q and a natural homomorphism Nk --t Nk Q9 Q Ilave been simultaneously constructed. It can be shown, by an argument using the spectral sequence, that this homomorphism induces ;tn isomorphism H* (Nk Q9 Q; Q) ~ H* (Nk:; Q) in Q cohomology. This completes the induction step. Geometrically, Nk Q9 Q is nothing but [he fundamental group of the rational homotopy type K(Nk , 1)0 (see ~i1.1.3) of the Eilenberg-MacLane space K(Nk, 1). 1.4.4. Fundamental groups and differential forms. The main theorem of the de Rham homotopy theory concerning the fundamental groups can be stated as follows. The Malcev completion of the fundamental group (tcnsored by lR) is equivalent to the I-minimal model of the de Rham complex. In this subsection, we briefly explain this theorem from the viewpoint of constructing both sides explicitly. Let M be a Coo manifold. We denote by the fundamental group Jrl(M) of M and let Ml be the I-minimal model of the de Rham complex A * (M). First we define Jrl (M) Q9 lR as follows. We consider the Malcev completion r Q9 Q and replace each nilpotent Lie group Nk Q9 Q by simply connected Lie group Nk Q9lR over R On the other lland, the I-minimal model Ml was obtained as an increasing series
r
Mo
= lR C Nl C
N2
C ... C Ml =
UNk k
of d.g.a.'s over JR. (see §1, 2, 3 (1.2) and §1.3.2, Definition 1.52 (i)). Now each Nk is a connected free d.g.a. generated by elements of degree 1. Hence, by taking the dual as in §1.2.1, Example 1.34, we obtain a Lie algebra nk over R Moreover, since nl ~ Hl(M;JR.)* is a commutative Lie algebra, it can be shown by induction on k that nk is a nilpotent Lie algebra for any k. In fact, nk can be obtained from nk-l by a central extension
(1.13) of Lie algebras. Here we have the identification Uk
= Hom(Ker(H 2 (Nk _d
--t
H 2 (r; lR)), lR).
Let N~ be the simply connected nilpotent Lie group corresponding to the nilpotent Lie algebra nk. Then the exponential map exp : nk --t is a bijection. Explicitly, it is described in terms of polynomial mappings by the Baker-Campbell-Hausdorff formula. Geometrically, Nk can be naturally identified with the d.g.a. consisting of left invariant differential forms on N~.
N:
48
1.
DE RHAM HOMOTOPY THEORY
The central extension (1.13) of nilpotent Lie algebras induces the associated central extension (1.14)
0 ~ A~ - , Nf
-->
Nf_l - , 1
of nilpotent Lie groups, which is equivalent to the former. Then the key fact, in the de Rham homotopy theory of fundamental groups, is that the above central extension is equivalent to the tensor product with IR of the central extension (1.8) which is associated to the lower central series of = 7fl (M). In particular, we have Nj;· ~ N k @ IR. This can be proved a..'l follows. To begin with, clearly HI (Nk ) ~ HI (r) = Nl for any k. I-Ience, by the long exact sequence of the cohomology of the group extension
r
1
-->
rk - 1 ~ r ~ N k - 1
--t
1
(cE. [BrD, we can deduce
H1(rk_l)r ~ Ker(H 2(Nk _ 1 )
......
H2(r»).
On the other hand, it is easy to see that HI (rk - 1 ; Q)r ~ (rk - 1 / I k)0 Q = Ak ® IQ. It follows that the tensor product with IR of the central extension (1.8) corresponds to
Ker(H 2 (Nk _ 1 ; lR) -, H2(r; IR»). On the other hand, the above central
exten~ion
(1.14) corresponds to
Ker(H 2 (Nk _d ...... H 2 (r;IR»). Then it can be proved by induction on k that there exists a natural isomorphism H 2 (Nk _ 1 ) ~ H 2 (N k _ 1 ;1R) ~ H 2(Nk _ 1 0Q;lR). The claim is now proved. Thus we obtain t.he following theorem. THEOREM 1.58 (Sullivan [Su3D. Let M be a Coo man~fold. Then the tower of nilpotent Lie groups {Nk®lRh, associated to the IR nilpotent completion 1fl (M) ® lR of the fundamental group of M, is canonically isomorphic to the tower of nilpotent Lie groups which is obtained by first d'ualizing the I-minimal model of the de Rham complex A*(M) and then taking the exponential. In the above, if we replace M and its de Rham complex A*(M) by a simplicial complex K and its de Rham complex AQ(K) of Q polynomial forms, then we obtain a similar theorem which holds over Q.
CHAPTER 2
Characteristic Classes of Flat Bundles Let G be a Lie group. One way of investigating global structure of prillcipal bundles with structure group G is by putting connections on them and then studying the associated curvature. This is the ingredient of the Chern-Weil theory. The most important cases are those where G is a general linear group, namely the cases of GL(n, JR) or GL(n, C). These correspond to considering real or complex vector bundles where characteristic classes like Pontrjagin classes or Chern classes play fundamental roles. Now a flat bundle, which we investigate in this chapter, is a principal bundle endoweci with a connection such that its curvature vanishes identically. Any characteristic class with real coefficients of such bundle is zero. Therefore it may appear that it is close to a trivial bundle. However, it will turn out that this is far from being true, depending on the fundamental group of the base space. We need new methods to study the structure of such bundles. The holonomy groups as well as the theory of characteristic classes of flat bundles, which is based on cohomology theory of Lie a.lgebras, can serve as those. In this chapter, we describe basic facts concerning these materials. We also give a brief account of the Gel'fand-F\lks cohomology theory ([GFl] [GF2]) which corresponds to the case where the structure group is infinite dimensionaL 2.1. Flat bundles 2.1.1. Chern-Weil theory. Let G be a Lie group and let 1r : P -) 1\11 be a principal G-bundle over a Coo manifold 1\11. Namely there is given a right action
of the structure group G on the total space P satisfying the following condition. 49
50
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
Local triviality: For any point p EM, there exist an open neighborhood U 3 P and a diffeomorphism r.p : 7r-l(U) S:! U x G such that
7r( ug)
= 7r( u),
r.p( ug)
= r.p('u)g
(u E 7r- l (U), 9 E G).
For example, the tangent frame bundle 7r : P(NJ) ---t M of NJ becomes a principal bundle with structure group GL(n,IR), where dim M = n, and it is a very important principal bundle for the investigation of the structure of M. In fact, for the study of manifolds, it is one of the main tools to consider various bundles over NJ, not merely the tangent frame bundle, and then to examine the structure of them. Now the Chern-Weil theory is the one which can analyze the structure of principal bundles. In this subsection, we review this theory briefly. Given a principal bundle 7r : P ---t NJ, in order to make certain connections between fibers 7r- 1 (p)(p EM), we first introduce what is called a connection on it. More precisely, at each point u E P in the total space, we give a direct sum decomposition (2.1) of the tangent space which is invariant under the action of G. Here Vu denotes the subspace consisting of all tangent vectors at u which are tangent to the fiber 7r- 1 (7r(u)) (these are called vertical vectors). Any tangent vector belonging to Hu is called a horizontal vector (with respect to this connection). The set of all Hu , namely 1-l = {Hu; u E P} becomes a distribution on P. In this terminology, we can say that a connection is nothing but a distribution on the total space which is G invariant and transverse to the fibers.
If we express connections in terms of differential forms, we obtain connection forms. Explicitly, if we denote by 9 the Lie algebra of G, then at each point u E P the projection TuP
--4
Vu S:! 9
induced by the direct sum decomposition (2.1) defines a I-form w E Al(p; g) on P with values in g. The form w thus obtained is called the connection form. Here 11" S:! 9 is the canonical identification. Clearly we have
Hu
= {X
E
TuP;w(X)
= O}.
Conversely, if we are given a I-form w E A1(P; g), then we have the associated distribution 1-l which is defined by the above equality.
51
2.1. FLAT BUNDLES
We can then express the condition that 1i becomes a connection by simple equalities of differential forms. Therefore connection forms are rrequently called simply connections. If there is given a connection W E A1(P; g) on a principal Gbundle, then we can define its curvature n E A 2 (P; g) by d~fferentiat ing it. These two forms satisfy the following fundamental equation: (2.2) which is called the structure equation. Now consider the k-th power
of the curvature form n and compose it with an invariant polynomial f E Jk (G) of G) namely a symmetric multilinear map f:gx···xg--d~
"----v----' k-times
which is invariant under the adjoint action of G. Then we obtain a 2k-form
on P. It can be shown that the form obtained in this way is always closed and moreover it is the pullback of a uniquely defined form on the base space under the projection 7f : P ---> 1\11. This procedure defines a homomorphism w: J(G)
----->
H*(M;JR)
(J(G)
=
Ef7klk(G))
which is called the Weil homomorphism. It can be proved that this homomorphism does not depend on the choice of a connection. Hence, for any element f E J(G), its image w(f) E H*(M;JR) expresses the way a given principal G-bundle is twisted in terms of real cohomology classes of the base space. We call these classes characteristic classes of principal G-bundles. If we unify connection, curvature and invariant polynomials in a single object, we obtain the Weil algebra
W(g)
=
A*g*18i S*g*.
It serves as a model for the de Rham algebra of the total space of
any principal bundle
7f :
P
--->
M endowed with a connection. More
52
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
precisely, the following commutative diagram is defined.
w W(g) - -
~
A*(P)
!~.
i! J(G)
._-..., A*(M) w
Here both i and 7r* are injective, and the homomorphism induced in cohomology by the map w in the bottom row is the Weil homomorphism mentioned above. The above is a brief summary of the Chern-Weil theory. 2.1.2. Definition of flat bundles. DEFINITION 2.1. A connection won a principal G-bundle is called a flat connection if its curvature n is identically O. A principal Gbundle equipped with a fiat connection is called a flat G-bundle. EXAMPLE 2.2. If we put the trivial connection on a product bundle M x G, it is clearly a flat bundle. This is called a trivial flat bundle. The connection form Wo of this bundle is given by Wo = q*O where q : M x G -+ G is the natural projection and () E Al(G; g) denotes the Maurer-Cartan form of G. EXAMPLE 2.3. Let 7r : P -+ M be a fiat G-bundle and let f N -+ M be a Coo map. Then the pullback bundle f* P -+ N by becomes a fiat G-bundle.
f
By virtue of the Chern-Weil theory, which we recalled in the previous subsection, any real characteristic class of a fiat bundle vanishes. However, such bundle is not necessarily a trivial bundle as a principal bundle and furthermore, even if it were so, the fiat connection on it is not necessarily a trivial one. Depending on the base :;pace M, it may happen that there are many fiat G-bllndles on it. In such a situation, it often becomes an important problem to consider all fiat bundles on M and then classify them. Accordingly we first give a criterion of classification of fiat bundles. DEFINITION 2.4. Let 1fi : Pi -+ Mi (i = 1,2) be two fiat bundles and let Wi E A 1 (Pi ; g)(i = 1,2) be their fiat connection forms. A
2.1. FLAT BUNDLES
53
bundle map
from PI to P2 is called a bundle map as fiat bundles if the condition WI is satisfied. This condition can be equally phrased that the differenti8l of .f sends horizontal vectors at any point u in PI to horizontal vectors at .fCu) E P2 . Two Hat G- bundles over the same base space M are said to be isomorphic if there exists a bundle map as fiat bundles between them over the identity map of M.
.f*W2 =
With this terminology, a fundamental problem is this: Determine the set of isomorphism classes of fiat G-bundles over a given manifold NI. 2.1.3. Flat bundles and completely integrable distributions. In this subsection and the next, we shall present a few mutually equivalent conditions which describe fiat G-bundles geometrically. Suppose that there is given a connection w on a principal Gbundle 7f : P ---t M and let ?t = {Hu; u E P} be the corresponding distribution on P. Let us rephrase the condition that this connection is fiat, namely the vanishing of its curvature, in terms of the distribution ?t. For that we recall the theorem of Frobenius. In a general setting, let M be a Coo manifold and suppose that there is given a distribution r on M, which is by definition a sub bundle of the tangent bundle TM. A submanifold N of M is called an integral manifold of T if at each point pEN we have TpN = Tp. A connected integral manifold which is not a proper subset of any connected integral manifold is called a maximal integral manifold. We denote by r( r) the set of all sections of T. Also let 1(r) denote the ideal of A*(M) generated by I-forms ex E AI(M) such that ex(X) = 0 for any X E r(T). Then, if we set
1k(T)
= {17
E Ak(M); 17(XI
,'"
clearly we have 1(T) = €Jk1k(r).
,Xk ) = 0 if Xi
E r(r)
for any i},
54
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
2.5. (i) A distribution T on M is called completely integrable if for any point on M there exists an integral manifold containing it. (ii) T is called involutive if the condition DEFINITION
X, Y
E r(T)
===}
[X, Y]
E r(T)
is satisfied. (iii) If dI(T) C I(T), I(T) is called a differential ideal. THEoREivr 2.6 (Theorem of Frobenius). Let T be a distribution on a Coo manifold M. Then the following three cond'itions are equivalent. (i) T is completely integrable. (ii) T is involntive. (iii) I (T) is a differential ideal.
If we use the theorem of Frobenius above, we can give an answer to the present problem as follows. PROPOSITION 2.7. A connection w on a principal bundle is fiat if and only if the corresponding distribution H is completely integrable. PROOF.
By virtue of the theorem of Frobenius, we have only to
show that w is fiat MxG given by Here
~(u)
epA'ljJ E Ck+f(g)
of two cochains which is explicitly given by ep A 'ljJ(XI , ... , Xk+e)
=
1
L
sgn(J" k!e! ep(Xa(I)"" , Xa(k»)'ljJ(Xa(k+l)"" , Xa(k+e»).
aE 6 kH
Now motivated by (2.11), we define a linear map d: Ck(g) - } Ck+l(g)
by setting d
V; alternating multilinear}
'--v-----' k-times
and define the coboundary operator
d: Ck(g; V)
--t
Ck+I (g; V)
2.2.
COHOMOLOGY OF LIE ALGEBRAS
65
by k+J
dcp(XJ,··· ,Xk+l) = 2)-1),+lXi CP(X I ,···
,Xi,···
,Xk+d
i=l
+ ""(-l)i+jcp([X ~
't)
Xl J ' Xl , ...
X ... 'J" X ... x k + J ) .
,tl
i<j
It can be checked that d 2 = O. Observe here a similarity between the above definition of d ancl the equality (2.10). DEFINITION 2.15. Let 9 be a Lie algebra and let V be a g-module. We denote by H* (g; V) the cohomology of the cochain complex C*(g; V) and call it the cohomology of 9 with coefficients in V.
2.2.4. Cohomology of .5[(2, IR). As an example of cohomology of Lie algebras, here we compute the cohomology of the Lie algebra
s[(2,IR)
=
{X E M(2, IR); TrX = O}
consisting of all 2 x 2 real matrices with Tr = O. We choose
Xo
=
G ~l)'
Xl
(~ ~),
=
X2
=
(~ ~)
for a basis of s[(2,IR) and let
CPo, CPI, CP2 E C I (£1[(2, IR))
=
£1[(2, IR)*
be its dual basis. Since
we have
Hence {
IR
k = 0,3
° ki
0,3,
and we see that [cpoCPjCP2l is a generator of H 3 (s[(2,IR)). Next we consider the maximal compact subgroup of 5L(2,IR), which is 50(2), and compute the relative cohomology H* (s[(2, IR), 50(2)) - H* (s[(2, IR), so(2))
with respect to it. We can take X = -Xl Then we have
i(X)cpo
=
0, i(X)CPI
=
+ X2
-1, i(X)CP2
as a basis of so(2). = l.
66
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We can deduce from these facts that any non-trivial element of C*(sf(2,1R),'s0(2)) is a multiple of CPo /\ (CPI + CP2). Hence we have Hk(sf(2 IR) so(2)) = {IR
"
k = 0,2 k # 0,2.
()
2.3. Characteristic classes of flat bundles 2.3.1. Characteristic classes of flat product bundles. Let P ~ M be a flat G-bundle and let wE Al(P; g) be the corresponding flat connection. As we saw in §2.2.1, W induces a homomorphism which commutes with the operation of taking exterior differentiation, namely a d.g.a. map 7f:
w: A*g*
----t
A*(P).
Hence, in view of the motivation for the definition of cohomology of Lie algebras mentioned there, we have a linear map (2.13)
w*: H*(g)
= H*(A*g*;d)
- \ H*(P;IR)
which can be regarded as giving characteristic classes of Hat bundles. However, characteristic classes are generally defined as cohomology classes of base spaces. Nevertheless, in the above we have cohomology classes of the total space rather than those of the base space. If we are given a section s : 1.1 ---> P, then we obtain cohomology classes of the base space by composing the induced homomorphism s* : H* (P; IR) ---> H* (M; IR) with (2.13). On the other hand, it is the same thing to give a section s to a principal G-bllndlc and to give a trivialization P '" 1\11 x G as principal G-bundlcs. This can be seen by the correspondence M x G 3 (p,g) t--+ s(p)g E P. With these facts in mind, we make the following definition. DEFINITION 2.16. Let 7f : P ~ M be a Hat G-bundle. If there is given a trivialization P ~ A1 x G as principal G-bundles, it is called a flat G-product bundle. Let 7ri : Pi ~ Mi (i = 1,2) be two flat G-product bundles. A bundle map
PI
(2.14)
j ----------t
1n2
nIl Ml
P2
------t
f
M2
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
67
as flat G-bundles is called a bundle map as flat G-product bundles if it satisfies the condition .f 0 S1 = S2 0 f. In particular, two flat Gproduct bundles 1ri : Pi -> M (i = 1,2) over the same base space are called isomorphic as flat G-product bundles if there exists a bundle map PI -> P 2 as flat G-product bundles over the identity map of M. With this preparation, we define characteristic classes of flat product bundles as follows. 2.17. Let 1r : P -> M be a flat G-product bundle and P he the corresponding section. Then the homomorphism
DEFINITION
let.') : Jv!
->
'W:
H*(g)
w'"
s*
---t
H*(P;JR.)"':""'" H*(M;JR)
obtained above is called the characteristic homomorphism. For each element a E Hk(g), the cohomology class w(a) E Hk(M; JR) is called the characteristic class of flat G-product bundles corresponding to a. It should be clear from the definition that the following proposition holds. PROPOSITION 2.18. Suppose that the-re is given a bundle map (2.14) as flat G-product bundles between two flat G-product bundles 1ri : Pi -> Mi. (i = 1,2). Then the diagram
H*(g)
w
---->
H*(Nh; JR)
ir
II H*(g)
---->
H*(NhlR)
w
is commutative. In particular, characteristic classes of mutually iso-
morphic flat G-product bundles coincide. The above property is called the naturality of characteristic classes of flat G-product bundles. 2.3.2. Definition of characteristic classes of flat bundles. By improving the consideration in the previous subsection, we define characteristic classes of any general flat G-bundle 1r : P -> M. In this case, a given G-bundle may not be a trivial bundle as a principal Gbundle. Hence we cannot assume the existence of a section s : M -> P, and so it is not possible to obtain cohomology classes of the base space as in Definition 2.17.
68
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We therefore take a maximal compact subgroup K of G and consider the quotient space G / K. Then the projection 'if : P -> M is the composition of two maps as shown by
P ---) P/K ---) M.
(2.15) Clearly the first map
P
(2.16)
-~
P/K
has the structure of a principal J(-bundle. Here we recall an important general fact concerning principal bundles. Let e be the Lie algebra of K. Then a differential form on P is the pullback of a form on P / J( if and only if the following two conditions are satisfied. (i) The interior product with respect to a fundamental vector field on P induced by any element of e is O. (ii) It is invariant under the action of K. By combining this fact with the considerations of §2.2.3, we obtain the following commutative diagram. w
C*(g) = A*g*
A*(P)
--}
r
(2.17)
r
C*(g, K) = A*(G/K)G
-
----->
A*(P/K)
w
Here both C' (g, K) -> C* (g) and A * (P / K) injections. Next we consider the second ma.p
->
A * (P) denote natural
P / J( ---) Jv!
(2.18)
appearing in (2.15). It is easy to see that this has the structure of a fiber bundle with fiber G / K. As is well known, G / K is diffeomorphic to a Euclidean space of suitable dimension, and in particular it is contractible. Hence the map (2.18) becomes a homotopy equivalence, and we obtain an isomorphism H*(P/J(;~) ~ H*(M;~)
(2.19)
in cohomology. From (2.17), (2.19), we obtain a linear map w: H*(g,K) ~ H'(P/K;~) ~ H*(M;JR). DEFINITION
2.19. Let
'if:
P
->
M be a flat G-bundle. The map-
ping w : H* (fj, K) ---) H* (M; 1R)
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
69
defined above is called the characteristic homomorphism. For any a E Hk(g,K), the cohomology class w(a) E Hk(Nf;JR) is called the characteristic class of fiat G-bundles corresponding to a. Similar to the case of fiat G-product bundles, the following result holds which shows that. characteristic classes of fiat G-bundles are natural with respect to bundle maps. PROPOSITION 2.20. Let 7fi : Pi - t Mi (i = 1,2) be two flat Gbundles. Suppose that there is given a bundle map as flat G-bundles as in Definition 2.4. Then the diagram
H*(g,K)
w
~
ir
II H*(g,K)
H*(M1; JR)
--->
w
H*(M2; JR)
is commutative. In particular, chamcteristic classes of mutually iso-
morphic flat G-bundles coincide. 2.3.3. Classifying spaces of flat bundles and characteristic classes. Here we briefiy mention the relation between characteristic classes of fiat bundles, which we considered in the previous two subsections §2.3.1 and §2.3.2, and the classifying spaces of flat bundles. For a Lie group G, the classifying space of principal G-bundles is usually denoted by BG. As is well known, in the case where G is a general linear group such as GL(n, JR) or GL(n, C), its classifying space can be explicitly described by (limits of) Grassmannian manifolds. Then how can we describe classifying spaces of fiat G-bundles? According to Theorem 2.9, to give a fiat G-bundle over a manifold M is the same thing as giving a representation 7f1 (M) - t G. If we express this fact from the topological point of view, we can say that the space BG Ii = K(G,l) serves as the classifying space of flat G-bundles. Here G li denotes the group G equipped with the discrete topology. Hence characteristic classes of flat G-bundles given in Definition 2.19 induce a homomorphism w: H*(g,K) ----> H* (BG Ii ; JR) = H*(GIi;JR). Next we consider flat G-product bundles. Recall that a flat Gproduct bundle is a principal G-bundle equipped with two structures:
70
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
one is a flat connection, and the other is a trivialization as a principal G-bundle. Hence the classifying space of flat G-product bundles is played by the homotopy fiber (see Example 1.5(iii), §1.1.1) of the natural map BG6 ~ BG. This space is usually denoted by BG, and we obtain a fibration BG
----T
BGo
----T
BG.
lf G is connected, G becomes a topological group explicitly described as G
=
{(g,e) E GO x JVIap(J,G);e(O)
= g,e(l) = e}
C G6
x Map(I,G).
In this case, Definition 2.17 gives rise to a homomorphism w: H*(g)
----T
H*(BG;IR).
Summarizing the above, we can say that characteristic classes of various G-bundles are given by the following commutative diagram. H*(g)
w
--------->
H*(BG;IR)
--------->
H*(BG 6 ;1R)
r
T (2.20)
H*(g,K)
w
r
T J*(K)
--------->
H*(BK;IR) ~ H*(BG;IR)
w
Here in the bottom row, we have used the well-known facts that G is diffeomorphic to the product of its maximal compact subgroup K and some Euclidean space and also its corollary that BG is homotopy equivalent to BK.
2.3.4. Chern-Simons forms and Chern-Simons invariants. In this subsection, we mention very briefly the Chern-Simons theory, which can be considered as a certain refinement of the Chern-Weil theory. For details, we refer the reader to the original paper [ChS] of Chern and Simons and also a closely related paper [CS] of Cheeger and Simons. Let f E Jk(G) be an invariant polynomial of a Lie group G. As was already mentioned in §2.1.1, to any principal G-bundle 7r : P ~ !VI with a connection w E A 1 (P; g), there corresponds a certain closed form f(D,k) E A 2k (p). Moreover, this form is the pullback of a closed form f(nk) E A2k(M) of the base space by 7r, and the de Rham cohomology class [f(D,k)] E H2k(M; IR) represents the characteristic
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
71
class corresponding to f. Now as will be easily checked, the form f(0, k ) E A 2k (p) itself is an exact form because of the following reason. By the naturality of characteristic classes with respect to bundle maps, the above form represents the characteristic class of the pullback bundle on the total space by the projection 7f. On the other hand, this bundle is clearly trivial. In the above cited paper, Chern and Simons constructed an explicit (2k -I)-form Tf(w) E A 2k - 1 (p) such that
This differential form is called the Chern-Simons form. In the context of the Weil algebra W(g), the Chern-Simons form is nothing but an element T f E W 2k - 1 (g) such that dT f = f E Ik(G) C W2k(g). From the triviality of the cohomology of W(g), we can conclude that such a form certainly exists and the difference of any two such forms is an exact form. It can be said that Chern and Simons specified one natural element among them. Here we briefly mention only the result. Let VI, ... ,Vm be a basis of 9 and let wI, ... ,w ffi be the dual basis of g*. Also let 0,1, ... ,0,7n be the corresponding basis of Slg*. Then the two elements Tn
w
=
L wi ® V., E g* ® 9 c W(g) ® 9 ;.=1 m
D =
L ni ®
Vi
E Blg* ® 9
c
W(g) ® 9
i=l
are the universal connection form and the universal curvature form, respectively, which are defined at the level of the Weil algebra. We can define natural maps 1\ :
(W(g) ® g01.:) ® (W(g) ® gOt')
[ , 1: (W(g) ® g) ® (W(g) ® g)
----+
----+
W(!J) ® g0(k+£)
W(g) ® g,
and the following two important equations: 1 dw = -2'[w,w] d0, =
[0"w]
+ 0,
(structure equation) (Bianchi identity)
hold in W(g) ® g. Now the Chern-Simons form Tf E W 2k - 1 (g) corresponding to an invariant polynomial f E Ik (G) c Hom(g0 k , lR)
72
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
is given by k-l
Tf =
L
Ai f(w /\ [w, w]i /\ nk -
i - 1)
i=O
where A _ ()i k! (k - I)! i -1 2i(k + i)!(k -1 - i)'· If T f (w) becomes a closed form for some reason, we can define the integral
1
Tf(w)
associated to any (2k - I)-dimensional cycle c in P. The invariants obtained in this way are called Chern-Simons invariants. For example, let M be a closed oriented 3-dimensional Riemannian manifold. As is well known, the tangent bundle T j\lJ is trivial so that the orthonormal frame bundle P(M) has a section s. Hence, by considering the Levi-Civita connection on P(l11) and the first Pontrjagin class, a Chern-Simons invariant is defined by CS(M)
=
r
./s(M)
~ TPI
E
IR/Z .
2
Here we set the value of the integral to be in IR/Z because if we change the section, the value changes by a certain integer. (However, later a close relationship with the 1J-invariant of Atiyah-Patodi-Singer [APS] was found and this indeterminacy disappears now.) As another example, let !vJ be a closed oriented 3-dimensional manifold and let AM denote the set of all connections on the trivial SU (2)-bundle over M. Then the Chern-Simons invariant corresponding to the second Chern class C2 induces a function CS:AM
-~R
After the celebrated work of Witten [W], this function has been playing a fundamental role in the recent development of low dimensional topology using the gauge theory (cf. e.g. [Koh]). Now if w is a flat connection, the Chern-Simons form Tf(w) becomes a closed form. Hence we can consider its de Rham cohomology class [Tf(w)] E H 2k - 1 (P; IR). This is called the Chern-Simons class. Also for a flat G-product bundle, if we pull back the ChernSimons class by the section s : !vI ---t P, we obtain a cohomology class
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
73
of the base space. More explicitly in the context of the classifying space, there exists a natural closed form Tf E A2k - 1 fl* such that
H 2k - 1 (fl)
[Tf]
3
~
[Tf]
E H2k-l(BG;~).
Here l' f is nothing but the projection of l' f, which was defined as an element of the Weil algebra, to A" fl". Furthermore, if the characteristic class [f] E H2k(BG;~) corresponding to f is the image of an integral class, then the Chern-Simons class is defined as a cohomology class
[Tf] E H 2k - 1 (BGo; ~/Z) with coefficients in
~/Z.
For example, we have a cohomology class
TCk E H 2k - 1 (BGL('I7" C)0; C/Z) which is defined for flat complex vector bundles. EXAMPLE 2.21. As an example, let us consider the case G = G L( 1, C) = C*. Although this case is almost trivial, it is suggestive of higher dimensional generalizations. In this case, it is easy to see that BG = K(Z,2),BG o = K(C*,l) and BG = K(C,l). Also we have H* (BG; Z) = Z[Cl]' Then it can be shown that the classes TCI E Hl(BGo;C/Z) ~ Homz(C*,C/Z) and TCl E Hl(BG;C) ~ Homz(C, C) are given by the following commutative diagram:
C (2.21)
cxp
X ..!~; ---->
1
C*
C
1
modZ
-----> 2~{log
C/Z
2.3.5. Nontriviality of characteristic classes of flat bundles. We consider the nontriviality of characteristic classes of fiat bundles (see Definition 2.19). Let G be a connected semisimple Lie group. Then, based on a deep theory due to Borel and HarishChandra, Borel [Bor] proved that there exists a discrete subgroup reG such that r\G becomes compact. By a well-known lemma of Selberg, we may assllme that r is torsion free. Hence, jf we denote by K a maximal compact subgroup of G, M = r\ G I K = K(r,l) becomes an oriented closed manifold. Then we define a fiat G-bundle over M by 11" : r\ « G I K) x G) -} M, where the action of on (G I K) x G is given by ,([g], h) = ([,g], ,h) h E G, [g] E GIK,h E G). Then the associated flat GIK-bundle r\(GIK x G I K) --> r\ G / K has a section induced by the natural diagonal
r
74
2.
CHARACTERISTIC CLASSES OF FLAT BUNDLES
embedding G / KeG / K x G / K. Clearly this section is a homotopy equivalence. It follows that the characteristic homomorphism H* (g, K) - H* (M; lR) of the bundle 7r is induced by the natural map A*(G/Kf ------> A'(M).
In the top degree which is equal to the dimension of M, this map is an isomorphism because both are generated by volume form::;. On the other hand, it is known by Koszul [Kos] that the cohomology H*(g, K) satisfies the Poincare duality theorem. Sillce the above map is a homomorphism as algebras which preserves products, we can deduce that the induced homomorphism in cohomology is injective. Thus we obtain the following theorem. THEOREM 2.22 (Borel, Harish-Chandra, Selberg, Koszul). Let G be a connected semisimple Lie group and let K be a ma:t"imal compact subgroup of it. Then the homomorphism
H*(g,K) _ H*(BG 6 ;lR) is injective. The following example is very important because it is related to various concepts such as characteristic classes of flat bundles, ChernSimons invariants, 1]-invariant, characterisitc classes of foliations (see §3.4) and geometry of negatively curved manifolds. EXAMPLE 2.23. Consider the Lie group PSL(2,C) which is defined as SL(2, q/{±l}. As is well known, it serves a..
PSL(2,q
------>
PSL(2,C)/SO(3)
is identified with the orthonormal frame bundle P(lHf3) of JH[3. Now the Lie algebra of PSL(2, q is s[(2, C), and explicit computation shows that H*(s[(2,q) ~ H*(S3 x S3;lR),
H*(sl(2,Q,SO(3)) ~ H*(S3;lR).
We can check that H3(S((2,Q,SO(3)) ~ lR is generated by the volume form v of JH[3. Furthermore, under the natural isomorphism
2.4. GEL'FAND-FUKS COHOMOLOGY
75
the real part is generated by the Chern-Simons form TPI on P(JH[3) corre::;ponding to the first Pontrjagin class, while the imaginary part is generated by v (d. [Y] for example). Now let r c PSL(2, C) be a torsion free discrete subgroup such that the quotient space r\p S L(2, C) is compact. It was proved by Thurston, in his theory of hyperbolic 3-manifold::;, that there are plenty of such groups. In fact, in the above ::;ituation M: r\PSL(2,C)/SO(3) becomes what is called a compact hyperbolic 3-manifold, and r\PSL(2, q can be identified with the orthonormal frame bundle P(Iv!) of Iv!. Since it is known that Iv! is parallclizable, P(M) is diffeomorphic to M x S3. Thus we obtain a homomorphism f/*(sl(2,C)) - .., H*(P(M);JR.) ~ H*(M x S3;JR.) which turns ont to be injective if the Chern-Simons invariant of M is nontrivial. Moreover, if we denote by PSO(2) C PSL(2, q the set of all diagonal matrices, then the corresponding quotient P(M)/ PSO(2) -> 111 becomes a Cpl-bundle over M where Cpl = SO(3)/ PSO(2). It can be shown that it has the structure of a foliated Cpl.-bundle, which will be defined in §3.1.1, and its structure group i::; P SL(2, C) acting holomorphically on Cpl. We refer the reader to [Mor2] fo), details. 2.4. Gel'fand-Fuks cohomology 2.4.1. Characteristic classes of flat bundles, continued. We recol1::;ider the construction of characteristic classes of :flat bundles given in §2.3.1 from a more geometrical point of view. Let 1r : P -> IvI be a flat G-prodllct bundle; namely there is given a trivialization P ~ M x G together with the corresponding section s : IvI -> P and a flat connection w. The composition
* A* 9 * ----; W A*(P) ----) .,. A*(M) sow: of two d.g.a. maps w, s* induces a linear map s* ow* :
H*(g) --> H*(M;JR.),
and this gives rise to the characteristic classes of flat G-product bundles. We examine the above map in detail. Any element tp E Ck(g) ~ Akg*; in other words any alternating multilinear map tp:gx .. ·xg-->JR. '-v----' k-tirnes
76
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
determines a certain element s*w(cp)
E
Ak(M). More ex.plicitly, if
Xl"",XkETpM
(pEM),
then we have
s'w(cp)(X1 ,··· , Xk) =w(cp)(S.Xl,··· , s.Xk )
(2.22)
=cp(w(s*Xd,'" ,w(s*Xk ))·
On the other hand, let Xi E T(p,e) (M x G) (e E G denotes the identity element) be the lift of Xi which is horizontal with respect to wand define E TeG by the equality
xl
-
Xi
=
-j
S.Xi + Xi .
xl
Then we see that is nothing but the projection of Xi to the fiber direction with respect to the product structure P = M x G (cf. Figure 2.6). (pIx G
MXG
M
)
p
FIGURE
Xi
2.6
Then, since _
-
-I _
-I
W(S.Xi) - W(Xi - Xi) - -w(Xi ), by substituting it in (2.22), we obtain (2.23)
xf
can be done at any point (p, g) on the Now the construction of fiber {p} x Gover p E M. Hence, by varying 9 in G, we can regard
2.4. GEL'FAND-FUKS COHOMOLOGY
xl
as a vector field on {p} X G invariant. Hence we can write
= G.
77
Clearly this vector field is left
xl E TeG = :£(G)G. With these facts in mind, let us denote (2.23) by ~(XI'''' ,Xk)'
It may appear that this makes the matter unnecessarily complicated. However, it will play an essential role in the next subsection where we shall consider general flat product bundles with infinite dimensional structure groups. Thus we have defi.ned a correspondence
Ck(g)
3 cp
f-.
~ E
Ak(M).
It can be easily checked by tracing the above argument that this correspondence is defined even if w is not flat. An important point here is the fact that if w is flat, then the equality
holds. Let us recall one more important fact that, as was mfmtioned in (2.11) of §2.2.1, the exterior derivative dcp E Ck+l(g) is defined purely algebraically by the equality dcp(Xl"" ,Xk+l) = ~(-l)i+j,n([X ~ ..,... 1.,
X·]J' Xl , ... ,1.., j(. ...
x· ...
'J"
X k +l ).
2.4.2. Flat bundles whose fibers are general manifolds. All the flat bundles which we considered up to the previous subsection have (fi.nite dimensional) Lie groups as their structure groups. We shall try to extend those considerations to the case of fiber bundles whose structure groups are infinite dimensional such as the diffeomorphism groups of manifolds. For any Coo manifold F, let Diff F denote the diffeomorphism group of F. We can regard Diff F as the structure group of general differentiable fiber bundle 1f : E - 7 M whose fiber is F. Henceforth we call such bundle simply an F-bundle. let
DEFINITION 2.24. Let /VI be an n-dimensional Coo manifold and E - 7 1M be an F-bundle. An n-dimensional distribution 'H =
1f :
78
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
{Hu;u E E} on E which is transverse to any fiber, or equivalently which satisfies the condition
Eu = Vu u H u
(11,
E
E),
is called a connection. Here Vu C TILE denotes the subspace consisting of all tangent vectors at u which are tangent to the fiber. If F is a closed manifold, for any smooth curve £ : [a, b] can define the parallel translation
--->
JI/l we
he : Ee(n) ~ Ee(b)
along £ in the same way as the case of principal bundles. It gives a diffeomorphism from the fiber Ee(a) over £(a) to the fiber Ee(l;) over £(b). However, if F is not compact, it is not always possible to define parallel translations. Then we define as follows. DEFINITION 2.25. A connection on a fiber bundle is called a strict connection if parallel translation is always defined along any smooth curve on the base space.
If there is given a connection on a principal bundle, then we can "differentiate" it to obtain the curvature. Then the condition that the curvature is identically zero is equivalent to the one that the distribution induced by the connection is completely integrable. With this fact in mind, we make the following definition. DEFINITION 2.26. A connection on a fiber bundle is called a Hat connection if it is complE:tcly integrable, An F-bnndle equipped with a flat connection is called a flat F-bundle, If in addition the connection is a strict connection, it is called a strictly flat F -bundle.
As was already remarked, in case F is a closed manifold, any flat F-bundle is always a strict fiat F-bundle. DEFINITION 2.27. Two flat F-bundles 1fi : Ei ---> M (i = 1,2) over the same base space are called isomorphic as flat F-bulldlcs if there exists an isomorphism El ~ E2 as F-bundles such that it sends the connection on El to that on E 2 .
Now let 1f : E --7111 be a strict flat F-bundle. Then, by applying the same argument as that of §2.1.4, we obtain a homomorphism
p: 1fl(M)
~
DiffF
2.4. GEL'FAND-FUKS COHOMOLOGY
79
such that it induces an isomorphism
E
~
M x F/IT1(M)
as flat F-bundles. This homomorphism is called the holonomy homomorphism or equivalently the monodromy homomorphism depending on the contexts. Here IT1 (M) acts on M as the universal covering transformation group and also on F through the homomorphism p. The following theorem can be proved in the same way as Theorem 2.9. THEOREM 2.28. Let M and F be Coo manifolds. Then the correspondence, 'Which sends any .str-ict fiat F -b'undle over !VI to its holonomy homornorph'lsm, induces a bijection
{isomorphism class of stTict fiat F -bundle over M} ~
{conj1Lgacy class of homomorphism ITdM)
---+
DiffF}.
2.4.3. Definition ofthe Gel'fand-Fuks cohomology. In this subsection, we define the Gel'fancl-F'uks cohomology group HCF(F) for any Coo manifold F. As is well known, the set l:(F) of all Coo vector fields on F has a structure of a Lie algebra with respect to the bracket. Hence its cohomology group as a Lie algebra is defined (cf, §2.2). However, it is too huge an object for the computation of cohomology group, and also it is difficult to give geometrical meaning to it. On the other hanel, it turns out that l:(F) has a natural topology, called the Coo topology, and it becomes a topological Lie algebra, Therefore we may cOIlHider only continuous cochains of l:(F) with ret:>pect to this topology and then take the cohomology of the subcomplex of C*(l:(F)) cOIlsisting of continuous co chains , We call this cohomology the continuous cohomology of l:(F) and denote it by H:(l:(F)). The Gel'fand-Fllks cohomology HCF(F) of F is nothing but this cohomology group. Before Htating the precise definition, we continue to consider flat bundles. DEFINITION 2.29. Let is given an isomorphism E F-product bundle.
IT :
~
E ---+ M be a flat F-bundle. If there !VI x F as F-bundles, we call it a flat
In other words, a flat F-product bundle is a trivial F-bundle M x F together with a completely integrable distribution on it whose
80
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
dimension is the same as that of M and which is transverse to the fibers. Then for each point p E ]\,1/ on the base space, we can define a linear map TpM 3 X f-----> Xf E X(F) as follows. For each point (p, u) E M x F on the fiber over p, we consider the lift X E T(p,u) of X which is horizontal with respect to the given connection and then set Xf (71.) E T"F to be its orthogonal projection to F (d. Figure 2.7). (p)XF
MxF
M
)
p
FIGURE
x 2.7
If we move the point u in the above construction, then we obtain a vector field on F, and this is the definition of Xf. Now suppose that there is given an element of Ck(X(F)), namely an alternating multilinear map (2.24)
r; : X(F) x ... x X(F) ---; R ,
v'---""'-'
k-times
Then, if we set
(2.25) for Xl,'" , X k E TpM, we obtain an alternating multilinear map ij : TpM x ... x TpM ---; R
,
" k-times
,
If ij defined in this way is of class Coo with respect to p, then we obtain a k-form ij E Ak(M) on M. This condition will be satisfied if r; is continuous with respect to the topology of X(F).
2.4. GEL'FAND-FUKS COHOMOLOGY
81
With these considerations in mind, we now define the Gel'fandFuks cohomology. First we define the C= topology of X(F). Let X E X(F) be a vector field on F. For any coordinate neighborhood U of F and its local coordinates Xl, ... ,X n , let n
X
=
a ax,
LJi(X)-. i=1
be the local expression of X on U. Let K be a compact subset of U, r a non-negative integer and E > 0 a positive number. For any vector field Y E X(F), we consider its local expression n
Y = Lgi(X)
a ax
i=1
'
on U and define
N(X;U,K,T,E) to be the set of all Y such that the inequality
alalJi I IaX~'alalgi ... ax;;:n (x) - aX~l ... ax~n (x) < E holds any point x E K for all
lal = a1 + ... + an
~ T.
DEFINITION 2.30. Let F be a Coo manifold. We define a topology on X( F) by requiring that the set of all N (X; U, K, r, E) as above with arbitrary U, K, I, E forms a subbase for open sets. It is called the C= topology.
It is easy to see that the bracket operation is continuous with respect to this topology. Therefore if we denote by A~(X(F)) the set of all alternating multilinear maps
X(F) x ... x X(F) -----.., '-
~
,/
k-times
which are continuous with respect to the Coo topology and set A~(X(F))
= EBkA~(X(F)),
then it becomes a sub complex of C*(X(F)). Now we make the following definition. DEFINITION 2.31. Let F be a Coo manifold and let X(F) be the Lie algebra of vector fields equipped with the Coo topology. Its continuous cohomology H* (A~ (X(F)) is called the Gel'fand-Fuks cohomology of F, denoted by HCF(F).
82
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
2.4.4. Characteristic classes of fiat F-product bundles. We can define bundle maps for flat F-product bundles similarly as in §2.3.1. PROPOSITION 2.32. Let 7r : E -> !vI be a flat F -p-rvduct bundle. Then for any", E A~(:r(F)) the element ij defined by (2.25) is of class COO . Hence it is an element of A k (Af). FurtheTmoTe we have
dTJ
=
dr,.
As a corollary to this proposition, we obtain the following theorem. THEOREM 2.33. Let 7r : E the cOTTespondence
->
A~(X(F)) 3
TJ
M be a fiat F -product bundle. Then ~ ij E
A'(M)
'is a d.g,a. map. Hence it induces a homomorphism
HCF(F)
---'>
H' (M; 1R).
This homomorphism is natural with respect to bundle maps. Hence any element of the Gelfand-Fub cohomology [JTOUp HCF(F) serves as a chamcteristic class of flat F -product b7mdlc8. If we generalize the considerations in §2.3.3 to the present situation, we can see that the homotopy fiber of the natural map BDiffli F -> BDiff F plays the role of the clatitiifying tipace of flat F-prodllct bllndIes. If we denote this space by BDiff F, then we obtain a fibration
BDiff F - • BDiff'" F -) BDifi F. Then the above theorem claims that there is a llOIllomorphisIll HCF(F) - , H*(BDiffF).
Proof of Proposition 2.32. We first show that r, becomes a Coo form on M. It is clear from the definition that r, is linear with reBpect to functions on M; namely for any f E coo(.M) and Xl,' ,. ) Xx: E X(M), we have 1)(1Xl,'"
,fX k )
= f7i(X 1,'"
,Xk ).
Hence it is enough to prove that ij is of cla..'ls Coo, Let X be a vector field defined on a neighborhood U of p E M. Then for each point q E U a vector field
2.4. GEL'FAND-FUKS COHOMOLOGY
83
on F is determined which clearly varies in a Coo fashion if we move q. It follows from this fact together with the definition of ij that we have only to show the following. Namely for any Coo family, with respect to the parameter t, of k vector fields Xi(t) E X(F) (i = 1" .. ,k), the function
(2.26) becomes a Coo function of t. This condition is guaranteed by the cont.inuity with respect to the Coo topology of vector fields which is contained in the very definition of the Gel'fand-Ful<s cohomology. Let us check this point hriefly. For any element 1] E A~(X(F)), the contiulli ty iIll plieH that
HellCC we call conclude first that (2.26) is of class Co. Next from the equality
I. T/(Xl(t),"', Xk(t)) - 1](X1 (O),··· ,XdO)) t-U t
In1~~~~~~~~--~~~~~~~~
k
= L 1](X1 (O),···
,X;(O),··· ,XJ.:{O)),
i=1
we s(~e that (2.26) is of dass C l . If we continue with a similar argument, we Hee eventually that (2.26) is of class Coo. Next we prove the equality (2.27)
d1]
• dij.
For each vector field X E- X(J\!I) on the base space, we denote by }i E X(E) the vector field 011 the total space E = 111 x F which is obtained hy cOllsidering horizontal lift at each point on it. Then, by virtue of the flatness of the connection together with the theorem of :Frobenimi, we have
(2.28)
[X,Y]
= [X,Y]
for any X, Y E X(M). Now to prove (2.27), it suffices to show that the equality
(2.29) holds for any Xl,'" ,Xk+l E X(J\!I). Since it is enough to show this locally, we consider the arbitrary local coordinate system (U; Yl,'" ,Ym)
84
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We know already the linearity with respect to functions. Hence we have only to prove (2.27) assuming that
Xi = -
[}
(i = 1,,,, ,k + 1).
[}Yi
Now set
Xi
+ ~i
= Xi
(here we write ~i instead of X/ to simplify the notation). ~i is a vertical vector field on the product manifold E = jVf x F so that for each point p E NI, we may regard ~i(P) E X(F). Since we have clearly [Xi, X j ] = 0, it follows from (2.28) that [Xi,Xj]
= O.
On the other hand, we have
Hence we obtain [~i,~j]
(2.30)
[}
=
a~j Yi
[}
- 8~i' Yj
If we write A for the left hand side of (2.29), then we have A =dT](X1 ,'" ,Xk+d
(2.31 )
=dr) (6 , ... '~k+l)
=L(_l)'i+jT]([~i'~JJ,6,'"
,t;.,"· ,t
j , .. ·
,~k+l)'
i<J
On the other hand, if we write B for the right hand side of (2.29), we can use the fact [Xi, X J ] = 0 to see that B =di/(X 1 ,'" ,Xk+d
(2.32)
_- 'L....-( " -1) i+l XiT](X ' 1 , " ' , Xi,"
. ,Xk + 1 )
Both A and B are functions on U, and we have to show A = B. Let U be any point and for simplicity assume that the local coordinate
p E
2.4. GEL'FAND-FUKS COHOMOLOGY
of that point corresponds to the origin Yi in (2.31), we obtain
= O.
85
If we substitute (2.30)
(2.33) A(P)
=L7)(~~'(O)- ~~j(0),6(0), ... i<j
'€i(O),.·· ,tj(O),· ..
y,
YJ
'~k+l(O»).
On the other hand, it follows from (2.32) that B(p)
= L(-1)Hl~7)(6,··· ,€i,···
,.
(2.34) =
0Yi
'~k+dl Yi=O
L( _l)i+l L 7)(6(0), ... , ~~J (0),··· '~k+l(O»). i
j"ii
Y.
If we compare A(p) with B(p) carefully, we see that they coincide.
This completes the proof.
0
CHAPTER 3
Characteristic Classes of Foliations In this chapter we treat the theory of characteristic classes of foliations. Foliations are certain patterns on manifolds. Locally they look thc same everywhere. However, globally we can draw various patterns. Characteristic classes of foliations represent their global behaviour in terms of cohomology. This theory was established during a short period from the end of the 1960's to the beginning of the 1970's. Here we first consider the Godbillon-Vey class, which is the representative among characteristic classes of foliations. We give the definition for the case of codimension 1 and describe Thurston's examples showing continuous variation of the class. VYe then give a survey of the general theory of characteristic classes of foliations from the viewpoints of the Gel'fand-Fuks theory mentioned in Chapter 2. For details, we refer the reader to [Bot2] [BH] [HI [KT]. In §3.5 we formulate a problem which arises inevitably from a significant feature of characteristic cla~ses of foliations, that is the continuous variation. 3.1. Foliations 3.1.1. Definition of foliations. Roughly speaking, a foliation on a Coo manifold means that we cover the whole manifold evenly by a. family of submanifolds whose dimensions are the same. Here by evenly we mean that locally the situation is the same as the decomposition lR n =
U lR
n- q
x {x}
x jRn = jRn-q X jRq such that for any a E A, the image under 'P of any connected component of Un La can be written as {x = (Xl,'" , Xn) E jRn; X n - q +l = Cn-q+l,'" ,Xn = Cn} (see Figure 3.1). Here each Cj is a constant. Each La is called a leaf of F.
FIGURE
3.1
EXAMPLE 3.2. Any n-dimensional Coo manifold M admits two trivial foliations. One is the co dimension n foliation on l\!J whose leaves are points on M, and the other is the codimension 0 foliation with a single leaf which is M itself. In general, any Coo fiber bundle 'Jr : E -> l\!J determines a foliation on the total space E whose leaves are fibers of 'Jr. The co dimension of this foliation is equal to the dimension of the base space M. EXAMPLE 3.3. As a simple example of foliations, we define linear foliations on a 2-dimensional torus T2 = jR2 I'll}. Fix a constant
3.1. FOLIATIONS
89
C E JR U {oo}. Then the set of all parallel translations of the line through the origin in JR2 whose slope is c defines a foliation on JR2 of co dimension one. Since this foliation is clearly invariant under parallel translations, it induces a foliation Fe on the quotient space T2. If cEQ U {oo}, all leaves of Fe are diffeomorphic to 8 1 . On the other hand, if c 1. Q U {oo}, then all leaves of Fe are diffeomorphic to JR and moreover they are dense in T2. Thus, although all Fe look the same locally, they have distinct global features according to the value of c. EXAMPLE 3.4. A codimension one foliation on JR x I is depicted in Figure 3.2 (i). If we rotate this foliated manifold around the axis JR x H}, then we obtain a foliation on JR x D2 as shown in (ii). This foliation is invariant under the translation by 1 in the direction of JR. Hence if we take the quotient by this, we obtain a codimension one foliation on 8 1 x D2 as shown in (iii). This is called the Reeb component. If we glue two Reeb components according to the wellknown decomposition 8 3 = Sl X D2 U D2 X S1 of the 3-sphere, we obtain a codimension one foliation on S:3. This is called the Reeb foliation. EXAMPLE 3.5. Flat bundles, which were defined in §2.1.2, serve as important examples of foliations. Let G be a Lie group and let 7r : P ---+ M be a flat G-bundle. Then the set F of all maximal integral manifolds in P defines a foliation whose codimension is equal to the dimension of G. Similarly, the total space of any flat F-bundle 7r: E ---+ B, defined in §2.4, admits a foliation whose leaves are maximal integral manifolds. Therefore such bundle is sometimes called a foliated F-bundle.
Let F be a foliation on M of codimension q. Then the set
r (F)
=
{X
E
T M; X is tangent to a leaf}
becomes a subbundle of the tangent bundle TNI of M. This is called the tangent bundle of:F. The quotient bundle
v(F) = TM/r(F) is called the normal bundle of:F. Clearly r(F) is an involutive subbundle of T M. Namely, if we denote by r(r(F)) the set of sections of r(F), we have
90
3. CHARACTERISTIC CLASSES OF FOLIATIONS
ii)CCCCCC (ii)
(iii)
FIGUR.E
3.2. Reeb component
Conversely, by the theorem of F'robenius (Theorem 2.6 ill !i2.1.3), any involutive subbulldle T of T!vI becomes the tangent bundle of a foliation. This is because snch subbundle is c:ompletcly integrable and the set of all maximal integral manifolds de tines a foliatioll, as is easily seen. Next we describe foliations in terms of differential forms. Let 7 C T ll,iI be a snbbundle and set v ~ T IYI/ 7. If we denote by v* the dual bundle of v, it becomes a subbundle of the cotangent bundle T* M of M. Explicitly, we can write
T*M:J v* = {w E T*MiW(X) = 0 for any X E T}. Then, as was already described in §2.1.3, we have another formulation of the F'robenius theorem as follows. Namely, T is completely integrable if and only if the ideal [(7) of the de Rham complex A*(!vI) of !vI generated by sections of r(v*) of v* is a differential ideal. This
3.2. THE GODBILLON-VEY CLASS
91
condition can be described locally as follows. Let U be a coordinate neighborhood of Nf and let wI, ... ,w q be elements of r( v*) which are linearly independent at any point of U. Then there exists a system of I-forms w} such that q
(3.1)
dJ..ui =
Lwj /\w j . j=l
Thi::; condition is called the integrability condition. Let F be a foliation on M of co dimension q. A Coo map f N --> M is called transverse to F if the composition 7r 0 f* : TpN - 7 Tf(p)M --> Vf(p)(F) is ~urjective at any point pEN. In this case, if we set f,
= {X
E TN;
f*(X)
E
T(F)},
it becomes a subbundle of TN of codimension q, and it is easy to see that it satisfies the integrability condition above. The corresponding foliation is denoted /*(F), and we call it the foliation induced from F by f. The leaves of /*(F) take the form of connected components of the inverse images of those of F by f. In particular, any submersion f : N - 7 M determines a foliation on N whose leaves are connected components of f-l(p) (p E A1). In the above, we described a minimum of materials which will be nece~sary henceforth in this book. For more details of foliations, see e.g. [L][T][To]. 3.2. The Godbillon-Vey class 3.2.1. Definition of the Godbillon-Vey class. If there is given a foliation F on a Coo manifold M, we simply write (M, F). DEFINITION 3.6. Suppose that for any given foliation (Nf, F) of codimen::;ion q, there is defined a cohomology class
Q(F) E H*(M; JR) which satisfies the following naturality condition. Namely, for any Coo map f : N --> 111 which is transverse to F, the equality
Q(f*(F)) holds. Then we say that of codimension q.
Q
=
J*(Q(F)) E H*(N; JR)
is a characteristic class of foliations
92
3. CHARACTERISTIC CLASSES OF FOLIATIONS
We can also consider coefficients other than lR. for characteristic classes of foliations. However, the case of real coefficients is the most important. The first example of characteristic classes of foliations is the one given in [GV], and it is now called the Godbillon-Vey class. Let F be a codimension 1 foliation on M and assume that the normal bundle I/(F) is trivial for simplicity. Then there exists a I-form W E r(//* (F)) such that it does not vanish at any point on M. It follows from the integrability condition (3.1) that there exists a I-form Tj such that
dw = Tj /\ w.
(3.2)
LEMMA 3.7. The 3-form H = Tj /\ dTj on M is closed, and its de Rham cohomology class [H] E H 3 (M;lR) does not depend on the choice of wand Tj. PROOF. We first show that 0 is closed. If we differentiate (3.2) and use the fact that." /\ dw = 0, we obtain dTj /\ w = O. Hence there exists a certain I-form ~ such that d." = ~ /\ w. Then we have
dO
= d." /\ d." = 0
so that 0 is closed. Next if we have dw = Tj' /\ w for another I-form .,,', then (Tj' - .,,) /\ w = 0 so that there exists a function f such that Tj' -." = fw. Hence we can concl ude
H' = .,,' 1\ d.,,' = (." + f w) 1\ (dTj + df /\ w = ." /\ drl + ." 1\ df /\ w = 0 - d(fdw).
+ f dw)
Thus we obtain [0'] = [0). If w' is another I-form which defines F, then there exists a nowhere vanishing function 9 sHch that w' = gw. Then dw' = dg 1\ w
+ gdw
) , ( dg =-+.,,/\w.
9
Hence we can set Tj'
= ~ + Tj,
0' = .,,'
1\
and we have
ciTj' =
(d g +.,,) g
1\
dTj
= 0 _ d (d g 1\.,,). 9
3.2. THE GODBlLLON- VEY CLASS
Thus we obtain [0'] DEFINITION
=
93
[0], completing the proof.
D
3.8. We denote by gv(F) the cohomology class [0]
E
H 3 (M;IR) obtained above and call it the Godbillon-Vey class. In the above definition, if we change W to -W, the form 0 remains unchanged. Hence we can conclude that the Godbillon-Vey class is defined even if the normal bundle v(F) is non-trivia.!. Let f : N --> M be a map transverse to F. If w is a 1-form which defines F, then clearly we can take j*w for a defining I-form of j*(F). It follows that the Godbillon-Vey clas::; is a characteristic class of co dimension 1 foliations. EXAMPLE
3.9 (Roussarie). Let M be the 3-dimensional Lie group
PSL(2,IR) =
{(~ ~)
;a,b,c,d E' IR,ad- bc= 1}/{±1}.
We take a basis of the corresponding Lie algebra 5((2, IR) to be
Xo =
G~1)'
Xl =
(~ ~),
X2 =
(~ ~).
Then we have
Hence, if we consider Xi as left invariant vector fields on M, the 2-clirnellsional subbulldlc T C TM spanned by Xo and Xl becomes completely integrable. Therefore it defines a foliation F on M of codirnension 1. To compute the Godbillon-Vey class of F, let Wi be the dual basis of Xi and consider them as left invariant I-forms on lvI. We can take W2 as a 1-form which defines F, and we have dw 2 = -2woAw2. From this we obtain n = -2wo A d( -2wo) = 4wo A WI A W2 which is nothing bllt a volume form of lvI. Unfortunately, lvI is diffeomorphic to Sl x ]R2 so that H3(M; IR) = 0 and hence the Godbillon-Vey class must be zero. However, as will be mentioned below, it is known that there exist plenty of torsion-free discrete subgroups r c lvI such that the qnotients r\M become 3-dimensional closed manifolds. Since F is clearly left invariant, we obtain a codimension 1 foliation r\F on r\M. Then r\O is a volume form of a closed manifold so that its d(~ Rham cohomology class is non-trivial. Hence we can conclude that gv(r\F) i 0 E H 3 (r\M; IR) ~ R Thus we obtain a foliation whose Godbillon-Vey class is non-trivial.
94
3. CHARACTERISTIC CLASSES OF FOLIATIONS
We add a few more facts concerning the above example. The group PSL(2,1R) acts on the upper half plane lHI = {z
= x + iy E C; Y > O}
by orientation preserving i!:iometries with respect to the Poincare metric. On the other hand, if I: denotes any closed surface of genus 9 2:: 2 equipped with a metric of constant negative cllrvature, then its universa.l covering manifold is isometric to 1HI. Hence a homomorphi!:im p: 1fl (I:) --> PSL(2, IR) is defined up to conjugacy. We can take !:iuch 1m p corresponding to any I: as which we used in the above exa.mple. In this case, r\M can be identified with the unit tangent bundle 11 I: = {X E TI:; II X II = I} of I:, and the foliation r\F 011 it is called the Anosov foliation, which is an important subject of the theory of dynamical systems.
r
3.2.2. Continuous variation of the Godbillon-Vey class. If there is given a codimension 1 foliation F on a dosed oriented 3dimensional manifold M, we can evaluate its Godbillon-Vey class on the fundamental cycle of M to obtain a number gv(F)[M]. This is called the Godbillon-Vey number or Godbillon-Vey invariant. In [Thl], Thurston proved the following theorem by analy;-;ing the above example 3.9 ill detail. THEOREM 3.10 (Thurston [Thl]). There exists a family F t (t E IR) of codirnen8'ion 1 foliations on S3 s·u.eh that the Godbillo7l,- Vey rmrnber' of F t is eqnal to t.
This theorem showed decisively that the Godbillon-Vey class is a cohomology class with essentially real coefficients. Here we shall briefly explain his ideas. The foliation all PSL(2, IR) described in the previous subsection can be expressed more geometrically as follows. As wa..':> already mentioned, PSL(2, IR) acts on the upper half plane IHI by isometries. Therefore it also acts on the unit tangent bundle TIlHI = {v E TIHI; Ilv II ---' I} of 1HI. If we fix a tangent vector v() E TIlHI at the point i E 1HI of length 1, the correspondence PSL(2, IR) 3 A r-> f(A) =
3.2. THE GODBILLON-VEY CLASS
Ava
E
95
TIIHI becomes a bijection, and we obtain a commutative dia-
gram PSL(2, JR)
11 1HI 1HI Here .f(A) = Ai E 1HI. It follows that we can identify the mctp f : PSL(2,lR) ----> IHI with the unit tangent bundle ?T : TIIHI ----> 1HI which is a principal SO(2)-bundle over 1HI. Under this identification, we can describe the foliation on PSL(2, JR), given in Example 3.9, in the context of geometry of TllHI as follows. Let va E TllHI be an arbitrary point and set. ?T( vo) = Zo E: JlI. As is well known, geodesics of IHI are nothing but the intersect.ions with 1HI of circles (or lines) which are perpendicular to the real line. Hence there exists a unique geodesic pa.%ing through Zo which is tangent to Vo E TIIHI at zoo Let 90 denote it and let :.co, Xoo be the intersection points of its closure 90 with lR U { 00 }. Here we assume that the direction from :1:0 to Xoo coincides with that of Va. Then the leaf LV(l passing through va can be descri bed as
LVI) . : {positive unit tangent vectors on all geodesics through xo} (d. Figure 3.3). Vo
Xo
FIGURE 3.3. The leaf Lvo passing through Vo For each v E TllHI with 71'( v) = Z E IHI, let 8 denote the angle at z which is measured counterclockwise from the perpendicular to the real line to v (cf. Figure 3.3). Then 8 is defined globally on the whole T1IHI, and the correspondence TIIHI 3 v f-> (7I'(v), 8(v)) E IHIxlRj271'Z becomes as an orthonormal frame a diffeomorphism. If we take -y y , y
t tx
3. CHARACTERISTIC CLASSES OF FOLIATIONS
96
field, then the corresponding dual l-forms are 6 = -~dY,6 = ldx. y y Hence the canonical I-forms (see §3.3.I) of TllHI are given by Wl
W2
= cos e 6
=-
+ sin e 6
sin e 6
1
= -
Y
(sin e dx - cos e dy)
+ cos e 6 = ~ (cos e dx + sin e dy). Y
If we denote by Wo the Riemannian connection form, then we see from
that the first structure equation is given by Wo = de
1
+ -dx. y
Then we have dw o = Wl 1\ W2 = y\ dx 1\ dy. Hence if Gaussian curvature, the second structure equation dw o
=
J(
denotes the
-J(Wl 1\ W2
implies that K == -1. Thus we have verified that lHI certainly has a constant negative curvature -l. Coming back to our original situation, for each v E Lvo, the angle between the line, joining Xo and z, and the positive direction of the real axis is ~. Therefore Lvo is represented by the equation (}
y
2
x - Xo
tan- = - - - .
(3.3)
If we differentiate (3.3) and arrange it, we obtain
de =
sin e dy y
_ 1-
cos e dx. y
Hence, as a I-form which defines the foliation on T1IHf in question, we can take W=
e d 1 - cos e. dx de - sin - - y+--
= Wo Since
y
- W2·
y
3.2. THE GODBILLON-VEY CLASS
97
if we set ry = -WI, we have dw = ry 1\ w. Hence the Godbillon-Vey form is give by
n = 'I] 1\ dry = WI
1\ Wo 1\ W2
1 = - 2 dO 1\ dx 1\ dy. Y
The results obtained above are invariant under the action of P SL(2, JR) from the left. Hence we can use them in the computation of the Godbillon-Vey number of the Anosov foliation F on the unit tangent bundle TILg of any surface Lg equipped with a metric of constant negative curvature. Since dx 1\ dy is the volume form of lHl, we obtain
;2
gv(F)[T1L g ] = {
n=
-
lT1Eg
=
-27rvol(Lg)
=
47r 2 (2 - 29).
~dO 1\ dx I\. dy
( lTlEg Y
Here vol denotes the volume (or rather areal) form, and we have used the fact vol(Lg) = 27r(29 - 2) which is a consequence of the theorem of Gauss-Bonnet. Thus we have seen the non-triviality of the Godbillon-Vey numbers quantitatively. However, the values obtained above are discrete. To show that they vary continuously, Thurston considered the following. Instead of the upper half plane 1HI we consider the unit disk ]])) = {z E C; Izl < I} which is isometric to 1HI, and inside it we take a "hyperbolic square" ABCD as depicted in Figure 3.4 where each edge is a geodesic. We denote it by K r , Here 0 < r < 1 is a suitable parameter such that the four vertices approach the boundary of IDl when r --+ 1. Let a( r) be the angle at the four vertices of Kr which have the same value. We have limr-->l a(r) = O. Now let Isom+1Dl be the orientation preserving isometry group of IDl which is the same as the group of all holomorphic transformations of 1Dl. Hence we have Isom+1Dl ~ Isom+lHI = PSL(2, JR). Let Ir E Isom+1Dl be the element which sends the edge AD to BC, and similarly let 9r E Isom+1Dl be the element which sends AB to DC. Then it can be shown that the composition
A~B~CSD~A is a rotation around the fixed point A by the angle 4 a( r). Then, for a sufficiently small c > 0, we remove from Kr the c-neighborhood of
98
3. CHARACTERISTIC CLASSES OF FOLIATIONS /\
})
FIGURE
3.4. "Hyperbolic square" Kr
each vertex and let Kr(c) be the remaining part. If we paste the four edges of Kr(c) by the two maps j,.,gr, then we obtain a figure To which is a torus with an embedded disk removed. On the other hand, since the unit tangent bundle 7r : T1Jl)) -, Jl)) of Jl)) is isometric to TllHI, the Anosov foliation is defined on it. We consider the restriction of the Anosov foliation to n-1(Kr(c)). Since this foliation is invariant under the action of Isom+Jl)), if we paste the boundary of n-1(Kr(C:)) by iT> gT> we obtain a codimension 1 foliation on To x S1. We denote it by:Fr . Now the boundary of To x Sl is a torus T2, and the restriction of:Fr to it is a linear foliation on T2 (cf. Example 3.3) corresponding to the angle 4 a( r). Let To X S1 :) (z,w) ~ (z,w n ) E To X S1 be the n-fold covering along Sl and let :F,~1I) be the pullback of :F,. by this map. Then the restriction of :F;n) to the boundary 8To x SJ = T2 is a linear foliation of angle ~4a(r). a(r) is a monotone decreasing function of rand lim"_d a(r) = O. Hence there exists a certain r.l with 1 > r' > r such that
a(r-')
= ~a(r-). n
Then the restrictions of the two foliations :F,., , :F;n) on To x Sl to the boundary are linear foliations with the same angle. Hence we can paste them along the boundaries. If we paste two copies of To along their boundaries, we obtain a genus 2 closed surface ~2' We
3.2. THE GODBILLON- VEY CLASS
99
thus obtain a foliation F(r, n) on E2 x 51. It is expected that the corresponding Godbillon-Vey number is given by gv(F(r, n))[E2 x 51]
=
-21f(vol(Kr') - n vol(K.,.))
= -21f{(21f - 4a(r,l)) - n(21f - 4a(r))}
1) - 81f(n 2 - l)a(r').
= 41f2(n -
If this is true, then we can conclude that the Godbillon-Vey numbers move continuously by choosing r, n appropriately. However, the I-forms w on each of the two copies of To x 51 which define the given foliation and which were used in the above computation of the Godbillon-Vcy number do not connect smoothly to each other. This disadvantage can be overcome as follows. First we denote by Fe(r, n) the above foliation becauxe F(r, n) depends on the choice of c as it stands. However, actually the foliation-preserving diffeomorphism class of F£(r, n) does not depend on E because the place where we paste the two copies of foliated manifolds is a torus with a linear foliation. Next it can be shown that, for some co > 0, we can take a I-form w;' on To x 51 which defines Fr for any c E (0, co) satisfying the following conditions. (i) (ii) (iii) (iv) (v) (vi)
w;' = wr near the boundary of 1(l x 51. w;' = w away from the boundary of 1() x 51.
w;.
d.w; = T};' 1\ r,; = 0 near the boundary of To x 51. 1/; = -WI away from the boundary of To x 51. lime~() ./~)XSI T}; 1\ dT}; WI 1\ dw 1 == -21fvol(Kr ).
IK"
Here in (i), w" is a closed I-form on a neighborhood of the boundary of To x 51 which definex the linear foliation there. From the above, we can conclude that gv(F(r,n))[E2 x 51] =
r
lToxsl
T}.~, 1\ dT};'
= lim ( /" 17;' £-.0 lToxsl
1\
- n
r
lToxsl
T}~ 1\ dT};
dT};' - n /"
lToxsl
T}; 1\ dT};)
= - 21fvol(Kr ,) - n(-21fvol(Kr )).
Therefore we see that the above rough computation was correct. We thus have proved the following theorem.
100
3. CHARACTERISTIC CLASSES OF FOLIATIONS
THEOREM 3.11 (Thurston). For any t E JR., there exists a codimension 1 real analytic foliation F t on ~2 x Sl such that qv(Ft )[I: 2 x S1] = t.
3.3. Canonical forms on frame bundles of higher orders 3.3.1. Canonical forms and connection. Here we recall some well-known facts about the tangent frame bundle 7r : peN) -> N of an n-dimensional Coo manifold N. First there is defined on the total space peN) of the tangent frame bundle certain I-form () E Al(p(N);JR.n) with values in JRn as follows. It is called the canonical form or canonical I-form. Each point u E peN) defines a linear isomorphism
O(X)
=
7r.(X) = ()l(X)Vl
+ ... + On(X)vn .
The canonical I-form is invariant under diffeomorphisms as described in the following proposition. The proof is easy and is left to the reader. 3.12. Let f : M -> N be a diffeomorphism and peN) be the diffeomorphism induced by f. Then j'()N = ()M where OM and ()N denote the canonical forms of M and N respectively. PROPOSITION
let
j : P(M)
->
Now suppose that an (affine) connection w E Al(p(N);gf(n;JR.» is given on peN). Ifn E A2(p(N)jgf(njJR» denotes the corresponding curvature form, the following second structure equation (3.4)
holds. Then how can we express dO ? Let X, Y E TuP(N) be tangent vectors at an arbitrary point U E peN). If we denote by X h , Y h the horizontal components of them, then we know that n(Xh' Y h) = dw(Xh' Y h ). We define 8 E A 2 (P(N)j JRn) by setting
8(X, Y) = dO(Xh, Y h )
3.3. FRAME BUNDLES OF HIGHER ORDERS
and call it the torsion form. equation
101
Then the following first structure
(3.5) holds. If we denote by ei , e i , WJ, OJ the components of e, e, w, 0 with respect to the standard basis of IR n and gf(n; IR), then the above two structure equations (3.4);(3.5) can be written as n
dwj
= -
LW~
/\ w; + OJ
k=l
n
de i
= -
L wj /\ e + e j
i.
j=l
A connection W with e = 0 is called a torsion free connection. If we have a connection w, it determines a closed form Pi(O) which expresses the i-th Pontrjagin class [Pi (N)] E H4i(N; IR). This differential form depends on the choice of connection. Nevertheless it may be natural to ask whether it may be possible to determine it more canonically. As preparation for giving an answer to this question, in the next subsection we introduce a concept of jet bundles due to Ehresmann. 3.3.2. Tangent frame bundles of higher orders. The tangent frame bundle P( N) of an n-dimensional Coo manifold N consists of all frames at all points of N. Here a frame at a point pEN is an ordered basis of 1~N and so it represents a first order approximation of local coordinates at p. If we consider higher order approximations, we obtain the concept of tangent frame bundles of higher orders or simply frame bundles of higher orders. 3.13. Let x be a point in IR n and let f : U ----7 IR n be a Coo function defined on some neighborhood (which may depend on f) of x. The k-jet of f at x, denoted by j;(f), is an equivalence class of all such functions divided by the following equivalence relation. Here two functions f and 9 are said to be equivalent (denoted by f ",k g) if they satisfy the following conditions. If k = 0, the condition is simply f(x) = g(x). If k ~ 1, DEFINITION
f '" k
9
olal f oxO'.
{==? - -
I Ix
olal 9 I oxO< x
=--
3. CHARACTERISTIC CLASSES OF FOLIATIONS
102
for any multi-index a = (al,··· , (Xn) with I(XI = (Xl + ... + an k. Here Xl,· .. , Xn are the standard coordinates of rn;n and ox'" oxr" ... O.T;;''' .
N.
->
106
3. CHARACTERISTIC CLASSES OF FOLIATIONS
Now put a torsion free connection w in Jl (N) and let n be its curvature form. For example, we put a Riemannian metric on N and consider the associated Levi-Civita connection. Then the i-th Pontrjagin form
Pi(n) is defined which is a closed form on Jl(N) (or actually on N). This differential form depends on the choice of a connection. However, we shall see below that the Pontrjagin forms are determined canonically on J2(N). The structure equation of the connection w is given by n
On the other hand, by Proposition 3.19, if we consider the section s : Jl(N) -+ J2(N) which corresponds to the torsion free conection w, then we have i *()i Wj = S j'
Now we define a 2-form R by the equality
=
(Rj) on .I2(N) with values in gl(n,R) n
(3.6)
d();
=-
L ()~ /\ ()j + R; . 1 A*(Jf(N)), which is induced by the projection p : Je(N) --> Je-l(N), is injective, it is enough to prove that the equality holds on Je(N) for a sufficiently large e. Then for any two tangent vectors X, Y E TuJf(N), let
3. CHARA CTERIS TIC CLASSE S OF FOLIAT IONS
112
X, Y
and also conside r obtain
dej(X, Y)
Then by equalit y (3.8), we
E ?£(Je(N)).
dej(X, Y) = Xej(Y) - yej(JY) - (1j([X, YJ)
=
(the i-th = -~ OXj
- ) (0) compon ent of Z[X,YI
(the i-th compon ent of [Zx, Zyl) (0) ~ OXj
=
OWi n a (l)Zk ~-
= -,-
OXj
UXk
k=l
OZ;) Wk ::.,.) UXk
= I:
0 2 Zi (02Wi - 'Wk Zk OXjOX k; OXjOXk k=l n
(0) OZi )
O'Wk OZk OWi --- + ~ax" OXj
ax"
(0)
OXj
n
I: (e"(X)ejk(Y) - (1"(Y)tijk(X)
=
k=l
+ ej(x)ek(Y) - e;(Y)(1k(X)) n
= -
:L(ejk
!\
ek + ek !\ ej)(x, Y).
k=l
o
This comple tes the proof. Now if we set JOO(N)
Ii!!! Jk(N),
=
e=
then the totality
w, (1;,
ej k> ... )
of canonic al forms can be regarde d as an element of n Then Propos ition 3.23 can be considered as the structu re equatio we that For which expresses de. We put it in a more tractab le form. choose a basis of an as follows. We set
a
e -.t
-
OXi"
=
e~
a
-:rJ'~,. , uXi
and more generally we define e j, t
.. jk -
~"" . '" (_l)k 2k~xJl syrn
. ~
x JkOX ' ~
3.4. CHARACTERlSTIC CLASSES OF FOLIATIONS
I I \
Here sym means that we take the symmetric sum with respecL I.., I I,,· indices. In other words, we set
(-I)k
Ok
(the coefficient of e:l-j,,) = l.
aXjl ... aXjk
If we denote the canonical form by
o= L i
Oi ei
+L
OJ eI
+L
O} k e{ k
+ . .. ,
i,j,k
i,j
then it becomes a I-form on Joo(N) with values in express the structure equation simply by
Un.
We C11.11
1111\\
1 dO = - -2 [0 ' OJ .
(3.9)
3.4. Bott vanishing theorem and characteristic claS8(~8 of foliations 3.4.1. Bott vanishing theorem. Let M be an n-dimeJll'jlllllll Coo manifold and let :F be a foliation on M of co dimension (/. 11,1' Definition 3.1 in §3.1.1, for any point p E M there exist all '1111'11 neighborhood U '3 P and a diffeomorphism
2q).
Pi, (v(F))··· Pic (//(F)): 0
PROOF. If we generalize the consideration in §3.3.3 to the case of .1 00 (F), we see that the following facts hold. N amdy, if we define 2-fo1'm8 Rj on J2(F) by setting q
Rj ... d()j +
L ()~; 1\ e.~ , k=l
then we can define canonical Pontrjagin forms
in terms of them. Moreover, if we choose a GL(q, lR)-eqllivariant section s : Jl(F) ~ J2(F), then (s*()j) becomes a connection form of Jl(F) and s*(p.;(R)) becomes a Pontrjagin form representing the Pontrjagil1 class Pi(l/(F)). Now recall that we have canonical forms
3.4. CHARACTERISTIC CLASSES OF FOLIATIONS
115
on .J3(F) and they satisfy the equality d(); = -
q
q
k=l
k=l
L ()1 !\ ()~' - L
Ojk !\ ()k
(see Proposition 3.23). If we choose a section follows from S*()iJ = (JiJ that
8 :
.J2(F)
-->
.J3(F), it
q
R; = d(); + L
()1 !\ ()J
k=l q
'C.o
+ L ()1 !\ ()j)
s*(d();
k~l
q
-8*
(L Ojk !\ ()k). 1.:-1
Hence if
e> q, we can conclude
(3.10) The claim of the theorem now follows from this.
o
3.4.2. Definition of characteristic classes of foliations. We defi.ne characteristic classes of foliations by making use of the equality (3.10) which played a fundamental role in the proof of the Bott vanishing theorem (Theorem 3.24). First of all, t.he Weil algebra (cf. 32.1.1) of g[(q, JR) can be decribed as W(g[(q, JR)), A*g[(q, IR)* ® S*g[(q, IR)* = E(w;) ® p[n;l.
As was mentioned earlier, here E and P denote the exterior and polynomial algebras respectively. Also and nj denote the components of the universal connection and the curvature forms with respect to the standard basis of g[(q, IR), respectively. By the definition of Rj, we sec that the correspondence
w;
w'i )
f-----4
Oi
J'
nJi
f-----4
Ri J
induces a d.g.a. map
(3.11)
that this map
q}
form a basis of H*(Wq) ~ H*(W(g!(q,JR))).
It is clear from (3.11) that all the above elements are closed under d. If the normal bundle v(F) of a foliation is trivialized, then by using the corresponding section s : M -> Jl(F), we obtain a homomorphism
3. CHARACTERISTIC CLASSES OF FOLIATIONS
l18
We can see that the image o:(F) E H*(M; 1R) of an arbitrary element 0: E H* CWq) under the above map becomes a characteristic class of foliations whose normal bundles are trivialized. In the case of general foliations whose normal bundles arc not necessarily trivial, we can construct characteristic classes as follows. The action of the orthogonal group O(q) on the Weil algebra W(gl(q,IR» through interior product and adjoint representation induces the action on the truncated Weil algebra W(gl(q, JR». Hence, if we denote by W(gl( q, 1R) )o(q) the subalgebra of W(g((IJ, 1R» consisting of all elemeut.s which are killed by thc int.erior product of any clement of O(q) anci which are also invariant under the adjoint action of it, then we obtain a homomorphism H* (W(gl(q, 1R»O(q»
-------+
H*(J 2 (F)jO(q); JR).
If i is odd, it can be shown that we can choose W(g((q, IR))O(q)' Then we set
Ui
from the set
WO q = E(Ul, U3,"') ® Pq[Cl,'" ,cq] C W(gl(q, 1R»O(q)· As for this, similar to Proposition 3.27 and Proposition 3.28, the following two propositions hold. PROPOSITION
3.29. Thf. above inclusion map induccs an isom.or-
phism in cohomology. PROPOSITION
3.30 (Vcy). The elements
{UTC.!; ·tk is odd [or any k:, 8
=j:. 0
=>
PI :::; q,
8
= 0 ~. all.ie are + 1.11 > q}
cven,
i 1 < the smallest odd .ie, i 1
[onn a IJO.o9is of H*(WO q ) ~ H*(W(gl(q, 1R»O(q)· Thlit; we obtain a homomorphism H*(WOq)
-------+
H*(J 2 (F)jO(q);JR) ~ H*(M;IR),
and we can see that the image o:(F) E H* (M; 1R) of an arbitrary element 0: E H*(WOq) under the above map becomes a characteristic class of foliations. In particular, we call the characteristic class represented by the element Ul ci the Godbillon-Vey dass of codimension q.
3.5. DISCONTINUOUS INVARIANTS
119
In the above definition of characteristic classes of foliations, we llse only information about J2 (F). It might be natural to expect that we cOllld obtain more characteristic classes other than the above by considering jet bundles Jk(F) of higher orders. However, the following theorem shows that unfortunately this is not the case. TUEOREM
3.31 (Gel'fand-Fuks [GF2]). The inclusion map
Wq C W(g[(q,JR)) c A~(uq) induces an isomorphism in cohomology. Here A;: (u,,) denotes the co chain complex consisting of continuous cochaills of the Lie algebra uq with respect to certain natural topology on it. 3.5. Discontinuous invariants In this section, we consider the sit.uation where we are given a real cohomology class Q E HI) (X; lR) of a topological space X or more generally, a system of real cohomology classes of it. We discuss how we can obtain infinitely many higher dimensional cohomology classes from O~ by twisting it using discontinuous endomorphisms of lR and thcn takillg cup products of them. In this book, we call the values of thelll on variolls integral cycles of X discontinuous invariants. As is well known, classical characteristic classes snch as Chern classes or Pontrjagin classes are integral cohomology cla.':;ses. In contrast with this, characteristic cla.
H.(BAi'Z).
We call this the discontinuolls invariants induced by f. Here the crucial point is that the ground field K is lR. In case K = Q, if we define similarly as above, we obtain nothing interesting. This is because of the following reason. On the classifying space BA, there is defined the tautological A differential system A
->
A*(BA).
The induced homomorphism H*(A)
->
H*(BA; K)
3.5. DISCONTINUOUS INVARIANTS
125
in cohomology turns out to be an isomorphism provided K = Q. In contrast with this, if K = JR, it is far from being an isomorphism. We shall show this by an example. EXAMPLE 3.38. Let K be a simplicial complex and let AQ(K) and A*(K) be the de Rham complex consisting of all Q polynomial forms and all Coo forms respectively (see §1.3.1). Let M~ and M~ be their minimal models (cf. §1,2). Then BM~ becomes the rational homotopy type Ko of K (cf. §1.1.3). Hence we have H*(BM~; Q) ~ H*(K;Q). However, though KJR = BM~ is, so to speak, the real homotopy type of K, it turns out that H*(KJRi JR) is much bigger than H* (K j JR). For example, in case K is an odd dimensional sphere s2n+1, we have Ko = K(Q,2n+1) and KJR. = K(.JR,2n+1). We know from the result in §3.5.1 that the latter space has a huge cohomology group. In general, it is a very difficult problem to compute H*(K!Ri Z).
Let G be a Lie group and let 9 be its Lie algebra. Let Bg denote the classifying space of the d.g.a. A" g* consisting of all left invariant differential forms on G. For example, we have
Bsl(2, JR) = K(JR, 3),
Bs[(2, C) = K(C, 3)
(cf. §2.2.4 and §2.3.5, Example 2.23). Then there arises a natural question of how many geometric cycles are in H. (Bgj Z). Here by geometric cycles we mean those cycles which come from flat Gproduct bundles over manifolds. This problem is unsolved even for the very first case where we do not take cup products. For example, in the case of .&[(2, C), the real part of
H3(Bs[(2, C)j Z)
~
C
corresponds to the values of the Chern-Simons class associated to the first Pontrjagin class, while its imaginary part corresponds to the values of the cohomology class induced by the volume form of the hyperbolic 3-space JHI3. It is unknown which values they can take on geometric cycles. This problem is closely related to the shape of the image of the map
(17, i vol) : {isometry class of closed hyperbolic 3-manifold}
-->
C
which is induced by the 17-invariant and the volume of closed hyperbolic 3-rnanifolds. In regard to foliations, we first consider the Godbillon-Vey class of codimension 1. Recall that for any foliation (M, F) of codimension 1, its Godbillon-Vey class gv(F) E H3(M; JR) is defined and Thurston's
126
3. CHARACTERISTIC CLASSES OF FOLIATIONS
result, mentioned at the beginning of this section, shows that it can take any real number on geometric 3-cycles. Then a natural question . arises as to the values of higher discontinuous invariants
gVk(!VI,F): H3dlVJ;7/,)
---7
A~(IR)
(k = 2,3,···)
which are induced by the Goclbillon-Vey class. Concerning this problem, even the non-trivialities of them are completely unknown. The situation is the same for the case of foliatiOJls of general co dimensions. For simplicity, we consider a foliation (lVI, F) of codirnensioll q whose normal bundle is triviali:;,ed. Then there is defined a system
8(F) = (8\ 11;, 8jk' 8jke,"') of canonical forms on JOO(F) (see §3, 4,1). If we pull back these forms by the section .s : !VI --; Jl(F) which is induced by the trivializatioll of the normal bundle together with the homotopy equivalence ,]1(F) ~ Jk(F) (k ~ 2), then we obtain a certain system of differential forms on !VI. In view of the structure equation (3.9) in §3.3.5 for 8(F), it is reasonable to denote the dassifying space of this differcntial system by Bu q . Then the problem is to determirH~ geometric cycles in H* (Bu q ; 7/,). If we use the terminology of Haefliger's classifying space Bfq (cf. [H]), then the above problem is equivalent to asking whether the canonical map Bfq ---7 BU q is a homotopy equivalence or not, as already mentioned in Sullivan's paper [Su2). This is also an extremely difficult question.
3.6. Characteristic classes of flat bundles II 3.6.1. Classifying space of foliated F-bundles. Let G be a Lie group. The relations among characteristic classes of various Gbundles were described ill the commutative diagram (2.20) in §2.3.3. It is an important problem to generalize this commutative diagram to the case of various F-bundles for a givcn closed Coo manifold F. As was already mentioned in §2.4.4, there exists a fibering (3.16)
BDiffF
---7
BDifF F --; BDiffF
consisting of classifying spaces of varions F -bundles. Here Diff F denotes the group Diff F equipped with the discrete topology. The fiber BDitf F is the classifying space of foliateel F-product bundles, that is foliated F-bundles together with trivializations as differentiable Fbundles. It was described in §2.4 that the Gel'fand-F'uks cohomology
3.6. CHARACTERISTIC CLASSES OF FLAT BUNDLES II
127
group HCF(F) of F plays the role of characteristic classes of fiat F-product bundles. Next BDiffo F serves as the classifying space of fiat F-bundles (see §2.4.2, Definition 2.26) or equivalently foliated F-bundles (cf. §3.1.1, Example 3.5), for, by Theorem 2.28 in §2.4.2, the isomorphism class of any flat F-bundle 7r : E -+ M is determined completely by its holonomy (or monodromy) homomorphism p: 7r1(M) ----. DiffF.
The third classifying space BDiff F in (3.16) classifies usual differentiable F-bundles. Then the problem mentioned above is to construct two unknown theories in the following commutative diagram: H*GF (F)
(3.17)
----+
H*(BDiffFiJR.)
r r
unknown theory
----. H* (BDiffO Fi JR.)
unknown theory
-; H*(BDiff Fi JR.)
In the case of commutative diagram (2.20) corresponding to a finite dimensional Lie group G, it is essential that G admits a maximal compact subgroup K. More precisely, the relative cohomology H*(g, K) of the Lie algebra of G with respect to K and the algebra J(K) of invariant polynomials of K occupy the two places corresponding to the blanks in the above diagram. Unfortunately, however, the diffeomorphism group Diff F in general does not have subgroups which can play the role of maximal compact subgroups except for a few cases like F = S1, S2, . . .. Therefore we cannot generalize the Chern-Weil theory, which concerns finite dimensional Lie groups, as it stand:::;. The theory of characteristic classes for differentiable fiber bundle:::; with general fibers is a vast unknown domain. We can make use of characteristic classes of foliations, which we considered in §3.4, to construct characteristic classes of foliated Fbundles, though only partially, as follows. Let 7r : E -+ M be a foliated F-bundle and let F be the corresponding foliation on E. We set dimF = q and let et E H*(WO q ) be a characteristic class of codimension q. Then et(F) E H*(Ei JR.) is defined. Here we consider the Gysin homomorphism (or integration along the fiber) 7r* :
H*(EiJR.) ----. H*-q(M;lR)
128
3. CHARACTERISTIC CLASSES OF FOLIATIONS
(see §4.2.3). Then it is easy to check that 7r*(a(F)) E H*- 0 by the equality f*v = p,(f)v. Then, ~f we define gv E Gq+1 (Diff+F; JR) by
gv(h, ... ,fq+d =
llogP,(fQ+l) dlogp,(fqfQ+d··· d log p,(!I '" f q +1) ,
it becomes a group cocycle representing the Godbillon- Vey class gv E HQ+l(Diff+F; JR) mentioned above.
In the case F = S1, the Thurston co cycle can be described more explicitly as follows. Express any element f E Diff+S1 by a periodic diffeomorphism f E Diff+JR and choose dt for the volume form. Then we have ,Af) = Df· Here D f denotes the derivative of .f. Hence gv(j, g) =
r log
lSI
Dg D log D(fg) dt
becomes the cocycle which represents gv. Now transform the above integral as g
v(f
r
)- ~ /IOgD9 ,g - 21sl DlogDg
logD(fg) / dt D log D(fg) .
Then we see that the Thurston cocycle is a globalization of the co cycle 0'. of Gel'fand-Fuks described in §3.6.3, and conversely the latter is an infinitesimal version of the former.
CHAPTER 4
Characteristic Classes of Surface Bundles In t.his chapter, we describe basics of the theory of characteristic classes of surface buncHes. By surface bundles we mean differentiable fiber buudles whose fibers are closed orientable surfaces. Roughly speaking, the theory Il...
f
M2
between two F-bundles 7ri : Ei ----) Mi (i
=
1,2), we have the equality
In the terminology of the classifying space, we can write 0: E Hk(BDiffF; A) and if f : X ----) BDiffF is the classifying map of the given bundle 7r : E ----) M, then we have 0:(7r) = 1*(0:). Namely
4.l. MAPPING CLASS GROUP
137
characteristic classes of F-bundles are nothing but elements of the cohomology group of BDiff F. It follows immediately from the definition that two F-bundles over the same base space which have a different characteristic class are not isomorphic to each other. Thus it is desirable to define as many characteristic classes as possible. 4.1.2. Surface bundles. A 2-dimensional Coo manifold, which is compact, connected and without boundary, will simply be called a closed surface. The classification of closed surfaces was done already in the beginning of the twentieth century. As is well known, the Euler number together with the property of being orientable or not can serve as a complete set of invariants. In particular, the set of all the diffeomorphism classes of closed orient able surfaces can be described by the series 52,
T2,
2:09 (g
= 2,3",,),
Here 52 and T2 denote the 2-dimensional sphere and torus, respectively, and 2: g stands for a closed orient able surface of genus g. Of course we have 2:0 = 5 2,2: 1 = T2. Henceforth we assume that an orientation is fixed on each 2: 09 , DEFINITION 4.2. A differentiable fiber bundle with fiber 2: g is called a surface bundle or a 2: g -bundle. Let 7r : E -} !vI be a 2: g -bundle. Then the set of all tangent vectors on the total space E which are tangent to the fibers, namely the set ~
= {X E TE; 7r*(X) = O},
becomes a 2-dirnensional vector bundle over E. '''Ie call ~ the tangent bundle along the fiber of the given 2: g -bundle. Sometimes the notation T7r will also be used for~. This concept is defined not only for surface bundles but also for general fiber bundles. DEFINITION 4.3. A surface bundle 7r : E --+ 111 is said to be orient able if its tangent bundle along the fiber T7r is orientable. If a specific orientation is given on T7r, then it is called an oriented surface bundle. Henceforth in this book, all surface bundles are assumed to be oriented and all bundle maps between them are assumed to preserve the orientation on each fiber.
138
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
DEFINITION 4.4. Two L:g-bundles 1Ti : Ei --+ M (i = 1,2) over the same manifold NI are said to be isomorphic to each other if there exists a bundle isomorphism El ~ E2 which preserves the orientation on each fiber. Our principal problem can now be stated as follows. Determine the i::iet of isomorphism clasiie:;; of L:g-bundles over a given manifold. Let Diff+L:g denote the group of all the orientation preserving diffeomorphiiims of L: g equipped with the Coo topology. It serves as the structure group of oriented L:g-bundles. In the case where 9 = 0, namely for the sphere, it was proved by Smale [8] that the natural inclm;ion
SO(3) C Diff+S2 is a homotopy equivalence. It follows from this fact that any S2_ bundle is isomorphic to the sphere bundle of somc uniquely defined 3-dimensional oriented vector bundle. Hence the classification of S2_ bundles over a given manifold NI is equivalent to that of 3-dimensional oriented vector bundles over M. Since the homotopy typcof the classifying space BSO(3) is known, wc may say that this problem is solved. Also for the 3-dimensioual sphere S3, Hatcher [Hat] proved that the inclusion SO(4) C Diff+S3 is a homotopy equivalence. Hence the problem is solved also in this case. However, it wa..
Sp(2g, Z).
This homomorphism was essentially obtained around the end of the nineteenth century and is a classically important one. Also it was already known then that p is surjective. The group Sp(2g, Z) is called the integral symplectic group or the Siegel modular group as it played a crucial role in the theory of automorphic functions due to Siegel. If 9 = 1, then
Sp(2,Z) = SL(2,Z) = {A E GL(2,Z);detA = I} and the representation p: Ml phism. Hence we have
Ml
->
S:!
Sp(2, Z) is known to be an isomorSL(2, Z).
However, in the cases where 9 > 1, it is known that p has a big kernel. In fact, the group Ig
= Ker p = {cp E
Mg; p(cp)
= id}
is called the Torelli group, which turns out to be an extremely important normal subgroup of the mapping class group. Recently active research has been done concerning this group. However, its structure remains largely uncovered so that we must say that it is still a mysterious group. We end this section by the following fundamental short exact sequence of groups 1
->
Ig - ) My ~ Sp(2g, Z)
->
1.
4.1.5. Classification of surface bundles. As was mentioned in §4.1.2, in the case 9 = 0 Smale showed that the inclusion SO(3) C Diff+S2 is a homotopy equivalence so that BDiff+S2 ::: BSO(3).
Next we consider the case 9 = 1, namely surface bundles whose fibers are diffeomorphic to the torus T2. If we identify T2 with ]R2 /Z2, then T2 acts on itself by diffeomorphisms. Hence T2 can be naturally
4.1. MAPPING CLASS GROUP
143
considered as a subgroup of DiffoT2 which is the identity component of Diff+T2. Moreover it is known by Earle-Eells [EE] that the inclusion
T2
C
DiffoT2
is a homotopy equivalence. On the other hand, we have an isomorphism Diff+T2 jDiffoT2 = M1 ~ SL(2, Z) which was already mentioned in the previous subsection. From these facts we obtain a fibration
BT2
-->
BDiff+T2
-->
BSL(2, Z) = K(SL(2, Z), 1).
The structure of the group SL(2, Z) is classically well known, and we have a homotopy equivalence BT2 c:: ClP'oo x ClP'oo. Based on these facts we can compute the cohomology of BDiff+ T2 which serve as the characteristic classes of T2-bundles. But here we omit the details. In the cases where g ;::: 2, the situation changes drastically. More precisely, Earle and Eells proved in the above cited paper [EE] that DiffoEg is contractible so that BDifl'+Eg = K(M g , 1).
It follows immediately from this that PROPOSITION 4.6.
Let g;::: 2. Then for any Coo
man~fold
M, we
have a natural bijection {isomorphism class of Eg-bundle over M} ~
{conjugacy class of homomorphism 7f1(M) ---) Mg}.
In particular if M is simply connected, then any Eg-bundle over it is trivial. However, in general, it is almost impossible to determine the set of all conjugacy classes of homomorphisms from a given group to Mg. It may be better to understand the above proposition as a starting point for the construction of a classification theory rather than a direct role. Now let a be a characteristic class of Eg-bundles of degree k with coefficients in an abelian group A. Then we can write 0:
E Hk(BDiffEg; A) = Hk(K(M g, 1); A) = Hk(Mg; A)
(cf. §l.4.2, Definition l.57). In other words, characteristic classes of surface bundles of genus 9 ;::: 2 are nothing but cohomology classes of the mapping class group Mg.
144
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
4.2. Characteristic classes of surface bundles 4.2.1. Definition of characteristic classes. Let I: g be an oriented closed surface of genus 9 and let 7r:E--7M be an oriented I:g-bundle. If we denote by ~ the tangent bundle along the fiber of 7r, then by Definition 4.3 ~ has a structure of an oriented 2-dimensional real vector bundle. Hence its Euler class
is defined. For each non-negative integer i, we consider the power ei + 1 E H 2 (i+l) (E; Z) of the Euler clast; e. We then apply the Gysin homomorphism (cf. §4.2.3) 7r* : H 2 (i+1)(E; Z) - - 7 H 2i (M; Z) to ei+1 and obtain a cohomology class of the base space M which we denote by ei(7r) = 7r*(ei +l) E H 2i (M;Z). DEFINITION 4.7. The cohomology class ei(1r) E H 2i (M; Z) which is defined for any I:g-bundle 7r : E ~ M as above is called the i-th characteristic class of surface bundles. The fact that ei in fact defines a characteristic class of surface bundles, namely that it is natural under the bundle maps, can be checked as follows. Let 7ri : Ei ~ Mi (i = 1,2) be two I:g-bundles and let
E1
~l
j
---->
1
M1
E2
1~2 ---->
f
M2
be a bundle map between them. Then by the definition of bundle maps, the restriction of .f to each fiber is an orientation preserving diffeomorphism. Hence if ~i denotes the tangent bundle along the fibers of 7ri, then we have
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
145
The naturality of the Gysin homomorphism (cf. Proposition 4.8(iii)) implies that
ei( 1f 1) = J*(ei( 1f2)) which shows that ei is indeed a characteristic class. Hence from the description of §4.L5, we can write ei E H2i(BDiff+~g; Z)
= H2i(Mg; Z)
for 9 ;:: 2. In other words, ei can be considered as a cohomology class of Mg of degree 2i. 4.2.2. Characteristic classes of surface bundles and the moduli space of Riemann surfaces. The mapping class group of surfaces has an intimate relationship with an important space called the moduli space of Riemann surfaces. In particular, over the rationals, any characteristic class of surface bundles can be considered as an element of the cohomology group of the moduli space. In this section, we briefly mention this matter. The concept Riemann surface was introduced by Riemann himself as the denomination indicates. It is a surface obtained by suitably modifying the domain of a many-valued analytic function in such a way that it becomes single valued. However, here we call a (real 2dimensional) surface which is given a fixed complex structure a Riemann surface. In particular, if a complex structure on ~g is specified, then it is called a genus 9 compact Riemann surface. We denote by Mg the space of all biholomorphism classes of genus 9 compact Riemann surfaces and call it the moduli space of Riemann surfaces of genus g. We can also define Mg as follows. Let Tg denote the space of isotopy classes of complex structures on ~9 whose induced orientations coincide with the fixed one. It is called the Teichmiiller space of genus g. The mapping class group Mg acts on Tg naturally, and there is a canonical identification Mg =
Tg/M g .
The Teichmi.iller space was introduced by Teichmi.iller in the 1930's. As was already mentioned in §4.L3, the mapping class group is also called the Teichmiiller modular group because of the above fact. Many things are known concerning Tg (see e.g. [IT]), but here we only mention the following two facts. One is that in the cases 9 2': 2, Tg is known to be homeomorphic to IR6g-6 and the other is that the
146
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
action of M 9 on Tg is properly discontinuous. It follows from these facts that there exists a continuous map BDiff+2::g
~
Mg
which is well defined up to homotopy such that it induces an isomorphism H*(Mg; Q) ~ H*(BDiff+2::g; Q) = H*(Mg; Q) in the rational cohomology. In this way we find that the two spaces, namely the moduli space of compact Riemann surfaces and the classifying space of surface bundles which play fundamental roles in the theory of Riemann surfaces and in topology respectively, are very close to each other. Let us reconsider the above matters from a somewhat different point of view. First of all, for a general Coo manifold F let R(F) denote the space of all the Riemannian structures on it. Recall here that a Riemannian structure is given by assigning an inner product, that is a positive definite symmetric bilinear form, on the tangent space TpF of each point p which varies smoothly with respect to p. By the way, any two points in the space of inner products on a finite dimensional vector space over JR: can be joined by a segment. This is because any linear combination (l-t)/tO+t/-l1 (t E [0,1]), where /-lo, /-l1 are two inner products, is again an inner product. It follows easily from this fact that R(F) is contractible. In the case where F = 2::g, we can consider the subspace Ro(2::g) of R(2::g) which consists of all the metrics of constant Gaussian curvature I{ == 1,0, -1 accordingly as 9 = 0, I, :::: 2. In the cases g :::: 2, it can be shown that Ro (2:: g) is naturally a strong deformation retract of R(2::g) as follows. To begin with, any metric on 2::g gives rise to a complex structure via the isothermal coordinates. Secondly, it is a classical result that any complex structure on 2::09 (g :::: 2) is obtained as a quotient space of the upper half plane JH[ with respect to some discrete representation 1[1(2:: 09 ) ---> PSL(2,JR). On the other hand, the group P SL(2, JR:) of biholomorphisms of JH[ can also serve as the group of orientation preserving isometries of JH[ with respect to the Poincare metric. Hence any complex structure on 2::09 is equivalent to the associated metric of constant negative curvature. Thus we find that any point in R(2::g), namely any metric on 2:: g, will uniquely determine an associated point in Ro(2::g), that is a metric of constant negative curvature, which is conformally equivalent to the original one.
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
147
Let 'if : E ---> M be a Eg-bundle and let us give a fiber metric on it. Then each fiber Ep (p E E) becomes an oriented closed surface equipped with a Riemannian structure. The above fact implies that each metric on Ep defines the associated structure of a Riemann surface. Keeping in mind these structures as a whole, we can consider 'if : E ---> 1\1/ as a family of Riemann surfaces over M. Byassociating the Riemann surface Ep E Mg to each point p EM, we obtain a continuous mapping f: M --+ Mg. It is easy to see that the homotopy class of this mapping does not
depend on the choice of the initial fiber metric. If we apply this operation to the universal Ey-bundle 'if : EDiff+Eg ---> BDiff+Eg, then we obtain a map BDiff+Eg --+ Mg. The moduli space Mg of compact Riemann surfaces is a very important space in algebraic geometry, and many deep results have been obtained by many mathematicians, notably by Mumford (cf. [HL][HM]). Among other things, Mumford began the theory of the Chow algebra A*(Mg) of Mg ([MuD. In pa.rticular, he defined a series of canonical elements
It follows easily from the definitions that, under the natural isomor-
phism H*(Mg;Q) ~ H*(Mg;Q), the cohomology class of Ki corresponds to the characteristic class ei E H2i(Mg; Q) of surface bundles introduced in §4.2.1 multiplied by the factor (_1)'i+1. The class Ki is in fact defined as an element of the Chow algebra A * (My) of the Deligne-Mumford compactification Mg. The classes Ki are called the tautological classes of the moduli space and their cohomology classes, namely the classes ei are called the Mumford-Morita-Miller classes. 4.2.3. The Gysin homomorphism. In §4.2.1 we defined characteristic classes of surface bundles where we made essential use of the Gysin homomorphism. This homomorphism is very important for the study of surface bundles as well as general manifolds. In this subsection we briefly summarize basic facts concerning it. Let F be an oriented closed manifold and let 1f:E--+M
148
4. CHARACTERlSTIC CLASSES OF SURFACE BUNDLES
be an F-bundle over NI. We assume that this bundle is oriented; that is the tangent bundle along the fiber of 7r, denoted by ~ = {X E TE; 7r*X = a}, is orientable and is given a specific orientation. Although we are only concerned with the case F = ~9' the Gysin homomorphism is defined for general F-bundles. If we denote by {E~,q} the spectral sequence for the cohomology of the above F-bundle, then its E2 term is given by
Eg,q
~
HP(M;7-i5).
Here Hq stands for the local coefficient system associated to the qdimensional cohomology Hq (7r- 1 (p); Z) (p E M) of the fibers. If F is n-dimensional, then clearly Hq = 0 for q > n so that
Eg,q
=a
(q> n).
By the assumption, 11.71 is isomorphic to the constant local system Z. Hence E~,n ~ HP(M; Z). On the other hand, the homomorphism dr : E;?-r,n+r-l -+ E~,71 is trivial for any p and r 2: 2 so that we obtain a series of monomorph isms Ef,;,n c ... c Ef,71 C E~,71 ~ HP(M; Z). Now we denote the homomorphism
HP(E; Z)
-----+ g~-n,n C E~-n,n ~
HP-n(M; Z)
which is the composition of the natural projection HP(E; Z) with the above monomorphism (with shifted degree) by
-+
E"g;;n,n
(4.1)
and call it the Gysin homomorphism of the F-bundle 7r : E -} M. Sometimes the symbol 7r! is used instead of 7r *. Note that this homomorphism goes in the opposite direction to the usual one which is induced by the projection 7r and also that it decreases the degree by n, namely the dimension of the fiber. Similarly we have the Gysin homomorphism (4.2)
in homology. The above method of defining the Gysin homomorphism using the spectral sequence is valid over Z, and we may say that it is theoretically the best one. However, it might not be easy to see its geometrical meaning. To cover this point, let us examine the Gysin
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
149
homomorphism in the context of de Rham cohomology, although the coefficients must be reduced to R Here it can be explained by means of an operation (4.3)
defined in the de Rham complex which is called the integration along the fiber. Any differential p-form on E can be expressed locally as a sum of the forms like w
=
L hj(x, y) dXi l 1\ ... 1\ dXis 1\ dYjl 1\ ... 1\ dYj,· i,j
Here Xl, ... , Xm (m = dim M) and Yl, ... ,Yn are local coordinates of M, F respectively and dYl 1\ ... 1\ dYn is assumed to coincide with the given orientation of F. Also the summation is taken over the multiindices i = (iI,'" ,is), i l < ... < is, j = (jl,'" ,jt), jl < ... < jt s + t = P with s + t = p. We now set 7r * (w)
=
L
i,j(t=n)
(1
iij (x, y)dYI 1\ ... 1\ dYn) dXi l 1\ ... 1\ dX"p_n'
F
It can be easily shown that 7r * is in fact uniquely defined by the above. The integration along the fiber commutes with the exterior differential d, namely d
0 7r
*
= 7r *
0
d.
Hence it induces a homomorphism
and it can be shown that this coincides with the Gysin homomorphism (4.1) which was defined by using the spectral sequence. In the cases where the base space 1\11 is an oriented closed manifold, there is one more interpretation of the Gysin homomorphism. Here we describe it simply. Suppose that there is given a continuous mapping i : N -4 N' between two oriented closed manifolds N, N'. Then a homomorphism
i* : HP(N; Z)
----+
HP-d(N'; Z)
which is also called the Gysin homomorphism is defined to be the composition
150
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Here n = dim N, d = dim N - dim N', and D and D' denote the Poincare duality maps of N, N' respectively. Similarly, the Gysin homomorphism
j* : Hp(N'; Z)
----4
Hp+d(N; Z)
in homology is defined by setting 1* = D-l 01* 0 D'. Now let us go back to the original situation where we are given an F-bundle 7r : E --+ M and assume that M is an oriented closed manifold. Then the total space E is also a closed manifold with the induced orientation which is locally equal to the one on thc product M x F. In this case, it can be shown that the Gysin homomorphisms 7r*
:HP(E; Z)
7r*
:Hp(M;Z)
----4 ----4
HP-n(M; Z) Hp+n(E;,£)
associated to the projection 7r defined above coincide with the former definitions (4.1), (4.2). The following proposition concerns basic properties of the Gysin homomorphism. We leave the proof of it to the reader because it is relatively easy.
(i) Let F be an oriented closed manifold M be an oriented F -bundle. Then for any ex E HP (M; '£) and {3 E Hq (E; '£), the equality
PROPOSITION 4.8.
and let
7r :
E
--+
7f* (7f* ex U (3) = a
U
7f * ((3)
holds. (ii) For any u E Hp(M; Z) and'Y E HP+n(E; Z), we have
("(, 7f*(u)) = (7f*("(), u). In particular, in the situation of (i), if we further assume that M is an oriented closed manifold and p + q = dim E, then
(7f* ex U {3, [ED
=
(ex U 7r * ((3), [M]).
(iii) The Gysin homomorphism is natural with respect to b'undle maps between oriented F -bundles. More precisely, the composition of the Gysin homomorphism followed by the induced homomorphism in cohomology of the base spaces is equal to the composition of the induced homomorphism in cohomology of the total spaces followed by the Gysin homomorphism, The following lemma treats a special case of the property of Gysin homomorphism of covering maps. We will use it in the next section.
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
151
LEMMA 4.9. Let M be an n-dimensional oriented closed manifold and let!!.-- : M ---> M be a finite covering. We put the orientation on M induced from M. Suppose that the Poincare dual [M] n a E Hn-k(M; Z) of a cohomology class a E Hk(M; Z) is represented by an (n - k)-dimensional oriented submanifold B of M. Then the Poincare dual of 1T*(a) E Hk(M; Z) can be represented by the oriented submanifold jj = 1T- 1(B) of M.
PROOF. Let N(B) be a closed tubular neighborhood of B in M. If we denote by v the normal bundle of B, then we can identify N(B) with the disk bundle D(v) with respect to a suitable metric. We set W M\Int N(B) and consider the following natural homomorphism:
Hk(D(//), aD(I/); Z) ~ Hk(N(B), aN(B); Z) ~
Hk(M, W;Z)
-7
Hk(M;Z).
If we denote by U E Hk(D(v), aD(v); Z) the Thorn class, then as is well known, the image of U under the above homomorphism is nothing but the Poincare dual of [B], namely a E Hk(N£; Z). On the other hand, 1T-1(N(B)) can serve as a closed tubular neighborhood N(B) of B, and moreover under the homomorphism
induced by the projection..'... the above Thorn class U clearly goes to that of the normal bundle of B. The claim follows from this immediately.
o 4.3. Non-triviality of the characteristic classes (1) 4.3.1. Ramified coverings. In this section as well as in the next, we prove the non-triviality of the characteristic classes of surface bundles defined in §4.2.1. The proof will be given by explicitly constructing surface bundles with non-zero characteristic classes. In this subsection, we briefly discuss ramified coverings, which are essential for us to use in our construction. The concept of ramified covering (or branched covering) is obtained by generalizing that of covering spaces, and there are various formulations in the framework of algebraic varieties, complex manifolds or differentiable manifolds. Roughly speaking, a submanifold, called the ramification locus or branch locus, is given in the base manifold, and away from there it is a usual covering space. Suitable
152
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
conditions are required on the ramification locus according to each framework mentioned above. Here we consider only the most simple type of ramified coverings, namely cyclic ramified coverings. Let m be a positive integer. An m-fold cyclic ramified covering is defined by taking the map (4.4) as a model. This map is the identity at the origin, and outside of there it is a usual covering map. From a slightly different viewpoint we can also interpret it as follows. The cyclic group 'LIm of order m acts on C naturally by
C '3
27ri
Z f---+
exp- z E C m
where ( denotes the generator of 'LIm. This action is free outside of the origin, and the quotient space can be canonically identified with
C/('Llm)
~
C
is equivalent to the above map (4.4). Now a ramified covering is defined, locally, by taking a direct product of this model with other manifolds. More concretely, assume that the cyclic group 'LIm acts on an oriented Coo manifold N by orientation preserving diffeomorphisms satisfying the following condition. Namely the fixed point set F
=:
{p EN; ((p) '-- p}
is a submanifold of N of co dimension 2, and the action is free outside of F. Then, it can be checked that the quotient space N =-= NI(Zlm) has a natural structure of an oriented Coo manifold by investigating the action of Zim on the normal bundle of each connected component of F. If we denote by 7r:N---+N the natural projection to the quotient space, then F = 7r(F) becomes a submanifold of N of codimension 2. Moreover the restrictioIl7r : F --> F is a diffeomorphism and 7r : N \ F --> N \ P is a covering map in the usual sense. Finally it is easy to see that the map 7r : N --> IV is equivalent to the above model F x C '3 (p, z) ~ (p, zm) E P x C near F.
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
153
E(v)
N(8)
-HH--t-iH--t-i-t-+-t-t-+ IJ
FIGURE
4.2
In such a situation, we call 7r : N -> N an m-fold cyclic ramified covering ramified along F. It is also called simply an m-fold ramified covering. 4.3.2. Construction of ramified coverings. Let M be an oriented closed COO manifold. Assume that there is given an oriented submanifold B c M of codimension 2. Following Atiyah [Aj and Hirzebruch [Hi], let us recall a sufficient condition to the existence of an m-fold ramified covering of M ramified along B. Let a E H2(M; Z) be the Poincare dual of the fundamental homology class [Bj E H n - 2 (M; Z) (n = dim M) of B. The first Chern class of complex line bundles induces a canonical bijection {isomorphism class of line bundles over M} ~ H2(M; Z). Hence there exists a complex line bundle 'f/ over M which corresponds to Q. This bundle can be constructed explicitly as follows. Let v be the normal bundle of B in M and denote by E(v) its total space. v is a 2-dimensional real vector bundle over B, and it has a natural orientation induced by those of .M, B. Hence we can consider vasa complex line bundle. Let N(B) be a closed tubular neighborhood of B. Then as is well known, by choosing a Hermitian metric on v, we can construct a diffeomorphism B be the projection and set 7r'
= 7r 0
~g." In other words, we fix a normal subgroup 1f1 (~g/) C 7rl (~g) of inci~l m. Since 1f1 (~g) has only finitely many normal subgroups of indlx- m, we can choose a normal subgroup Tl c Aut+7r1(~g) of finite index such that any element of it preserves 1fl (~gl ). Consider the natural homomorphism 1':
Tl
---+ Allt+1fl(~91) ---+
Mgl
where Aut+1fl(~gl) -> Mgl is the projection. It is easy to see that for any element 'Y E Inn1fl(~g) n T I , the oreler of r("() is finite. Now choose a normal subgroup T2 c Mgl of finite index which is torsion free. Clearly we have r-1(n) nInn1fI(~g) c KerT. If 7r : Aut+1f1(~g) -> Mg denotes the projection, then 1f(T- 1 (T2 )) is a subgroup of Mg of finite index. Therefore, if we set T3 to be the intersection of all the subgroups which are conjugate to this subgroup, then it becomes a normal subgroup of'M.9 of finite index. From the construction, there is a natural homomorphism T3 -) Mgl. Now let h : 1fl (M) -> Mg be the holonomy homomorphism of the given ~g-bundle 7r : E -> Iv! anel let Ml -> Iv! be the finite covering induced by the kernel of the composition 1f1 (M) -> M 9 -> M g / T3 . Then it can be shown that the ~gl-bundle, which is defined by the homomorphism 1fl (IvIt) -> T3 -> Mgl, satisfies the required conditions. D The third operation, associated to any cohomology dass u E
H2(M; Z/m) of the base space of a given ~g-bundle 1f : E
->
M,
is to construct an appropriate finite covering p : Iv! -> M such that p* (u) = O. This is not always possible as can be seen easily by considering the case where 111 = S2 for example. We need to impose certain conditions on the base space. DEFINITION 4.14. For each non-negative integer n, we define a class Cn consisting of 2n-dimensional connected Coo manifolds recursively as follows. Elements of Co are O-dimensional manifolds each of
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
163
which is a single point, and C1 is the class consisting of closed orientable surfaces of genus g 2: 2. In general, Cn + 1 is defined to be the class consisting of any finite covering of the total space of an oriented I:g-bundle with g 2: 2 whose base space belongs to Cn- We denote by C the disjoint union of all Cn and call members of it iterated surface bundles. PROPOSITION 4.15. Let E be a manifold belonging to the class Cn and let m be a positive integer. Then for any cohomology class u E H2(E; '!LIm) there exists a finite covering p : E ---+ E such that p*('u) o. PROOF. We use induction on n. The statement is clear for n = 0,1. So assume that n> 1 and let E be a manifold belonging to By definition, there exists a manifold Mo E Cn - 1 and a I:g-bundle Eo ---+ NIa over Ma such that E is a finite covering of Eo. It is easy to see that the composition E ---+ Eo ---+ Mo is a fiber bundle. Its fiber, denoted bJ'\I:, is not necessarily connected but consists of some pieces of finite coverings of :E g . 7rl(Mo) acts on the finite set 7ro(:E) of connected comnonents of :E. We take a suitable finite covering NIa ---+ Mo of Mo/which kills this action and let E' ---+ Ma be the pullback of E ---->Mo by this mapping. Clearly we have Ma E Cn - 1 · If we let Eb be a connected component of E', then from the construction the fiber of the map Eb ---+ Ma is connected so that it is a :Eg-bundle for some g 2: 2. On the other hand, since Eb is a finite covering of E, the pullback of u defines a cohomology class u' E H2(Ea; '!LIm). If we apply Lemma 4.13 here, we can conclude that there is a finite covering 1Vh ----> MIl such that the pullback :Eg-bundle El ----> Ml has an m-fold covering Ei ----> 1Vh along the fibers. Then Ei ---+ Ml is an :E g ,bundle for some :E g , which is a certain m-fold covering of :E g . 7rl(M1) acts on the finite group HI (:E g ,; '!LIm). We take a finite covering NIi ----> Nh which kills this action, and we further take another finite covering M2 ---+ Mi such that the homomorphism Hl(Mi;'!Llm) ----> H 1 (Nfz;'!Llm) is trivial. Let E2 ---+ M2 be the pullback :Eg,-bundle by the map Adz ----> MI. Summarizing our construction so far, we obtain the following commutative diagram:
en.
E2 (4.9)
1f
---->
1
Eo'
M2
---->
E'1 - .. -- '.... -+ El
1
Eg'
---->
1
1
Eg
M1=Nh
E'0 Eo
---->
M'0
164
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Let us write p : E2 ~ Eo for the composition of the three mappings in the upper row of the above diagram. We show that there is an element v E H 2(M2;'Ljm) such that p*(u') = 7f*(v). For that we consider the Serre spectral sequence {E;.,q, dr } for 'Ljm-cohomology of the fiber bundle 7f : E2 ~ M 2. The local system over N/2 defined by the 'Ljm-cohomology of the fibers is clearly trivial. Hence the E 2 term is given as E~,(j = HP(Nh; Hq(L,gl; Zjm)). p*(u') is an element of H2(E2; Zjm), and we have the following two short exact seqllences:
o ---+ K
---+
o --> EC;;}
H2(E2; Zjm)
---+
Er;;}
K --> E;~/
---+
0
-->
.-->
0
where K = Ker(H2(E2; Zjm) ~ H2(L,g'; Zjm)). Now since L,gl ~ L,g is an m-fold covering, the image of p*( u') in E~2 is O. Hence p*(u') belongs to K. On the other hand, by the construction of two finite coverings M{ ~ Nh and M2 ~ M{, the image of p* (u') in E~1 is also O. Hence from the above short exact sequence we see that p*(u') E E~2. But since E~2 = Im(H 2(M2;Zjm) ~ H 2 (E2;'Ljm)), there exists an element v E H2(Nh;Zjm) such that p*(u') = 7f*(v). Now since M2 is a finite covering of M~ E Cn - 1 , M2 also belongs to the class Cn - 1 . Therefore by the induction assumption, there is a finite covering p : M3 ~ M2 such that p* (v) = O. If we let E ~ M3 be the pullback L,gl-bundle by the map M3 ~ M 2, then the composition E - t E2 - t E is a finite covering. We can now conclude that this bundle satisfies the required condition; namely the pullback of the cohomology class u E:: H2(E; 'Ljm) to E is O. This completes the ~~ 0 With the above preparation, out of any L,g-bundle 7r : E ~ M whose total space belongs to the class C, we construct a new surface bundle as follows. To begin with, we apply the first operation; namely we consider the pullback bundle 7r:
E*
-->
E
by the projection 7r. This bundle is equipped with a canonical section s : E --+ E*, and its image D c E* is an oriented sllbmanifold of codimension 2. We denoted by Vm E H2(E*; 'Ljm) the mod m reduction of the Poincare dual of its homology class. Now we consider the
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
165
following commutative diagram consisting of various surface bundles:
E*
--7
E3
1
1
E ---
E
'B.q "
--7
'B., --7
E2
--7
1
'By'
E2
--7
'E~
1
--7
'B.,
Ei
--7
E*
1
'B.
E1 - - - E1
-----..;
E
Here the four columns from the right are the same as those of Diagram (4.9) in the proof of Proposition 4.15 where we replace Eb ~ M6 by "E* --t E and u' E H2(Eb; Z/m) by /.1m E H2(E*; Z/m), respec'tively. In particular, 'Ei -) Ei is an m-fold covering along the fibers (cf. Lemma 4.13). The image in H2(E2; Z/m) of the cohomology class /.1 m is equal to the pullback to E2 of some element v E H2(E2; Z/m). Now since E2 is a finite covering of E, it belongs to the class C. 'Hence, again by Proposition 4.15 there is a finite covering p : E --.:;\ E2 such that p* (v) = O. Therefore if we let E3 ~ E be the pullpack ~g,-bundle, then the image of /.1m in H2(E3;Z/m) is O. It y>llows that, if we denote by D* C E3 the inverse image of D by..the map E3 ~ E*, then the pair (E 3 , D*) satisfies the conditions ~f Proposition 4.10. Hence there exists an m-fold cyclic ramified covering E* --t E3 ramified along D*. The projection E* --t E has a structure of a Eg,,-bundle where E g" is an m-fold cyclic covering of E g , ramified at m points on it. In particular we have g" = mg' + ~(m2 - 3m) + 1 = m 2 g - ~m(m + 1) + 1. We call t.he above process by which we obtain a Eg,,-bundle fro!?, a given Eg-bundle E ~ M an m-construction. The total space E* of the new surface bundle clearly belongs to the class C. Hence for any positive integer m' we can apply the m'-construction on E* ~ E. We have thus proved that, starting with any surface bundle whose total space belongs to the class C, we can construct various surface bundles by applying mj-constructions successively for j = 1,2, .... The construction due to Kodaira and Atiyah mentioned in §4.3.3 is nothing but the 2-construction on the trivial Eg-bundle Eg ~ pt over a single point.
4.4.2. Non-triviality of ei. We compute the characteristic classes of surface bundles constructed in the preceding subsection. Let 7r : E --t M be a Eg-bundle such that E belongs to the class C
166
4.
CHARACTERISTIC CLASSES OF SURFACE BUNDLES
and let E* ~E* ~ E 7r
1E~ E
7r'
-->
r
1
Eg
E
7r
----->
1
Eg
M
7r
be the m-construction on it. We denote by e E H2(E; 71.,) and e E H2(E*; 71.,) the Euler classes of 7f : E ---> M and if : E* ---> E, respectively, and let D be the image of the section s : E ---> E*. Furthermore we set D = r- 1 (D) C E* and write v E H2(E*;7I.,), v E H2(E*;7I.,) for the Poincare duals of D, D, respectively. PROPOSITION 4.16. We have the following equalities. (i) r*(v) = mv and hence v = ~ r*(v). (ii) v 2 = q*(e) v = (7f')*(e) v. (iii)
e = r* ( q* (e) - (1 - ~) v).
PROOF. The Euler class of 7f' : E* ---> E is clearly q* (e). Hence (i) and (iii) follow from Proposition 4.12. On the other hand, if i : D ---> E* is the inclusion, it is easy to see that i*q*(e) = i*(7f')*(E) is equal to the Euler class of the normal bundle of D in E*. If we use the Thom isomorphism theorem, (ii) follows. D PROPOSITION 4.17. If we denote by ek, ek the k-th characteristic classes of 7f : E ---> M and if : E* ---> E, respectively, then we have
ek = m 2 f* (7f*(e k ) -
(1 - m-(k+1l)e k ).
PROOF. By Proposition 4.16, we have (4.10)
ek+1
= r*(q*(e) _ (1- ~) v)k+1 =
r*(q*(e k+1) - (1- m,-(k+1l)(7f')*(e k )v).
By applying the Gysin homomorphism
7f*: H 2 (k+1l(E*;Q)
--->
fJ2k(E;Q)
to (4.10), we obtain
ek =
m,2f*(7f*(ek) -
This completes the proof.
(1- m-(k+l l )ek ). D
NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES
4.4.
(2)
167
If the base space M of a I:;g-bundle 7r : E --> M is an oriented 2n-dimensional closed manifold, then by evaluating any polynomial in ei of degree 2n on the fundamental cycle of M, we obtain various numbers. More precisely, associated to any partition I = {iI, ... ,ir } of n, the corresponding number
is defined. These numbers are called characteristic numbers of surface bundles. Let I = {i1' ... ,i r } be a partition of some natural number. For any subset J = {jl,'" ,js} of I, we describe its complement JC = I \ I as JC = {k 1 , ... ,kt } (s + t = r). PROPOSITION 4.18. Let 7r : E --> M be an oriented I:;g-b~ndle over an oriented 2n-dimensional closed manifold 111 and let if : E* --> E be an m-construction on it.\ Then for any" partit~on I = {iI, ... ,i r } of n + 1 the characteristic n)U1nber of 7T : E* --> E associated to I is given by eI[E] = dm 2r
L( _l)t (1 - m~(kl+1)) ... j
Here d is the mapping degree of E of I.
-->
E and J runs through all subsets
PROOF. By proposition 4.17, we have
eI[E] = =
eij ...
edE]
m 2 f* (7r*(eiJ -
(1 -
m~(il+!))eil) ...
m 2 f*Cn'*(e;J -
(1- m~(ir+I))eir)[E]
= dm 2 (7r*(eil) - (1- m~(il+l))eil) ... (7r*(eiJ - (1 - m-(tr+l))eir)[El 1'
= dm?1'L(-1)t(1-m-(k 1 +l)) ... j
(1 -
m-(k'+!))ejek1+.+k,_1 [M].
The last equality follows from §4.2.3, Proposition 4.8 (ii).
0
In particular, it follows from the above proposition that if en[Ml is non-zero, then en+! [E] is also non-zero for any m > 1. Hence by
168
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
induction we can conclude that all the characteristic classes non-trivial.
ei
are
4.4.3. Algebraic independence of characteristic classes of surface bundles. In view of the description of §4.1.5, the characteristic classes ei of surface bundles can be regarded as cohomology classes of the mapping class group. Namely we can write ei
E H2i(Mg; Z)
(i = 1,2,··· ).
In this subsection, we give an outline of the proof of the following theorem which shows the algebraic independence of these characteristic classes. This theorem was proved by Miller [Mi] and the author [Mor3] independently. However, the range of injectivity below is due to Harer's improved stability theorem [Har3] which was obtained later. THEOREM 4.19. Let n be any positive integer. Then, for any g with 9 :2 3n, the homomorphism
Q[el' e2,···]
-~
H*(Mg; Q)
is injective up to degree 2n. Sketch of Proof. If we examine the formula of Proposition 4.18, it is almost clear that there is no algebraic relations between the characteristic classes of surface bundles which are obtained by successively applying mrconstructions for appropriate mj (j = 1,2, ... ). However, the theorem does not follow from this because it is not obvious whether algebraic independence for one specific genus implies the same property for all larger genera or not. To solve this problem, we introduce mapping class groups of surfaces with base points or boundaries. First assume that a base point * E :E g is given on :Eg . In this case we denote by Diff+ (:E g , *) the group of orientation and base point preserving diffeomorphisms of :E g and set
M
g ,.
= 7fo(Diff+(:Eg, *)).
We call it the mapping class group of :E g relative to the base point. By forgetting the base point, we obtain a projection 7f : M g ,. --+ M g " Furthermore, by the assumption that g ;::: 2, it can be easily shown that there is a canonical isomorphism Ker 7f ~ 7f1 (:Eg). Thus we obtain the following short exact sequence: 1 ---7 7fl(:Eg) ~ M g,. ~ Mg
--+
1.
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
169
Here we can regard any 'Y E 1f1 (~g) to be an element of M g,. as follows. Choose a closed curve e based at the base point which represents 'Y- Then i(ry) is the isotopy class, keeping the base point fixed, of a diffeomorphism defined by pulling the base point along e in the reverse direction and coming back to the initial point. Next let D2 C ~g be an embedded disk and let Diff (~g, D2) be the subgroup of Diff+~g consisting of all elements which are the identity on D"2:-We set
M g ,1
= 1fo(Diff(~g,D2))
and call it the mapping class group of ~g relative to D2. It is also called the mapping class woup of the surface ~g \ IntD2 relative to the boundary. By taking t\he base p¢nt * on D2, a natural projection 1f : M g ,1 --> M g ,. is definel:LJt.cin be shown that the kernel of this projection is isomorphic to an infinite cyclic group generated by the isotopy cla.ss, keeping D2 fixed, of a diffeomorphism which rotates D2 by 360 degrees. Thus we obtain the following short exact sequence:
o ----+ Z ----+ M g, 1
----+
M g, *
----+
L
We can deduce from the result of Earle-Eells [EE] that BDiff+(~g, *)
= K(M g,., 1),
BDiff(~g, D2)
= K(Mg,l, 1).
BDiff+ (~g, *) is the classifying space of surface bundles equipped with a section, and BDiff(~g, D2) serves as the classifying space of surface bundles with a section together with a trivialization of the normal bundle of its image. Now if we regard Diff (~g, D2) as the group of diffeomorphisrns of 1: y \IntD2 which restrict to the identity on the boundary, then any element of it defines an associated diffeomorphism of ~g+l which is the identity on the last handle. This operation induces a homomorphism
M g ,1
----+
M g +1,1.
Hence if we are given a surface bundle 1f : E --> M whose holonomy group (or monodromy group) can be lifted to Mg,l, we can modify 1f to obtain another surface bundle the genus of whose fiber is greater than the original by one (and hence any number). More precisely, we can perform this operation as follows. By the assumption, a section s : lvI -, E and a diffeomorphism between a closed tubular neighborhood N(Ims) of its image and M x D2 are given. We now set To = T2 \ Int D2 and E = (E\IntN(Ims)) uM x To.
170
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Here the right hand side represents the manifold which is obtained by pasting the two pieces along their boundaries by the identification 8N(Im s) ~ M X 8 1 = 8(M x To). Then the natural projection 'IT : E -+ 1\1/ becomes a surface bundle whose genus goes up by one and whose holonomy group is contained in M g + 1 ,1. This operation can be generalized as follows. Suppose that there are given k genera 91, ... ,9k with L 9j ::; g. In this case, once we fix an embedding Uj E.g] \ Int D2 C E.g of disjoint union of E gj \ Iut D2 into Eg, we obtain a homomorphism
(4.11)
L : A1 yl ,1
X ... X
Mgk,l
----->
Mg,l.
Under such a situation, it is not hard to see that the characteristic classes behave as follows. Namely, with respect to the homomorphism
L* : H*(M g,l;Z) induced by
L
----->
H*(Mg1,1 x ...
X
Mgk,l;Z)
in cohomology, we have k
(4.12)
L*(ei)
= 2:Pj(e'i)
(i
= 1,2",,),
j=l
Here Pj : M gl ,l X ... X Mg"l --+ Mgj,l is the projection to the ,i-th factor. With the above preparation, we can prove the theorem as follows. In the preceding subsection, we showed ei =f. 0 for any i by computing the characteristic classes of iterated surface bundles which are obtained by applying the m-constructions successively. We can enhance this construction to obtain surface bundles which have sections with trivial normal bundles such that Ci =f. 0 in the rational cohomology group. The holonomy group of such bundles lies in Mg,l for some g. Hence we can conclude that Ci =f. 0 E H 2i (M g ,1;Q). It then follows from equation (4.12) that the same fact holds for any genus ~ g. We can now prove that the homomorphism in the statement of the theorem is injective up to degree 2n as follows. For each Cj U = 1, ... ,n) choose gj U = 1,,,' ,n) such that ej is non-zero in H*(M gj ,l; Q). Next, for each j choose d j such that jdj ~ n and set 9 =~ L j djg j . Then we obtain a homomorphism
L: (M g1 ,1)d1 x ... x (M gn ,l)d"
----->
Mg,l
similarly to (4.11). If we use (4.12) here again and the theorem of Kiinneth, we see that the characteristic classes are algebraically
4.5. APPLICATIONS OF CHARACTERISTIC CLASSES
171
independent in H*(Mg,liQ) up to degree 2n. This completes the ~~
0
The above is an outline of the proof of Theorem 4.19 given in the author's paper ~or31. The proof in [Mi] is basically the same except for the fact tnatit uses the following fundamental result of Harer [Har2] to show the existence of 9 with ei # 0 E H 2i (M g ,li Q) without giving explicit surface bundles whose holonomy groups lie in Mg,l. It is called Harer's stability theorem which claims that the natural homomorphisms
H*(Mg,liZ)
-+
H*(MgiZ),
H*(Mg+1,liZ)
-+
H*(Mg,liZ)
are isomorphisms in a certain stable range (up to degree approximately t g). It follows that the stable cohomology algebra of the mapping class group lim H*(Mg; Q) g->oo
is defined. Later it was ~j?)YHarer [Har3] that the stable range can be doubled for the rational cohomology. As mentioned already, the range of injectivity in the theorem is due to this improved result. Using the theorem of Milnor-Moore regarding the structure of Hopf algebras, Miller [Mi] further proved that the above stable cohomology algebra is isomorphic to the tensor product of polynomial algebra on primitive elements of even degrees and exterior algebra on primitive elements of odd degrees. 4.5. Applications of characteristic classes 4.5.1. The Nielsen realization problem. The mapping class group Mg has many finite subgroups. For any fixed g, it is an interesting problem to classify conjugacy classes of finite subgroups of Mg and also to give upper bounds of orders of them. By the way, there was a classical problem concerning finite subgroups of mapping class groups called the Nielsen realization problem which can be described as follows. Let 7r : Diff+Eg -+ Mg be the natural projection. A finite subgroup G C Mg is called realizable as a transformation group of Eg if it can be lifted with respect to the projection 7r, namely if there exists a subgroup G C Diff+Eg such that the restriction of 7r to G gives rise to an isomorphism 7r : G ~ G. Now the above problem is the question Is it true that any finite subgroup of Mg is realizable?
172
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
and it originates from the works of Nielsen in the 1940's. Nielsen himself gave an affirmative answer to this problem for the cases where G is cyclic, and there have been obtained many results for various other cases. Finally Kerckhoff [Ke] solved it for general cases affirmatively. The above problem is meaningful also for infinite subgroups of Mg. In the case where 9 = 1, namely for the torus, if we take T2 = ]R2/'l} as its model, then it is easy to construct a right inverse homomorphism to the projection 7T:
Diff+T2
----+
M1 ~ 8L(2,Z).
More precisely, by letting 8L(2, Z) act on T2 linearly, we obtain a homomorphism s : 8L(2, Z) --> Diff+T2 which clearly satisfies 7T08 = id. It follows that any subgroup of M1 is realizable as a diffeomorphism group of T2. There arises naturally the question whether the same thing is true for the cases g ~ 2 or not. In the next subsection, we give a negative answer to this problem by using the characteristic classes of surface bundles. 4.5.2. The Nielsen realization problem for infinite subgroups. We prove the following theorem. THEOREM 4.20 ([Mor3]). Let 7T : DifJ'+L.g --> Mg be the natural projection and let 7T* : H*(Mg;Q) --> H*(Diff!L.g;Q) be the induced homomorphism where Diff!L.g denotes the group DifJ+"Eg equipped with the discrete topology. Then for any i ~ 3, we have 7T*(ei)
= O.
On the other hand, we know from Harer's improved stability theorem [Har3] together with Faber's work [F2] that e3 i- 0 in H6(M g ; Q) for all g ~ 5. Hence as a corollary to the above Theorem 4.20, we obtain the following result. THEOREM 4.21. The natuml projection 7T : Diff+L.g not have a right inverse homomorphism for all 9 ~ 5.
-->
Mg does
This can be shown as follows. Assume that there is a homomorphism s : M.9 --> Diff+"E g such that 7T 0 8 = id. Then we have s*1r*(e3) = e3 i- O. But this contradicts the conclusion 7T*(e3) = 0 of Theorem 4.20.
4.5. APPLICATIONS OF CHARACTERISTIC CLASSES
173
Proof of Theorem 4.20. Suppose that there is a homomorphism s : Mg ---. Diff+Eg which is a right inverse to the projection 7r. For any Eg-bundle 7r: E ---+ M, consider the composition of s and the holonomy homomorphism p : 7rl(M) ---. Mg sop: 11"1 (M) ---+ Diff+Eg. Then we can conclude that 11" : E ~ !vI has a structure of a foliated bundle with fiber E.g (cf. §3.1, Example 3.5 and §2.4, Definition 2.26). In other words, on the total space E there is a foliation F of codirnension 2 such that any leaf of F is transverse to the fibers of 11". Clearly the normal bundle u(F) of:F can be naturally identified with the tangent bundle ~ alor(g'the fibers of 11". If we apply here the Bott vanishing theorem (§3.4.\Theorem 3.24), then we have
pi(//(F)) == ptr6 = o. On the other hand, we have Pl(~)
= X2(~) = e2 so
that
e4 = 0 E H 8 (E; «:)1). It follows that ei = 0 for all i ~ 3. Since this holds for any foliated Eg-bundle, we can conclude 11"* (ei) = O. This completes the proof. 0 It is well known that the mapping class group is also isomorphic to the group 1I"0(Homeo+Eg) of connected components where Homeo+E g denotes the group of orientation preserving homeomorphisms of E g • Henee we have a projection
7r : Homeo+Eg
---+
Mg.
Similar to the above ease of diffeomorphisms, we can also ask whether there exists a right inverse homomorphism to this projection or not. However, since the Bott vanishing theorem does not hold for CO foliations, our proof above is not applicable anymore. III Fad, Thurston [Th2] proved that the projection 11" induces an isoUlorphism 11"* :
H*(HomeotE g ) ~ H*(M y)
in homology. Hence there is no cohomologicnJ oh:4rllctions to the existence of a right inverse homomorphism. Nevert.lu,jpss I.he following conjecture seems to be reasonable. CONJECTURE 4.22. The natural projection "lr : /lomeo+Eg /V/ 9 does not have a right inverse homomorphism..
---+
174
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
In other words, we expect that M 9 would not act naturally on by homeomorphisms. In contrast with this, Cheeger and Gromov constructed a natural action of Mg on the unit tangent bundle T1~g of ~9 by homeomorphisms such that the following diagram is commutative: ~g
Mg C O\lt7fl(~g)
+ - - Out7fl(T1~g)
Here the homomorphism in the lower row is induced by the isomorphism 7f1 (Tl ~g) /Z ~ 7fl (~g) where Z C 7f1 (Tl ~9) is the center of 7f1(Tl~9)'
Directions and Problems for Future Research Here we summarize th, material in each chapter treated in this book, and based on them ws:ention a few open problems which are left for future research. In Chapter 1, we describe t_@J>asic part of the de Rham homotopy theory following Sullivan's original paper [Su3] and the book [GM] of Griffiths and Morgan. The fundamental theorem of this theory is that the minimal model of the de Rham complex of a simply connected manifold is equivalent to the real homotopy type of it. Also in the case of non-simply connected manifolds, the I-minimal model of the de Rham complex is naturally isomorphic to the Malcev completion of the fundamental group tensored with lR. This theory is closely related to Quillen's rational homotopy theory [Q] as well as Chen's theory [eh] of iterated integrals which were introduced in nearly the same decade. Since the beginning, these theories have been developed from various points of view until the present time. Of particular importance is the refinement of them to the cases of Kahler manifolds or algebraic varieties. Already in [DGSM], there was given an important application of this theory to the topology of compact Kahler manifolds. Through the works of Morgan [Mol and Hain [Hal] the refined rational homotopy theory is still making progress. On the other hand, some of the problems presented in the original paper [Su3] remain open_ In particular, it is expected that one can construct a theory which could obtain more information about manifolds than the nilpotent ones using cohomology of local systems on them. In Chapter 2, we described characteristic classes of fiat bundles with finite dimensional Lie groups as fibers in terms of cohomology of the corresponding Lie algebras. We then introduced the theory 175
176
DIRECTIONS AND PROBLEMS FOR FUTURE RESEARCH
of Gel'fand and Fuks from the viewpoint of characteristic classes of flat bundles whose structure group is the diffeomorphism group Diff F of a given manifold F, namely foliated F-bundles. Dupont's lecture note [Dul] contains compact and clear descriptions starting from the Chern-Weil theory reaching to characteristic classes of flat bundles. The theory of characteristic classes of flat bundles has been developed continuously from various points of view. In particular, several deep results have been obtained in connection with algebraic geometry or number theory, and it is very likely that this situation will continue in the future. We refer the reader to papers [BI], [Du2], [R], [BE], [DHZ] and references therein. In Chapter 3, we described the general theory of characteristic classes of foliations. Roughly speaking, there are two approaches to this theory: one is from the viewpoint of the Chern-Weil theory and the other is from that of the Gel'fand-Fuks cohomology theory. In this book, we adopted the latter following [Bot2], [BH], [H]. In the 1970's, many results were obtained concerning non-triviality of characteristic classes of foliations. However, there are still many problems that remain unsolved. Most of the results obtained so far depend on classical theory concerning Lie groups and geometry of homogeneous spaces. It is expected that one could construct essentially new examples or discover new phenomena that go beyond the known ones. The problem of determining whether the discontinuous invariants induced by characteristic classes with real coefficients, described in §3.5, is in general a completely open problem not only for characteristic classes of foliations but also for characteristic classes of flat bundles treated in Chapter 3. The only known results are those given in [Mor2] and Tsuboi's result in [Ts], which solves this problem for the case of piecewise linear foliations of codimension 1. The most interesting cases include the Godbillon-Vey class of Coo or real analytic foliations of codirnension 1 and also the Cheeger-Chern-Simons classes of flat GL(n, C)-bundles. However, we n