Ÿ 
Cech­De Rham theory for leaf spaces of foliations \Lambda 
Marius Crainic and Ieke Moerdijk 
Introduction 
This paper is concerned with characteristic classes in the cohomology of leaf spaces of 
foliations. For a manifold M equipped with a foliation F it is well­known that the 
coarse (naive) leaf space M=F , obtained from M by identifying each leaf to a point, 
contains very little information. In the literature, various models for a finer leaf space 
M=F are used for defining its cohomology. For example, one considers the cohomology 
of the classifying space of the foliation [2, 13, 17, 22], the sheaf cohomology of its holon­ 
omy groupoid [10, 18, 26], or the cyclic cohomology of its convolution algebra [7, 8]. 
Each of these methods has considerable drawbacks. E.g. they all involve non­Hausdorff 
spaces in an essential way. More specifically, the classifying space, which is probably 
the most common model for the ``fine'' leaf space, is a space which in general is infi­ 
nite dimensional and non­Hausdorff, it is not a CW­complex, and it has lost all the 
smooth structure of the original foliation. In particular, it is not suitable for construct­ 
ing cohomology theories with compact support. For this reason, the construction of 
characteristic classes in the cohomology of the classifying space of the foliation proceeds 
in a very indirect way, and many of the standard geometrical constructions have to be 
replaced by or supplied with abstract non­trivial arguments. The same applies to the 
construction of ``universal'' characteristic classes in the cohomology of the classifying 
space of the Haefliger groupoid 
over foliations by explicit geometrical methods [2, 20], 
but these classes are constructed in the cohomology of the manifold M rather than that 
of the leaf space M=F . 
The purpose of this paper is to present a `` Ÿ 
Cech­De Rham'' model for the cohomol­ 
ogy of leaf spaces (Section 2), which circumvents the problems mentioned above. This 
Ÿ 
Cech­De Rham model lends itself to the construction of (known) characteristic classes, 
now by explicit geometrical constructions which are immediate extensions of the stan­ 
dard constructions for manifolds (Section 3). As a consequence, for any transversal 
principal bundle over a foliated manifold (M;F ), we are able to lift the characteristic 
classes constructed in H \Lambda (M) by the methods of [21], to the Ÿ 
Cech­De Rham cohomol­ 
ogy H \Lambda (M=F ), and establish all the relations, such as the Bott vanishing theorem, at 
the level of H \Lambda (M=F) (see Theorem 2 below). 
We want to emphasize that the construction of the Ÿ 
Cech­De Rham model and of 
the characteristic classes makes no reference to (holonomy) groupoids or classifying 
\Lambda Research supported by NWO 
1 

2 
spaces. In particular, there are no non­Hausdorffness problems, and these construc­ 
tions can be understood by anyone having some background in differential geometry, 
including familiarity with the very basic definitions concerning foliations. 
To prove that our Ÿ 
Cech­De Rham model gives in fact the same cohomology as 
the other models (Theorem 1), we use 'etale groupoids (Section 4). In fact, our model 
and the associated method for constructing characteristic classes applies to any 'etale 
groupoid, not just to holonomy groupoids (see Theorem 3, and 4.6). In particular, 
when used in the context of the Haefliger groupoid 
characteristic classes (as a map from Gelfand­ 
Fuchs cohomology into the cohomology of B 
monodromy groupoids of foliations. Our 
methods also show that the characteristic classes of foliated bundles [21] actually live in 
the cohomology of the monodromy groupoid of the foliation, rather in the cohomology 
of M itself. 
Our Ÿ 
Cech­De Rham cohomology also has a natural version with compact supports, 
which is related to the one with arbitrary supports by an obvious duality. When passing 
to the cohomology of holonomy groupoids, this duality becomes the Poincar'e duality 
of [10] (Proposition 3). This new proof of Poincar'e duality for leaf spaces appears as a 
straightforward extension of the standard arguments [4] from manifolds to leaf spaces. 
Moreover, this duality extends the known one for basic cohomology of Riemannian fo­ 
liations [30]. 
There are several other cohomology theories associated to foliations which are easier 
to describe and are perhaps more familiar, such as basic cohomology (see e.g. [16, 30]) 
and foliated cohomology (see e.g. [1, 19, 20, 29]). In the last two sections of our pa­ 
per, we use our Ÿ 
Cech­De Rham model to explicitly describe the relations between the 
cohomology of leaf spaces and the basic and foliated cohomology. 
1 Transverse structures on foliations 
In this section we recall some basic notions concerning the transverse structures of 
foliations, which formalize the idea of structures over the leaf space. Throughout, we 
will work in the smooth context. 
1.1 Holonomy Let M be a manifold of dimension n, equipped with a foliation F of 
codimension q. A transversal section of F is an embedded q­dimensional submanifold 
U ae M which is everywhere transverse to the leaves. Recall that if ff is a path between 
two points x and y on the same leaf, and if U and V are transversal sections through 
x and y, then ff defines a transport along the leaves from a neighborhood of x in U to 
a neighborhood of y in V , hence a germ of a diffeomorphism hol(ff) : (U; x) 
. Embeddings of this form will be called holonomy 

3 
embeddings. Note that composition of paths also induces an operation of composition 
on those holonomy embeddings. (In section 4 below we will present a more general 
definition of the so­called ``embedding category''). 
1.2 Transversal basis Transversal sections U through x as above should be thought 
of as neighborhoods of the leaf through x in the leaf space. This motivates the defini­ 
tion of a transversal basis for (M;F) as a family U of transversal sections U ae M with 
the property that, if V is any transversal section through a given point y 2 M , there 
exists a holonomy embedding h : U ,! V with U 2 U and y 2 h(U ). 
Typically, a transversal section is a q­disk given by a chart for the foliation. Ac­ 
cordingly, we can construct a transversal basis U out of a basis ~ 
U of M by domains of 
foliation charts OE U : ~ 
U ~ 
plaque in ~ 
U through x is contained in the plaque in ~ 
V through 
h U;V (x). 
1.3 Transversal bundles Let G be a Lie group and let ß : P 
ers of ß and mapped by ß to those of F . The vectors tangent to ~ 
F define a flat 
partial connection on P . In particular, any path ff in a leaf L from x to y defines a 
parallel transport P x 
associated principal 
GL r ­bundle is foliated (transversal). By the usual relation between Cartan­Ehresmann 
connections and Koszul connections, we see that a foliated vector bundle is a vector 
bundle E over M endowed with a ``flat F­connection'', i.e. an operator 
r : 
[X;Y ] 
= [rX ; r Y ], for all X;Y 2 
which is functorial 
in h. We will usually just write h : P j U 
It is a transversal bundle by the very definition of (linear) holonomy. 
1.4 Transversal sheaves Analogous definitions apply to sheaves. A sheaf A on M 
is called foliated if its restriction to each leaf is locally constant. Thus, (the homotopy 
class of) a path ff from x to y in a leaf L induces an isomorphism between stalks 
ff \Lambda : A x 
y) in the notations above. 
If A is a transversal sheaf, any holonomy embedding h : U ,! V gives a well­defined 
restriction h \Lambda : 
4 
An example of a transversal sheaf is the 
sheaf\Omega 0 
bas of smooth functions which are 
locally constant along the leaves. One similarly has the transversal 
sheaves\Omega k 
bas of 
germs of basic differential k­forms. More generally, any foliated vector bundle E in­ 
duces a foliated sheaf E) the space 
of sections of E which are r­constant. Over M , 
it is foliated. 
Clearly 
by O. When restricted to a transversal open U , 
2 The transversal Ÿ 
Cech­De Rham complex 
Let (M;F) be a foliated manifold and let U be a transversal basis. Consider the double 
complex which in bi­degree k; l is the vector space 
C k;l = Ÿ 
C k (U 
;\Omega l ) = 
Y 
U 0 
h 1 
of differential k­forms on U 0 . 
For elements ! 2 C k;l , we denote its components by !(h 1 ; : : : ; h k ) 
2\Omega k (U 0 ). The 
vertical differential C k;l 
h k+1 ) = 
8 ! 
: 
h \Lambda 
1 !(h 2 ; : : : ; h k+1 ) if i = 0 
!(h 1 ; : : : ; h i+1 h i ; : : : ; h k+1 ) if 0 ! i ! k + 1 
!(h 1 ; : : : ; h k ) if i = k + 1 
(1) 
This double complex is actually a bigraded differential algebra, with the usual product 
(! \Delta j)(h 1 ; : : : ; h k+k 0 ) = ( 
will also write Ÿ 
C(U ; \Omega\Gamma for the associated total 
complex, and refer to it as the Ÿ 
Cech­De Rham complex of the foliation. The associated 
cohomology is denoted 
Ÿ 
H \Lambda 
U (M=F); 
and referred to as the Ÿ 
Cech­De Rham cohomology of the leaf space M=F , w.r.t. the 
cover U . 
Note that, when F is the codimension n foliation by points, then U is a basis for 
the topology of M , and C k;l is the usual Ÿ 
Cech­De Rham complex [4]. Thus in this case 
Ÿ 
H \Lambda 
U (M=F) = H \Lambda (M) is the usual De Rham cohomology of M . 
In general, choosing a transversal basis U and a basis ~ 
U of M as in 1.2, there is an 
obvious map of double complexes C k;l (U) 
R) 
; (2) 

5 
which should be thought of as the pull­back along the ``quotient map'' ß : M 
interpreted as a Ÿ 
Cech­De Rham theorem for leaf spaces. 
Theorem 1 There is a natural isomorphism 
Ÿ 
H \Lambda 
U (M=F) ¸ = H \Lambda (BHol(M;F ); R) ; 
between the Ÿ 
Cech­De Rham cohomology and the cohomology of the classifying space. In 
particular, the left hand side is independent of the choice of a transversal basis U . 
For the proof of this theorem, we choose a complete transversal section T which 
contains every U 2 U , and we consider the ``reduced holonomy groupoid'' Hol T (M;F ), 
defined as the restriction of Hol(M;F) to T . We may assume that U is a basis for 
the topology of T . By a well known Morita­invariance argument, the classifying spaces 
BHol(M;F) and BHol T (M;F) are weakly homotopyc equivalent. The advantage 
of passing to a complete transversal is that Hol T (M;F) becomes an 'etale groupoid 
(see section 4 for the precise definitions). For such groupoids G there is a standard 
cohomology H \Lambda (G; 
from the following proposition, which is a particular case of the Theorem 3 
below. 
Proposition 1 For any complete transversal T and any basis U of T , there is a natural 
isomorphism 
Ÿ 
H \Lambda 
U (M=F) ¸ = H \Lambda (Hol T (M;F ); R) : 
We mention here that there are several variations of Theorem 1. For instance, for 
any transversal sheaf A there is a Ÿ 
Cech complex Ÿ 
C(U ; A). In degree k, 
Ÿ 
C k (U ; A) = 
Y 
U 0 
h 1 
boundary ffi = 
P ( 
y groupoid) 
with coefficients in a sheaf ~ 
A naturally associated to A. 
Another variation applies to the cohomology with compact supports (see section 4). 
Note that all these are actually extensions of the usual `` Ÿ 
Cech­De Rham isomorphisms'' 
[4] from manifolds to leaf space. Accordingly, an immediate consequence will be the 
Poincar'e duality for leaf spaces (see Section 6), which is one of the main results of [10]. 
With Theorem 1 and its analogue for compact supports available, the new proof of 
Poincar'e duality is this time a rather straightforward extension of the classical proof 
[4] from manifolds to leaf spaces. 

6 
3 The transversal Chern­Weil map 
To illustrate the usefulness of the transversal Ÿ 
Cech­De Rham complex we will adapt 
the standard geometric construction of characteristic classes of principal bundles to 
transversal bundles, so as to obtain explicit classes in this complex. We will use the 
Weil­complex formulation, which we recall first (for an extensive exposition, see [21, 
12]). 
3.1 Classical Chern­Weil: [5] Recall that the Weil algebra of the Lie algebra g (of 
a Lie group G) is the algebra 
W (g) = S(g \Lambda 
)\Omega \Lambda(g \Lambda ): 
It is a graded commutative dga (graded as W (g) n = \Phi 2p+q=n S p (g \Lambda 
)\Omega \Lambda q (g \Lambda )), equipped 
with operations i X and LX (linear in X 2 g) which satisfy the usual Cartan identities. 
In the language of [21], this means that W (g) is a g­dga. If P is a principal G­ 
bundle over a manifold M , the 
algebra\Omega \Lambda (P ) of differential forms on P with its usual 
operations i X and LX is another example of a g­dga. A connection r on P is uniquely 
determined by its connection form ! 
2\Omega 1 (P 
)\Omega g. This can be viewed as a map 
! : W (g) 1 = g 
\Lambda 
= d! + 1 
2 
[!; !].) The restric­ 
tion of this map (3) to basic elements (elements annihilated by i X and G­invariant) 
gives a map of dga's 
S(g \Lambda ) G 
map for the principal G­bundle P . Because of the 2p in the 
grading of the Weil algebra, k(r) maps invariant polynomials of degree p to degree 
2p cohomology classes. Moreover, k(r) does not depend on r. This follows from the 
Chern­Simons construction (see below). A more refined characteristic map is obtained 
if one uses a maximal compact subgroup K of G. Since P=K 
of K­basic elements, one obtains a characteristic map 
H \Lambda (W (g; K)) 
principal bundle \Delta k \Theta P over \Delta k \Theta M , where \Delta k = 
f(t 0 ; : : : ; t k ) : t i – 0; 
P t i = 1g is the standard k­simplex. We define 
~ k(r 0 ; : : : ; r k ) = ( 
k 
~ k(r) : W (g) 
7 
(i) the map (6) commutes with the action of G, and with the operators i X , LX , and it 
vanishes on all elements 
ff\Omega fi 2 W (g) with ff a polynomial of degree ? dim(M ). 
(ii) 
[ ~ k(r 0 ; : : : ; r k ); d] = 
k 
X 
i=0 
( 
Proof: (ii) is just a version of Stokes' formula (see also [2]), while (iii) is obvious. 
We prove the vanishing result of (i). Denote by d the degree of the polynomial ff and 
by q the dimension of M . We prove that when d ! k or 2d ? q + k, our expression 
` = ~ k(r 0 ; : : : ; r k 
)(ff\Omega fi) 
vanishes (note that if d ? q, then at least one of these two equalities holds). First 
assume that d ! k. We have ~ 
k(r)(ff\Omega fi) = ff(\Omega\Gamma “ fi(!), where r is the affine 
combination (5), ! is the associated 1­form, 
and\Omega is its curvature. Let us say that a 
homogeneous form fdt i 1 
: : : dt i r 
dx j 1 
: : : dx js on \Delta k \Theta P has bi­degree (r; s). Since ! has 
bi­degree (0; 
1),\Omega is a sum of forms of bi­degree (1; 1) and (0; 2), so 
R 
\Delta k 
ff(\Omega\Gamma “fi(!) = 0 
because no bi­degree (r; s) with r = k will be involved. 
We now turn to the case 2d ? q + k. Let l be the degree of fi. Because of the similar 
property for fi, we have i X 1 : : : i X l+1 
` = 0 for any vertical vector fields X i . On the other 
hand, i Y 1 
: : : i Y q+1 
` = 0 for any horizontal vector fields Y i . Since deg(`) = 2d + l 
k (U 
;\Omega l (P )) = 
Y 
U 0 
h 1 
the maps h 1 : P j U 0 
holonomy embeddings h : U 
1 
g) 
U 0 
) : (9) 

8 
This map respects the total degree, and it is obviously compatible with the operations 
i X and the G­action. So, by restricting to basic elements it yields a map into the 
transversal Ÿ 
Cech­De Rham complex 
~ k(!) : S(g \Lambda ) G 
Theorem 2 The Chern­Weil map of a transversal principal G­bundle P over (M;F) 
has the following properties: 
(i) The maps (9) and (10) respect the differential, hence they induce a map 
k P := k(r) : S(g \Lambda ) G 
Ÿ 
H \Lambda 
U (M=F) 
of F . 
The classical Bott vanishing theorem [2] (for the normal bundle of the foliation) 
and its extensions to foliated bundles [21] are at the level of H \Lambda (M ). The point of The­ 
orem 2 is that, using classical geometrical arguments, one can prove these vanishing 
results and construct the resulting characteristic classes at the level of the leaf space, 
i.e. in the cohomology of the classifying space (cf. Theorem 1). 
Proof of Theorem 2: (i) and (iv) clearly follow from the main properties of the 
Chern­Simons construction 3.2. Also (iii) will follow from the independence of the 
connections. Indeed, it suffices to check that, if F is the foliation by points, then the 
resulting map kr : S(g \Lambda ) G 
map. But this is clear even at the chain level, provided we choose rU = rjU 
for some globally defined connection r. 
(ii) For two different choices r = frU g and r 0 = frU 0 g of connections, the map 
H : W (g) 
; : : : ; h k ) = 
k 
X 
i=0 
( 2 h 1 ; : : : ; r h k :::h 2 h 1 ; r 0 
U 0 ; r 0 
h 1 ; : : : ; r 0 
h i :::h 2 h 1 )(w) : 
provides an explicit chain homotopy. To prove the compatibility with the products, 
one can either proceed as in [21] using the simplicial Weil complex (see [9] for de­ 
tails), or, since the characteristic map is constructed through the double complex 
Ÿ 
C p (U 
;\Omega p+q (\Delta q \Theta P )) by integration over the simplices, one can use the simplicial De 
Rham complex and Theorem 2.14 of [13]. 

9 
3.4 Exotic characteristic classes: The vanishing result of Theorem 2 shows that 
the construction of the ``exotic'' classes also lifts to the Ÿ 
Cech­De Rham complex. To 
describe all the relevant characeristic classes, we consider the complex W (g; K) of K­ 
basic elements described in 3.1, together with its q­th truncation W q (g; K) defined as 
the quotient by the ideal generated by the elements of polynomial degree ? q. By the 
vanishing result (more precisely from the proof above), the map (9) induces a chain 
map W q (g; K) G=K as in 3.1, we 
obtain the following refinement of the characteristic map of Theorem 2. 
Corollary 1 The Chern­Weil construction of 3.3 gives a well­defined algebra map 
k ex 
P := k ex (r) : H \Lambda (W q (g; K)) 
(2)), it gives the exotic characteristic map of the 
foliated bundle P [21]. 
4 The Ÿ 
Cech­De Rham complex of an 'etale groupoid 
In this section we prove Theorem 1, as well as some generalizations and variants, in 
the context of 'etale groupoids. Our general goal is to describe the (hyper­) homology 
and cohomology of 'etale groupoids in terms of the (hyper­) homology and cohomology 
of small categories. We begin by introducing some standard terminology. 
4.1 Smooth 'etale groupoids: A smooth groupoid is a groupoid G for which the 
sets G ( 0 ) and G ( 1 ) of objects and arrows have the structure of a smooth manifold, all 
the structure maps are smooth, and the source and the target maps are moreover 
submersions. The holonomy groupoid Hol(M;F) of a foliation is an example of a 
smooth groupoid. Such a smooth groupoid is said to be 'etale if the source and the 
target maps are local diffeomorphisms. In this case the manifolds G ( 0 ) and G ( 1 ) have 
the same dimension, to which we refer as the dimension of G. An example of an 
'etale groupoid of dimension q is the universal Haefliger groupoid 
equivalent to an 'etale groupoid, namely to its restriction to any complete 
transversal T , denoted Hol T (M;F ). A Morita equivalence between smooth groupoids 
induces a weak homotopy equivalence between their classifying spaces. 
4.2 Sheaves and cohomology: For a smooth 'etale groupoid G, a G­sheaf is a sheaf 
A over the space G ( 0 ) , equipped with a continuous G­action. For any such sheaf there 
are natural cohomology groups H n (G; A) whose definition we recall. Denote by G ( k ) 
the space of composable arrows 
x 0 
g 1 
: G ( k ) 
c(g 1 ; : : : ; g k ) 2 A x 0 
. The boundary is 

10 
ffi = 
P ( d'' in the sense that A and 
its pull­backs ffl \Lambda 
k A are injective sheaves, then H n (G; A) is computed by the bar com­ 
plex B(G; A). In general, one chooses a resolution S \Lambda of A by ``good'' G­sheaves, and 
H n (G; A) is computed by the double complex B k (G; S l ). In general, these cohomology 
groups coincide with the cohomology groups of the classifying space BG [26]. 
Similarly, using compact supports and direct sums in the definition of the bar com­ 
plex, one defines the homology groups H \Lambda (G; A) [10] (sometimes denoted H \Lambda 
c (G; A) = 
H 
4.3 Ÿ 
Cech complexes: Let G be an 'etale groupoid and let U be a basis of opens 
in G ( 0 ) . A G­embedding oe : U 
should define an embedding of U into V . As in the first section, we 
can now define the Ÿ 
Cech complex Ÿ 
C(U ; A) for any G­sheaf A, 
Ÿ 
C k (U ; A) = 
Y 
U 0 
duct is over all strings of G­embeddings between opens U 2 U , and the 
boundary ffi = 
P ( 
U (G; A) as the cohomology of Ÿ 
C(U ; A). In general, we define Ÿ 
H \Lambda 
U (G; A) as 
the cohomology of the double complex Ÿ 
C k (U ; S l ), where 0 
the 
choice of the resolution. 
4.4 Examples: The 
G­sheaf\Omega l 
G of l­differential forms with its natural G­action is 
always U­acyclic, as is any soft G­sheaf. We obtain the Ÿ 
Cech­De Rham (double) com­ 
plex of G, Ÿ 
C(U ; \Omega\Gamma, computing Ÿ 
H \Lambda 
U (G; R). If the basis U consists of contractible opens 
(balls), then any locally constant G­sheaf A is U­acyclic, hence Ÿ 
H \Lambda 
U (G; A) is computed 
by Ÿ 
C(U ; A). 
Similarly one defines the Ÿ 
Cech complex with compact supports Ÿ 
C \Lambda 
c (U ; A) using 
M 
U 0 
U (G; A) is defined by Ÿ 
C c (U ; A). In 
general, one uses a resolution 0 
bedding category: The notion of G­embedding originates in [25], where 
the second author has introduced a small category Emb U (G) for each basis U of open 
sets. The objects of Emb U (G) are the members U of U , and the arrows are the G­ 
embeddings between the opens of U . The main result of [25] was that the classifying 

11 
space BG is weakly homotopy equivalent to the CW­complex BEmb U (G), provided 
each of the basic opens in U is contractible. 
Now any G­sheaf A defines an obvious contravariant functor 
with coefficients. Hence 
[25] proves that H \Lambda (G; A) = Ÿ 
H \Lambda 
U (G; A) provided all the opens U 2 U are contractible 
and A is (locally) constant. We now prove a stronger `` Ÿ 
Cech­De Rham isomorphism'' 
which applies to more general coefficients, and also to compact supports. 
Theorem 3 Let G be an 'etale groupoid, and let U be a basis for G ( 0 ) 
as above. Then 
for any G­sheaf A, there are natural isomorphisms 
H n (G; A) = Ÿ 
H n 
U (G; A); H n 
c (G; A) = Ÿ 
H n 
c; U (G; A) : 
Proof: The proofs of the isomorphisms in the statement are similar, and we only 
prove the first one (an explicit proof of the second one also occurs in [9]). By comparing 
resolutions of the G­sheaf A, it suffices to find a suitable complex C(A) and explicit 
quasi­isomorphisms 
B(G; A) / 
1 
that the target of g is in U 0 . The topology on S p;q is the 
topology induced from the topology on G, 
S p;q = 
a 
U 0 
p;q ) : 
For a fixed p, the complex C p;\Lambda is a product of complexes, namely, for each string 
U 0 
this cohomology is H \Lambda (U 0 ; A). Since A is assumed to be 
good, H \Lambda (U 0 ; A) vanishes in positive degrees, and we conclude that the canonical map 
Ÿ 
C p (U ; A) 
stalk of 
(ß p;q ) \Lambda (A p;q ) at x 0 
12 
where the colimit is taken over all basic open neighborhoods U of x q . For a fixed U , 
the complex inside the lim in (15) computes the cohomology of the (discrete) comma 
category U=Emb U (G) with coefficients in the constant group 
( q ) . Thus the natural map 
B q (G; A) = 
following immediate consequence, which is an improvement of the result 
of [25]. 
Corollary 2 If G is an 'etale groupoid, ~ 
U is a basis of opens of G ( 0 ) 
, and A is a G­sheaf 
with the property that H k (U; Aj U ) = 0 for all U 2 U , k – 1, then 
H \Lambda (G; A) ¸ = H \Lambda (Emb U (G); 
U 2 U is contractible, and A is locally constant as a sheaf on G ( 0 ) 
, then 
H \Lambda (G; A) ¸ = H \Lambda (Emb U (G); 
groupoids: Clearly all the constructions of Section 3 
apply to any 'etale groupoid G, provided we use the Ÿ 
Cech­De Rham complexes mentioned 
in 4.4. Hence, for any principal G­bundle P endowed with a smooth action of G, one 
has an associated Chern­Weil map 
S(g \Lambda ) G 
= dim(G). The refined characteristic 
map, 
H \Lambda (W d (g; K)) 
, 
k G : H \Lambda (WO d ) 
13 
When G = Hol T (M;F) this is the map discussed in section 3. But this is not 
the only interesting example. For instance, if one works with foliated bundles which 
are not necessarily transversal (as e.g. in [21]), then one has to replace the holonomy 
groupoid Hol T (M;F) by the monodromy groupoid Mon T (M;F ). The new versions 
of Theorem 2 and Corollary 1 for foliated bundles then yield characteristic classes 
in H \Lambda (BMon T (M;F)). These classes are refinements of the characteristic classes in 
H \Lambda (M ), already constructed in [21]. 
Another interesting example is when G is Haefliger's 
5 Explicit formulas 
In this section we illustrate our constructions in the case of normal bundles. In partic­ 
ular we deduce Bott's formulas for cocycles associated to group actions [3], as well as 
Thurston's formula. 
5.1 Explicit formulas for the normal bundle: We now apply the construction of 
the exotic characteristic map of Section 3 to the normal bundle š. Corollary 1 applied 
to the (principal GL q ­bundle associated to) š provides us with a characteristic map 
kF : H \Lambda (WO q ) 
standard [2] simplification of 
the truncated relative Weil complex W q (gl q ; O(q)) that we now recall. The idea is 
that the relative Weil complex W (gl q ; O(q)) (see 3.1) is quasi­isomorphic to a smaller 
subcomplex, namely the dg algebra S[c 1 ; : : : ; c q 
]\Omega E(h 1 ; h 3 ; : : : ; h 2[ q+1 
2 ] 
d(c i ) = 0; d(h 2i+1 ) = c 2i+1 : 
Truncating by polynomials of degree ? q, the resulting inclusion 
WO q := S q [c 1 ; : : : ; c q 
]\Omega E(h 1 ; h 3 ; : : : ; h 2[ q+1 
2 ] 
to [14] for the complete 
description of H \Lambda (WO q ), we recall here that the simplest such class is the Godbillon­ 
Vey class gv = [h 1 c q 
1 ] 2 H 2q+1 (WO q ). We denote by gvF 2 Ÿ 
H \Lambda 
U (M=F) the resulting 
cohomology class kF (gv). Its pull­back to H \Lambda (M) is the usual Godbillon­Vey class of F . 
More generally, the Bott­Godbillon­Vey classes gv ff = [u 1 c ff 1 : : : c ff t 
] (and their images 
gv ff 
F ) are defined for any partition ff = (ff 1 ; : : : ; ff t ) of q (i.e. q = 
P ff i ). 

14 
For explicit formulas, let us choose a basis U so that ~ 
U are also domains of trivial­ 
ization charts for š (as in 1.2). Let J h : U 
The 
corresponding r(h) are then given by the connection 1­forms: 
! h := J gl q ); 
for h : U 
Z 
t 0 +t 1 +:::+tpŸ1 
exp( (t 1 ! h 1 
+t 2 ! h 2 h 1 
+: : :+t p ! hp :::h 2 h 1 
) 2 )dt 0 dt 1 :::dt p : 
For instance, the first class C 1 = ch 1 (š) 2 Ÿ 
C \Lambda (U 
;\Omega \Lambda ) has the components 
C ( 1;1 ) 
1 (h) = T r(J 
;1 ) 
1 = 0; U ( 1;0 ) 
1 (h) = log(j det(J h ) j) 
transgresses C 1 . Computing the resulting closed cocycle U 1 C q 
1 we see that 
Corollary 3 The Godbillon­Vey class gvF 2 Ÿ 
H 2q+1 (M=F) is represented in the Ÿ 
Cech­ 
De Rham complex by the cocycle gvF living in bi­degree (q + 1; q): 
gvF (h 1 ; : : : ; h q+1 ) = log(j det(J h 1 
) j)h \Lambda 
1 T r(! h 2 
)h \Lambda 
1 h \Lambda 
2 T r(! h 3 
) : : : h \Lambda 
1 :::h \Lambda 
q T r(! h q+1 
): (17) 
Similarly, computing U 1 C ff 1 : : : C ff t 
for a partition ff = (ff 1 ; : : : ; ff t ) of q, we obtain 
the following formula, which explains Bott's definition of the cocycles associated to 
group actions [3]. 
Corollary 4 The Bott­Godbillon­Vey class gv ff 
F 2 Ÿ 
H 2q+1 (M=F) is represented in the 
Ÿ 
Cech­ DeRham complex by the closed cocycle gv ff 
F living in bi­degree (q + 1; q): 
gv ff 
F (h 1 ; : : : ; h q+1 ) = log(j det(J h 1 
) j) \Delta h \Lambda 
1 fT r[ ! h 2 
\Delta h \Lambda 
2 (! h 3 
) \Delta : : : (h( ff 1 
\Delta \Delta (h( ff 1 +ff 2 
oid. Due to its classifying 
properties, the case of the Haefliger groupoid 
dimension q foliations. We emphasize that all these properties are now 
part of the folklore on characteristic classes for foliations, but they are usually derived 
by non­trivial abstract arguments at the level of classifying spaces. 
First of all we make a slight simplification of the Ÿ 
Cech­De Rham complex of 
, we see 
that the category Emb U 
15 
and all the embeddings R q as in the 
previous sections, except that we take products only over strings 
R q oe 1 
R q . The main theorem of this section implies 
Corollary 5 The Ÿ 
Cech­De Rham complex Ÿ 
C 
we can describe the main (cohomological) universal properties of 
obvious map Ÿ 
H 
the map induced in cohomology by the classifying map M 
maps for codimension q foliations are just 
the composition of the (18)'s with a universal map 
k q : H \Lambda (WO q ) 
of the trivial connection on R q (compare to [2]). In particular, all the formulas 
of section 3 come from similar universal formulas in Ÿ 
C 
q ) is also computed by the Ÿ 
Cech (subcomplex) with 
constant coefficients 
Ÿ 
C 
R 
q oe 1 
lengthy but straightforward computation (for the details see Lemma 3.3.8 in [9]) we 
obtain: 
Lemma 1 An n­cocycle in the Cech­De Rham complex: 
u = u 0 + u 1 + : : : + u n ; u k 2 Ÿ 
C k 
n 
X 
s=0 
( 
16 
If we apply this to the Godbillon­Vey cocycle (i.e. to the formula (17) in the Ÿ 
Cech­ 
De Rham complex Ÿ 
C 
Corollary 6 The universal Godbillon­Vey class GV 2 H 3 (B 
Z oe 1 (0) 
0 
log(j oe 0 
2 (t) j) oe 00 
3 (oe 2 (t)) 
oe 0 
3 (oe 2 (t)) oe 0 
2 (t)dt: 
6 Relations to basic cohomology 
In the previous sections we have seen various models for the cohomology of the leaf 
space, all canonically isomorphic. Let us put 
H \Lambda (M=F) = H \Lambda (Hol T (M;F)); H \Lambda 
c (M=F) = H \Lambda 
c (Hol T (M;F)) : (20) 
The reader may choose one of the many models: Haefliger's model (as indicated by 
the above notations) i.e. 4.2 applied to the holonomy groupoid reduced to any com­ 
plete transversal T , the Ÿ 
Cech­De Rham model that we have described in section 2 (cf. 
Proposition 1), or the classifying­space model (cf. Theorem 1). We emphasize however 
that the last model only works for the cohomology without restriction on the supports! 
Here and in the next section we explain why these cohomology theories are suitable 
theories for the leaf space. We first compare them to the more familiar basic cohomology 
(see e.g. [16, 30]), which is a different cohomology theory for leaf spaces. 
6.1 Basic cohomology. Choosing a basis U of opens of a complete transversal T (or 
any transversal basis for F ), one 
defines\Omega k 
bas (T=F) as the cohomology of Ÿ 
C \Lambda (U 
;\Omega k ) 
in degree \Lambda = 0. This complex consists on k­forms on T which are invariant under 
holonomy, hence it does not depend on the choice of T (up to canonical isomorphisms, 
of course). The resulting cohomology is denoted H \Lambda 
bas (M=F ). There is an obvious map 
(induced by an inclusion of complexes) 
j : H \Lambda 
bas (M=F) 
in degree \Lambda = 0, 
i.e., as in [16], the quotient of \Phi 
U2U\Omega 
k 
c (U) by the span of elements of type ! 
general, the maps (21) and (22) are not isomorphisms. The basic cohomologies are 
much smaller then H \Lambda (M=F ); for instance H \Lambda 
bas (M=F) = 0 in degrees \Lambda ? q, and they 
are finite dimensional if F is riemannian and M is compact. The price to pay is the 
failure of most of the familiar properties from algebraic topology (e.g., as discussed 
below, Poincare duality and characteristic classes). However we point out that (21) 
and (22) are isomorphisms when the naive leaf space is an orbifold. This was explained 

17 
in 4.9 of [10], but the reader should think about the similar statement for actions of 
finite groups on manifolds, and the fact that the cohomology (over R) of finite groups 
is trivial. In particular (see also [27]), we have 
Proposition 2 If (M;F) is a riemannian foliation with compact leaves, then (21) and 
(22) are isomorphisms. 
Another fundamental property of our cohomologies (20) is 
Proposition 3 (Poincar'e duality) For any codimension q foliation (M;F), 
H \Lambda (M=F ; O) ¸ = H q 
oincar'e duality can be viewed as a rather straightforward extension of the classical 
proof for manifolds [4] (and can also be interpreted as the obvious duality between the 
homology and the cohomology of the discrete category Emb U (G), cf. 4.5). In contrast, 
the basic cohomologies H \Lambda 
bas (M=F) and H \Lambda 
c;bas (M=F) satisfy Poincar'e duality only in 
the riemannian case [30]. In this case these dualities are compatible via (21) and (22), 
and they coincide if the leaves are compact (see Proposition 2). 
6.2 Characteristic classes. As we have seen, one of the main features of H \Lambda (M=F) 
is that it contains the characteristic classes of the bundles over the leaf space (i.e. 
transversal bundles), and the Bott vanishing theorem and the construction of the exotic 
classes hold at this level. Regarding the groups H \Lambda 
bas (M=F ), again, they are too small 
to contain these characteristic classes. But, as before, this is not seen in the case of 
riemannian foliations. The reason is that, if F is riemannian, then the transversal metric 
induces a transversal connection, i.e. a connection which is invariant under holonomy. 
Using this type of connections in the construction the characteristic maps k š of the 
normal bundle š, we see that k š : S(gl \Lambda 
q ) inv 
we see that k š (and its exotic versions) factors through the 
basic cohomology groups. This obviously applies to general transversal bundles. In 
conclusion, 
Proposition 4 If P is a transversal principal G­bundles over (M;F) which admits a 
transversal connection then the characteristic map k P : S(g \Lambda ) G 
** 
k P 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U 
U H \Lambda 
bas (M=F) 
 
j 
H \Lambda (M=F) 

18 
6.3 Integration along the leaves. Haefliger's original motivation [16] for introduc­ 
ing H \Lambda 
c;bas (M=F) is the existence of an integration over the leaves map 
R 0 
F : H \Lambda 
c (M) 
(with the induced orientation), 
R 
:\Omega \Lambda 
c ( ~ 
U ) 
algebraic topology (``integration over the fiber'' as an 
edge map). The Hochschild­Serre spectral sequence (i.e. Theorem 4.4 of [10] applied to 
ß : M +t 
c (M ), where L t is a transversal 
sheaf whose stalk above a leaf L is H t 
c ( ~ 
L). This second description provides us with 
qualitative information. E.g., if the holonomy covers of the leaves are k­connected, we 
find that 
R 
F is isomorphism in degrees n 
7 Relations to foliated cohomology 
Another standard cohomology theory in foliation theory is the foliated cohomology of 
foliations (see e.g. [1, 19, 20, 29]). In contrast to the other cohomologies that we have 
seen so far (transversal cohomologies), the foliated cohomology contains a great deal of 
longitudinal information. In this section we describe its relation to our Ÿ 
Cech­De Rham 
cohomology. 
7.1 Foliated cohomologies. The foliated cohomology H \Lambda (F) is defined in analogy 
with the De Rham cohomology of M , which we recover if F has only one leaf. The 
defining complex 
is\Omega \Lambda (M;F) = 
i ; : : : ; “ 
X j ; : : : X p+1 )) 
+ 
p+1 
X 
i=1 
( i ; : : : ; X p+1 )): (25) 
Here LX (f) = X(f ). For later reference, we note the existence of an obvious (restriction 
to F) 
r : H \Lambda (M) 
X i 
in the previous formula, by the derivatives rX i 
w.r.t. 
the Koszul connection of E (see 1.3). 

19 
7.2 Remarks. In [29], the cohomology H \Lambda (F) is called ``tangential cohomology'', and 
is denoted H \Lambda 
ø (M ). The groups H \Lambda (F ; š) with coefficients in the normal bundle (see 
1.3) first appeared in [19] in the study of deformations of foliations, while those with 
coefficients in the exterior powers \Lambdaš show up e.g. in the spectral sequence relating 
the foliated cohomology with De Rham cohomology [1, 20]. The groups H \Lambda (F ; E) 
with general coefficients can also be viewed as an instance of algebroid cohomology 
[23]. Regarding the characteristic classes, since the Bott connection (see 1.3) is flat, it 
follows that the characteristic classes of the normal bundle are annihilated by r. This 
new vanishing result at the level of foliated cohomology produces new (``secondary'') 
classes, u 4k 
F 
;\Omega 0 
bas ) (via the map (27) below). Still 
related to [11], let us mention that if F is the foliation induced by a regular Poisson 
structure on M , then one has an induced foliated bundle K (the kernel of the anchor 
map), and H 2 (F ; K) contains obstructions to the integrability of the Poisson structure. 
As explained in [29] in the case of trivial coefficients, and in [19] in the case of 
the normal bundle as coefficients, the foliated cohomology can be expressed as the 
cohomology of certain sheaves on M . For general coefficients E we consider the sheaf 
R p \Theta R q . Since always 
For any foliated vector bundle E over (M;F), H \Lambda (F ; E) is isomorphic 
to H \Lambda (M ; 
7.3 Comparison. We now note the existence of a canonical map 
\Phi : H \Lambda (M=F 
;\Omega 0 
bas ) 
version with coefficients of the pull­back map (2) (cf. also Proposition 5). Accordingly, 
the simplest description is in terms of the Ÿ 
Cech­De Rham model. Choosing U and ~ 
U as 
in 1.2, the left hand side of (27) is computed by the cochain complex Ÿ 
C \Lambda (U ; C 1 (U)), 
which is obviously a subcomplex of the t = 0 column of Ÿ 
C s ( ~ 
U 
;\Omega t ( ~ 
U ; F ). Now (27) is 
the map induced in cohomology. Alternatively, at least when the holonomy groupoid 
is Hausdorff, H \Lambda (M=F 
;\Omega 0 
bas ) coincide with the differentiable cohomology [18] of the 
holonomy groupoid of F , and \Phi is precisely the associated Van Est map described in 
[31]. It then follows from one of the main results of [11] (applied to the holonomy 
groupoid) that \Phi is an isomorphism in degrees Ÿ k provided the leaves (or their holon­ 
omy covers) are k­connected. As in the previous section (see 6.3), the same result 
follows e.g. from the spectral sequences of [10]. 

20 
7.4 Integration along the leaves. If F is oriented, then we have an integration 
map Z 
F 
: H \Lambda 
c (F) 
This map is dual to the Van Est map (27) and can be viewed as a version of the inte­ 
gration map (24) with coefficients in the normal bundle (accordingly, there are similar 
maps for any transversal vector bundle E over M , cf. also 1.4 and Proposition 5). 
Again, as in the previous section (see 6.3), this map (28) becomes obvious if one uses 
the Ÿ 
Cech­De Rham model. 
We want to point out here that the integration over the fibers that we have de­ 
scribed clarifies the construction of the Ruelle­Sullivan current of a measured foliation 
(cf. e.g. Section 3 of [6], or [29] p 126), and also gives new qualitative information 
about it. Fix a transversal basis U for F . A smooth transverse measure ¯ is just a 
measure on each U 2 U , which is invariant w.r.t. holonomy embeddings. Hence the 
integration against ¯ is simply a linear map 
Z 
¯ 
:\Omega 0 
c;bas (M=F) = H 0 
c (M=F 
;\Omega 0 
bas ) 
arrange our 
maps into a diagram (see also (26)) 
H p 
c (M) 
// 
R 
F 
 
r 
H 0 
c (M=F) 
 
H p 
c (F) 
// 
R 
F 
H 0 
c (M=F 
;\Omega 0 
bas ) 
// 
R 
¯ 
R 
The resulting map 
R 
C 
: H p 
c (M) 
In terms of our diagram this simply means that 
R 
C 
factors through H p 
c (F ). 
7.5 Spectral sequences. Almost all of the maps that we have described in the last 
two sections figure in certain the spectral sequences. First of all, the filtration 
on\Omega \Lambda (M) 
induced by F (cf. e.g. [1, 20]) induces a spectral sequence 
E s;t 
1 = H s (F ; \Lambda t š) =) H s+t (M) : 
Similarly, the filtration of the Ÿ 
Cech­De Rham double complex induces a spectral se­ 
quence 
¯ 
E s;t 
1 
= H s (M=F 
;\Omega t 
bas ) =) H s+t (M=F) : 
Note that E 0;t 
2 = H t 
bas (M=F ). These two spectral sequences are related by the pull­ 
back map (2), and by the Van Est map (27) with coefficients, \Phi : H s (M=F 
;\Omega t ) 
involves (24) and the integrations 
R 
F : H s 
c (F ; \Lambda t š) 
21 
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