A characteristic map for the holonomy groupoid of a foliation

Abstract

We prove a generalisation of Bott's vanishing theorem for the full transverse frame holonomy groupoid of any transversely orientable foliated manifold. As a consequence we obtain a characteristic map encoding both primary and secondary characteristic classes. Previous descriptions of this characteristic map are formulated for the Morita equivalent \'{e}tale groupoid obtained via a choice of complete transversal. By working with the full holonomy groupoid we obtain novel geometric representatives of characteristic classes. In particular we give a geometric, non-\'{e}tale analogue of the codimension 1 Godbillon-Vey cyclic cocycle of Connes and Moscovici in terms of path integrals of the curvature form of a Bott connection.

Full text

A characteristic map for the holonomy groupoid of a foliation
Lachlan E. MacDonald
School of Mathematics and Applied Statistics
University of Wollongong
Northfields Ave, Wollongong, NSW, 2522
November 2019
Abstract
We prove a generalisation of Bott’s vanishing theorem for the full transverse frame holon-
omy groupoid of any transversely orientable foliated manifold. As a consequence we obtain a
characteristic map encoding both primary and secondary characteristic classes. Previous de-
scriptions of this characteristic map are formulated for the Morita equivalent ´etale groupoid
obtained via a choice of complete transversal. By working with the full holonomy groupoid
we obtain novel geometric representatives of characteristic classes. In particular we give a
geometric, non-´etale analogue of the codimension 1 Godbillon-Vey cyclic cocycle of Connes
and Moscovici in terms of path integrals of the curvature form of a Bott connection.
1 Introduction
Characteristic classes for foliation groupoids have been studied in an ´etale context in [9, 14,
20, 11, 21, 16, 31, 12, 30]. In this paper, we give analogous constructions in the context of
the non-´etale, or “full”, holonomy groupoid of a foliation. In working with such groupoids we
stay close to Bott’s Chern-Weil construction of the characteristic classes of a foliation [3], and
obtain a novel, global geometric interpretation of the Godbillon-Vey cyclic cocycle of Connes
and Moscovici [12, Proposition 19].
Associated to any (regular) foliated manifold (M, F), of codimension q, with leafwise tangent
bundle TFand normal bundle N=T M/T F, are characteristic classes associated to Nliving in
the de Rham cohomology H∗
dR(M) of M. Among these classes are the usual Pontryagin classes
for the real vector bundle N, as well as certain secondary classes, such as the Godbillon-Vey
invariant, which is tied to the dynamical behaviour of the foliation F[24].
Classically, representatives of these classes are obtained either via Gelfand-Fuks cohomology
[4] or via Chern-Weil theory [3]. The Chern-Weil theory in particular makes critical use of
connections on Nthat, in any foliated coordinate system, coincide with the trivial connection
on Nalong leaves. Such connections are called Bott connections, named after their originator
R. Bott [2]. Bott connections give rise to Bott’s vanishing theorem, which states that elements
of degree greater than 2qin the Pontryagin ring of Nmust vanish. It is precisely this vanishing
phenomenon that guarantees the existence of the secondary classes.
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To any such foliated manifold (M, F) is associated its holonomy groupoid G. As a space, G
may be thought of as a quotient of the space of all smooth paths in leaves of F, with two such
paths identifying in Gif and only if their parallel transport maps, defined with respect to any
Bott connection on N, coincide. The holonomy groupoid Gcarries a natural, locally Hausdorff
differentiable structure, and is thus a Lie groupoid [34]. Moreover the normal bundle Ncarries a
natural action of G, and one may therefore be interested in extending the “static” characteristic
classes appearing in H∗
dR(M) to “dynamic” characteristic classes for the holonomy groupoid G.
In order to obtain such classes, it has become standard practice in the literature [9, 14, 20, 11,
21, 16, 31, 12, 30] to “´etalify” the holonomy groupoid as follows. Letting qbe the codimension
of (M, F), one takes any q-dimensional submanifold T ⊂ Mwhich intersects each leaf of Fat
least once, and which is everywhere transverse to Fin the sense that TxT ⊕TxF=TxMfor
all x∈ T . Such a submanifold is called a complete transversal for (M , F). Having chosen such
a complete transversal Twe consider the subgroupoid
GT
T:= {u∈ G :r(u), s(u)∈ T }
of G. The subgroupoid GT
Tinherits from Ga differential topology for which it is a (generally
non-Hausdorff) ´etale Lie groupoid [15, Lemma 2] - that is, a Lie groupoid whose range (and
therefore source) are local diffeomorphisms. For any choice of complete transversal T, the C∗-
algebras of the groupoids Gand GT
Tare Morita equivalent [25, 1], so are the same as far as
K-theory is concerned. Moreover, the groupoids Gand GT
Tare themselves Morita equivalent
[15, Lemma 2], and consequently they are (co)homologically identical [15, 16] also.
This paper provides an analogous characteristic map to those defined in [11, 16] in the
context of the full holonomy groupoid Gof a foliated manifold. Our characteristic map will be
constructed in a Chern-Weil fashion from Bott connections for N, staying as close as possible
to the classical geometric approach. Section 2 consists of necessary background on differential
graded algebras, Weil algebras, classical characteristic classes for foliations, and Chern-Weil
theory for Lie groupoids as developed in [27]. This background material suffices to access the
Pontryagin classes of N, for which a choice of connection form αon the positively oriented frame
bundle Fr+(N) of Ndetermines a characteristic map ψG
αsending the invariant polynomials on
gl(q, R) to the de Rham cohomology Ω∗(G(∗)
1) of the action groupoid G1:= GnFr+(N).
In order to access the secondary classes of the foliation, such as the Godbillon-Vey invariant,
in Section 3 we introduce and prove the following generalisation of Bott’s vanishing theorem.
Theorem 1.1. Let α[be the connection form associated to a Bott connection on N. Then the
image of ψG
α[in Ω∗(G(∗)
1)vanishes in total degree greater than 2q.
The vanishing of certain cocycles implied by Theorem 1.1 enables us to refine ψα[to a
characteristic map of the truncated Weil algebra so as to obtain secondary classes, in a manner
entirely analogous to the classical setting. More precisely we have the following theorem.
Theorem 1.2. Let W Oqdenote the truncated Weil algebra. If α[is the connection form
associated to any Bott connection on N, then we obtain a characteristic map
ψG
α[:W Oq→Ω∗(G(∗)
1/SO(q, R)).
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Theorems 1.1 and 1.2 should be thought of as the non-´etale analogues of [16, Theorem 2
(iv)] and [11, Lemma 17] respectively. We conclude Section 3 by discussing how the lack of an
invariant Euclidean structure on Nobstructs the na¨ıve construction of a characteristic map for
the de Rham complex Ω∗(G(∗)) associated to G.
Finally in Section 4, we restrict ourselves to the codimension 1 case, and use the explicit
formulae provided by Theorem 1.2 to derive a Godbillon-Vey cyclic cocycle for the convolution
algebra C∞
c(G1; Ω 1
2) of smooth leafwise half-densities Ω 1
2associated to G1. Our cocycle should
be thought of as a non-´etale analogue of the Connes-Moscovici formula [12, Proposition 19]. In
contrast with the Connes-Moscovici formula obtained in the ´etale setting, the cocycle we obtain
in this paper has a novel geometric interpretation in terms of path integrals of the curvature
form associated to a Bott connection. These facts can be summarised as follows.
Theorem 1.3. Let R[be the curvature form on Fr+(N)associated to a Bott connection on N,
and let RGdenote the differential 1-form on G1defined by
RG
u:= Zγ
R[, u ∈ G1(1)
where γis any leafwise path in Fr+(N)representing u. Assume that Fr+(N)∼
=M×R∗
+has
been trivialised by a choice of trivialisation for N. Then the formula
ϕgv (a0, a1) := Z(x,t)∈Fr+(N)Zu∈(G1)(x,t)
a0(u−1)a1(u)dt
t∧RG
u, a0, a1∈C∞
c(G1; Ω 1
2)
defines a cyclic cocycle ϕgv for the convolution algebra C∞
c(G1; Ω 1
2).
We conclude the paper by demonstrating that the cyclic cocycle ϕgv coincides with that
obtained as Chern character of a semifinite spectral triple constructed using groupoid equivariant
KK-theory in [28, Section 4.3]. In doing so we give a (non-´etale) geometric interpretation for
the off-diagonal term appearing in the triangular structures considered by Connes [9, Lemma
5.2] and Connes-Moscovici [13, Part I], in terms of the integrated curvature of Equation (1).
Let us stress that the approach taken in this paper has the advantage of being intrinsically
geometric, giving representatives of cohomological data that are expressed in terms of global
geometric data for (M, F). For instance, the Godbillon-Vey cyclic cocycle obtained in this
paper has a completely novel interpretation in terms of line integrals of the Bott curvature
over paths in Frepresenting elements of G(see Proposition 4.1). This is to be contrasted with
the approaches taken in the ´etale context, in which the geometry of Mhas necessarily been
lost by “chopping up” Ginto GT
T. In the ´etale context, explicit formulae have so far tended to
be obtained by tracking the displacement of local geometric data (trivial connections in local
transversals) [16, 12, Section 5.1, p. 47] under the action of GT
T, which will in general not be
easily relatable to the global geometry of M.
1.1 Acknowledgements
I wish to thank the Australian Federal Government for a Research Training Program scholarship.
I also thank Moulay Benameur for supporting a visit to Montpellier in late 2018, and Magnus
3

Goffeng for supporting a visit to Gothenburg in early 2019, where parts of this research were
conducted. I also thank Magnus Goffeng and James Stasheff for helpful comments on the
paper. Finally, I would like to extend deep thanks to Adam Rennie, whose consistent (but never
overbearing) guidance and support have greatly benefited my growth as a mathematician.
2 Background
2.1 Differential graded algebras and the Weil algebra
For the entirety of this subsection, denote by Ga Lie group with Lie algebra g. One of the key
tools in Chern-Weil theory is the notion of a G-differential graded algebra.
Definition 2.1. AG-differential graded algebra (A, d, i)is a differential graded algebra
(A, d)equipped with a G-action that preserves the grading of Aand commutes with the differ-
ential, and a linear map isending gto the derivations of degree -1 on (A, d)such that for all
X, Y ∈gand for all g∈Gone has
iX◦iY=−iY◦iX,
g◦iX◦g−1=iAdg(X),
and
iX◦d+d◦iX=LX,
where LXdenotes the infinitesimal G-action defined by
LX(a) := d
dtt=0
(exp(tX)·a), a ∈A.
If (A, d, iA)and (B, b, iB)are two G-differential graded algebras, a homomorphism φ:A→B
of differential graded algebras is said to be a homomorphism of G-differential graded
algebras if it commutes with the action of Gand with the maps iAand iB
Since G-differential graded algebras are in particular differential graded algebras, they admit
a natural cochain complex and associated cohomology. For geometric applications however, one
is usually more interested in the associated basic cohomology.
Definition 2.2. Let (A, d, i)be a G-differential graded algebra, and suppose that Kis a Lie
subgroup of G, with Lie algebra k. We say that an element a∈Ais G-invariant if g·a=a
for all g∈G. We say that a∈Ais K-basic if it is G-invariant and if iXa= 0 for all X∈k,
and denote the space of K-basic elements by AK−basic. The G-basic elements will be referred to
simply as basic.
If (A, d, i) is a G-differential graded algebra, then it is an easy consequence of the commu-
tativity of dwith the action of G, as well as the fact that iX◦d+d◦iX=LXfor all X∈g,
that dpreserves the space AK−basic. Therefore we obtain a cochain complex
0→A0
K−basic
d
−→ A1
K−basic
d
−→ A2
K−basic
d
−→ · · ·
4

whose cohomology we will be mostly interested in for geometric applications.
Definition 2.3. Let (A, d, i)be a G-differential graded algebra, and let Kbe a Lie subgroup of
G. For each n∈N, the nth K-basic cohomology group of (A, d, i)is the group
Hn
K−basic(A) := ker(d:An
K−basic →An+1
K−basic)/im(d:An−1
K−basic →An
K−basic).
The K-basic cohomology of (A, d, i)is the collection H∗
K−basic(A)of all K-basic cohomology
groups.
Because a homomorphism φ:A1→A2of G-differential graded algebras commutes with
the respective actions of Gand g, it naturally induces a homomorphism φ:H∗
K−basic(A1)→
H∗
K−basic(A2) of their K-basic cohomologies for any Lie subgroup Kof G. It will be useful to
have an analogue of cochain homotopy to decide when the maps of cohomology induced by two
such G-differential graded algebra homomorphisms coincide.
Definition 2.4. Let φ0, φ1:A1→A2be two homomorphisms of G-differential graded algebras
(A1, d1, i1)and (A2, d2, i2). We say that φ0and φ1are G-cochain homotopic if there exists
a cochain homotopy C:A∗
1→A∗−1
2for which C◦g=g◦Cand C◦i1
X=i2
X◦Cfor all g∈G
and X∈g.
The equivariance properties of G-cochain homotopies can be used to verify the following
homotopy invariance result.
Proposition 2.5. If φ0and φ1are G-cochain homotopic homomorphisms A1→A2of G-
differential graded algebras then the maps H∗
K−basic(A1)→H∗
K−basic(A2)induced by φ0and φ1
coincide for all Lie subgroups Kof G.
Let us now consider some important examples.
Example 2.6. Just as the immediate geometric example of a differential graded algebra is
the algebra of differential forms on a manifold, the first geometric example of a G-differential
graded algebra is given by the algebra of differential forms Ω∗(P) on the total space of a principal
G-bundle π:P→Mover a manifold M.
The algebra (Ω∗(P), d) is regarded as a differential graded algebra in the usual way, while
the action of Gon Ω∗(P) is obtained by pulling back differential forms under the canonical
right action R:P×G→P. That is, the action of g∈Gon ω∈Ω∗(P) is given by
g·ω:= R∗
g−1ω.
To obtain the linear map ifrom ginto the derivations of degree -1 on Ω∗(P) we recall that any
X∈gis associated with the fundamental vector field VXon Pdefined by
VX
p:= d
dtt=0
(p·exp(tX)), p ∈P.
For any X∈gwe then define iXon Ω∗(P) to be the interior product operator with VX. The
5

usual properties of the interior product together with the fact that
(dRg)p(VX
p) = VAdg−1(X)
p·g
for all p∈Pand g∈Gthen show that (Ω∗(P), d, i) is indeed a G-differential graded algebra.
Since P/G ∼
=M, the basic complex (Ω∗(P)basic, d) identifies naturally with (Ω∗(M), d). More
generally, if Kis a Lie subgroup of Gwith Lie algebra k, then K-basic elements of Ω∗(P) of
degree mare precisely those forms ˜ω∈Ωm(P) for which
˜ω(X1+k, . . . , Xm+k) = ˜ω(X1, . . . , Xm)
is well-defined for all X1, . . . , Xm∈T P . Any ω∈Ω∗(P/K) pulls back therefore to a K-basic
form on P, while any K-basic form ˜ωon Pdetermines a form ωon P /K by the formula
ω(X1, . . . , Xm) := ˜ω(X1+k, . . . , Xm+k).
Thus the space of K-basic elements in Ω∗(P) coincides with the differential graded algebra
Ω∗(P/K).
Chern-Weil theory is obtained for principal G-bundles over manifolds by considering con-
nection and curvature forms. Our next example will be key in formalising the properties of such
forms.
Example 2.7. One of the most important examples of a G-differential graded algebra is the
Weil algebra W(g) associated to the Lie algebra gof G[23, Chapter 3]. The Weil algebra
should be thought of as being the home of “universal” connection and curvature forms, and is
constructed as follows.
For each k∈N, denote by Sk(g∗) the space of functions
g× · · · × g
| {z }
ktimes
→R
which are invariant under the action of the symmetric group on the kfactors. Denote by S(g∗)
the sum over k∈Nof the Sk(g∗). Consider also the exterior algebra Λ(g∗) defined in the usual
way. The Weil algebra associated to gis
W(g) := S(g∗)⊗Λ(g∗).
We endow W(g) with a grading by declaring any element a⊗b∈Sk(g∗)⊗Λl(g∗) to have degree
2k+l, under which W(g) is a graded-commutative algebra.
To define a differential on W(g) we choose a basis (Xi)dim(g)
i=1 for g(for a basis-free definition,
see [23, Section 3.1]), with associated structure constants fi
jk defined by the equation
[Xi, Xj] = X
i
fk
ijXk.
The corresponding dual basis (ξi)dim(g)
i=1 of g∗determines generators ωi:= 1⊗ξi∈S0(g∗)⊗Λ1(g∗)
6

of degree 1 and Ωi:= ξi⊗1∈S1(g∗)⊗Λ0(g∗) of degree 2, with respect to which the differential
dis defined by
dΩi:= X
j,k
fi
jk Ωjωkdωi:= Ωi−1
2X
j,k
fi
jk ωjωk
Extending dto all of W(g) turns (W(g), d) into the graded-commutative differential graded
algebra that is freely generated by the ωiand Ωi. Note that by definition of dωi, we can equally
regard W(g) as being freely generated by the ωiand dωi.
The coadjoint action of Gon g∗given by
(g·ξ)(X) := (Ad∗
g−1ξ)(X) = ξ(Adg−1X)
for g∈G,ξ∈g∗and X∈gextends to an action of Gon the generators αi, Ωiand hence to
an action of Gon all of W(g). For X∈g, we define a derivation iXof degree -1 by
iX(Ωi) := 0 iX(ωi) := ωi(X),
and the corresponding map ifrom gto the derivations of degree -1 of W(g) satisfies the required
properties to make (W(g), d, i) a G-differential graded algebra.
By definition of i, the basic elements of W(g) identify with the space I(G) = S(g∗)Gof
symmetric polynomials that are invariant under the coadjoint action of G. If more generally K
is a Lie subgroup of G, then we use W(g, K) to denote the subalgebra of K-basic elements.
The notions of connection and curvature can be formulated at the abstract algebraic level
of G-differential graded algebras.
Definition 2.8. Let (A, d, i)be a G-differential graded algebra. A connection on Ais a degree
1 element α∈A1⊗gsuch that:
1. g·α=αfor all g∈G, where the action of gon A⊗gis given by g·(a⊗X) = (g·a)⊗Adg(X)
and,
2. iXα= 1 ⊗Xfor all X∈g.
If αis a connection on A, then its curvature is the degree 2 element R∈A2⊗gdefined by
the formula
R:= dα +1
2[α, α],
where [·,·]denotes graded Lie bracket of elements in A⊗g.
Example 2.9. Let π:P→Mbe a principal G-bundle, with associated G-differential graded
algebra (Ω∗(P), d, i) as defined in Example 2.6. Then a connection αon Ω∗(P) is precisely a
connection 1-form α∈Ω1(P;g). By definition, such a connection 1-form is invariant under the
action of G
g·α= Adg(R∗
g−1α) = α, g ∈G,
and is vertical in the sense that α(VX) = Xfor all X∈g, where VXdenotes the fundamental
vector field on Passociated to X. The curvature of αis of course just the usual curvature
2-form R∈Ω2(P;g) defined by α.
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Example 2.10. The Weil algebra W(g) constructed in Example 2.7 admits a canonical con-
nection. Given a basis (Xi)dim(g)
i=1 for g, with associated dual basis (ξi)dim(g)
i=1 and generators
ωi= 1 ⊗ξiof degree 1 and Ωi=ξi⊗1 of degree 2 respectively, we define ω∈W(g)1⊗gby the
formula
ω:= X
i
ωi⊗Xi.
Because the Xitransform covariantly and the ωicontravariantly this ωdoes not depend on the
basis chosen, and for the same reason is invariant under the action of G. By construction we
have iXω= 1 ⊗Xfor all X∈g.
The Weil algebra enjoys the following universal property as a classifying algebra for connec-
tions on G-differential graded algebras.
Theorem 2.11. [23, Theorem 3.3.1] Let (A, d)be a G-differential graded algebra, and suppose
that α∈A1⊗gis a connection on A. Then there exists a unique homomorphism φα:W(g)→A
of G-differential graded algebras such that φ(ω) = α. Moreover if α0,α1are two different
connections on A, the corresponding maps φα0and φα1are G-cochain homotopic.
In this paper we will mostly be interested in the case G= GL+(q, R) of invertible q×q
matrices with positive determinant, and where K= SO(q, R). Regard an element ξof gl(q, R)∗
as a matrix in the usual way, and let Ω := ξ⊗1∈S1(gl(q, R)∗)⊗Λ0(gl(q, R)∗) and ω:= 1 ⊗ξ∈
S0(gl(q, R)∗)⊗Λ1(gl(q, R)∗) denote the corresponding elements of W(gl(q, R)) of degrees 2 and
1 respectively. Using matrix multiplication, the differential on W(gl(q, R)) acts by the simple
formulae
dω = Ω −ω2, dΩ=Ωω−ωΩ.
It is well-known [22, p. 187] that for 1 ≤i≤q, the elements ci∈W(gl(q, R)) defined by
ci:= Tr(Ωi) (2)
are all cocycles and are all GL+(q, R)-basic. With respect to the decomposition gl(q, R) =
so(q, R)⊕s(q, R) of all q×qmatrices into antisymmetric matrices and symmetric matrices
respectively, our elements ωand Ω defined above decompose as
ω=ωo+ωs,Ω=Ωo+ Ωs.
Here the subscript odenotes the antisymmetric part, while the subscript sdenotes the symmetric
part. It can then be shown [22, Proposition 5] that for 1 ≤i≤q, the elements
hi:= iTr Z1
0
ωs(tΩs+ Ωo+ (t2−1)ω2
s)i−1dt
of W(gl(q, R)) satisfy dhi=ci, and are SO(q, R)-basic for iodd. We can assemble the ciand
hiinto a new, simpler subalgebra of W(gl(q, R),SO(q, R)) which is useful for obtaining explicit
formulae.
Definition 2.12. Let q≥1and let obe the largest odd integer that is less than or equal to
q. We denote by W Oqthe differential graded subalgebra of W(gl(q, R),SO(q, R)) generated by
8

odd degree elements h1, h3, . . . , ho, with deg(hi) = i, and even degree elements c1, c2, . . . , cqwith
deg(ci)=2i.
We will abuse terminology in referring to the algebra W Oqas the Weil algebra. We pre-empt
its use for foliations with the following refinement of Theorem 2.11.
Corollary 2.13. Let (A, d)be a GL+(q, R)-differential graded algebra, and suppose that α∈
A⊗gl(q, R)is a connection on A, with curvature R. Decompose α=αo+αsand R=Ro+Rs
into their antisymmetric and symmetric components respectively. Then the formulae
ci7→ Tr(Ri),for 1≤i≤q,
hi7→ iTr Z1
0
αs(tRs+Ro+ (t2−1)α2
s)i−1dt,for 1≤i≤q,iodd,
define a homomorphism ψα:W Oq→ASO(q,R)−basic of differential graded algebras. Moreover
if βis any other choice of connection on A, the maps induced by ψαand ψβon cohomology
coincide.
2.2 The classical Chern-Weil homomorphism for foliations
Recall that a foliated manifold (M, F) of codimension qis transversely orientable if its normal
bundle πN:N:= T M/T F → Mis an orientable vector bundle. Given such a foliated manifold,
we can mimic the classical Chern-Weil construction of Bott [3] using Corollary 2.13 as follows.
Let πFr+(N): Fr+(N)→Mdenote the positively oriented transverse frame bundle of N, a
principal GL+(q, R)-bundle whose fibre Fr+(N)xover x∈Mconsists of all positively oriented
linear isomorphisms φ:Rq→Nx. Letting Gdenote the holonomy groupoid [34] of (M, F),
recall [28, Section 2.2] that there is a natural action G ×s,πNN→Nof Gon Nby linear
isomorphisms, which we denote
G ×s,πNN3(u, n)7→ u∗n∈N.
We obtain an induced action G ×s,πFr+(N)Fr+(N)→Fr+(N) of Gon Fr+(N) defined by
u·φ:= u∗◦φ:Rq→Nr(u),(u, φ)∈ G ×s,πFr+(N)Fr+(N).
By associativity of composition, this action of Gcommutes with the canonical right action of
GL+(q, R) on the principal GL+(q, R)-bundle Fr+(N). Moreover, the orbits of Gin Fr+(N)
define a foliation FFr+(N)of Fr+(N) for which the differential of the projection πFr+(N)maps
TFFr+(N)fibrewise-isomorphically onto TF. The following definition does not appear explicitly
in the literature, but as we will show it is nonetheless essentially classical.
Definition 2.14. A connection form α[∈Ω∗(Fr+(N); gl(q, R)) is called a Bott connection
form if TFFr+(N)⊂ker(α[).
Let us justify this terminology. First, let p:T M →Ndenote the projection onto the normal
bundle, and recall that a connection ∇[: Γ∞(M;N)→Γ∞(M;T∗M⊗N) for Nis called a
9

Bott connection if it satisfies
∇[
Xp(Y) = p[X, Y ], Y ∈Γ∞(M;T M ),(3)
whenever Xis a leafwise vector field. The next result establishes the relationship between Bott
connections in the sense of Equation (3) and Bott connection forms in the sense of Definition
2.14. This relationship is essentially classical, so although the next result does not appear
explicitly in the literature and its proof is independent, we make no claim to originality.
Proposition 2.15. Bott connections ∇[on Nare in bijective correspondence with Bott con-
nection forms α[on Fr+(N). Moreover, any Bott connection form α[on Fr+(N)canonically
determines a Bott connection ∇Fr+(N)for the foliated manifold (Fr+(N),FFr+(N)).
Proof. For the first part, let ∇be a connection on Nand let αbe the connection form on Fr+(N)
determined by ∇. Notice that in foliated coordinates (x1, . . . , xp;z1, . . . , zq) over U⊂M,
defining a local section χU:U→Fr+(N)|Uof πFr+(N),∇can be written
∇=d+αU
where αU:= χ∗
Uα∈Ω1(U;gl(q, R)). Let σ=σi∂
∂zibe a normal vector field over U, and let X
be a leafwise vector field over U. Then
∇Xσ=dσi(X)∂
∂zi+αU(X)σ=p[X, σ] + αU(X)σ
is equal to p[X, σ] if and only if αU(X) vanishes. Thus a connection ∇on Nis a Bott connection
if and only if its local connection form αUvanishes on leafwise vectors in any foliated coordinate
neighbourhood U. Since πFr+(N)maps TFFr+(N)fibrewise isomorphically to TFwe see that
every αUvanishes on TFif and only if αvanishes on TFFr+(N). Consequently ∇=∇[is a
Bott connection on Nif and only if its associated connection form α=α[on Fr+(N) is a Bott
connection form.
For the second part, suppose that α[is a Bott connection form on Fr+(N). By hypothesis
we have TFFr+(N)⊂ker(α[), and since ker(α[) projects fibrewise-isomorphically onto T M , the
quotient bundle H:= ker(α[)/T FFr+(N)projects fibrewise-isomorphically onto N=T M/T F.
Consequently, Hadmits a tautological trivialisation H∼
=Fr+(N)×Rqdefined by
Hφ3h7→ φ−1(dπFr+(N)(h)) ∈Rq, φ ∈Fr+(N).
The vertical bundle V:= ker(dπFr+(N)) over Fr+(N) is canonically trivialised by the fundamen-
tal vector fields, so we obtain the canonical trivialisation
NFr+(N):= TFr+(N)/T FFr+(N)=V⊕H∼
=Fr+(N)×(Rq2⊕Rq)
of the normal bundle for the foliated manifold (Fr+(N),FFr+(N)). With respect to this global
trivialisation, the vanishing of α[on TFFr+(N)implies that
∇Fr+(N):= d+ idRq2⊕α[
10

defines a Bott connection on NFr+(N).
Bott connections are important because of the following vanishing result, known as Bott’s
vanishing theorem [2, p. 34]. Its proof follows from an easy local coordinate calculation.
Theorem 2.16 (Bott’s vanishing theorem).Let α[∈Ω1(Fr+(N); gl(q, R)) be a Bott connection
form, and let R[:= dα[+α[∧α[be its curvature. Then any polynomial of degree greater than
qin the components of R[vanishes.
Bott’s vanishing theorem motivates the following refinement of the Weil algebra W Oq.
Definition 2.17. Let Jbe the differential ideal in W Oqconsisting of elements of R[c1, . . . , cq]⊂
W Oqthat are of degree greater than 2q. The truncated Weil algebra is the quotient W Oq:=
W Oq/J.
The following result is now an immediate consequence of Bott’s vanishing theorem, together
with Corollary 2.13.
Theorem 2.18. A choice of Bott connection form α[determines a homomorphism ψα[:
W Oq→Ω∗(Fr+(N)/SO(q, R)) of differential graded algebras, that factors through the trun-
cated Weil algebra W Oq. That is, letting p:W Oq→W Oqdenote the projection, there is a
homomorphism φα[:W Oq→Ω∗(Fr+(N)/SO(q, R)) of differential graded algebras such that
the diagram
W OqΩ∗(Fr+(N)/SO(q, R))
W Oq
ψα[
p
φα[
commutes. The maps on cohomology induced by ψα[and φα[do not depend on the Bott con-
nection chosen.
Given a choice of Bott connection form α[on Fr+(N), those classes determined by the
range of φα[that are not contained in the Pontryagin ring [φα(R[c1, . . . , cq])] ⊂Hev
dR(M) for N
are called secondary characteristic classes. In particular, the Godbillon-Vey class is the class
[φα[(h1cq
1)] ∈H2q+1
dR (Fr+(N)/SO(q, R)).
Remark 2.19. The fibre GL+(q, R)/SO(q, R) of Fr+(N)/SO(q, R) is contractible, so the total
space of Fr+(N)/SO(q, R) has the same cohomology as M. More specifically, a choice of
Euclidean metric on Ndetermines a smooth section σ:M→Fr+(N)/SO(q, R), which, together
with a choice of Bott connection α[on Fr+(N), determines a characteristic map σ∗◦φα[:
W Oq→Ω∗(M) for M. The arguments of [22, Remarque (c)] show that this characteristic map
agrees on the level of differential forms with that defined by Bott [3].
2.3 Chern-Weil homomorphism for Lie groupoids
The groupoid Chern-Weil material we present in this subsection is sourced primarily from the
paper [27], whose historical antecedents are to be found in the papers [18, 5].
11

Just as the classical Chern-Weil theory can be simplified and systematised by using principal
G-bundles, Chern-Weil theory at the level of Lie groupoids is most easily studied using principal
bundles over groupoids. For the entirety of this section we let Gbe a (not necessarily Hausdorff)
Lie groupoid, with unit space G(0) and range and source maps r, s respectively, and let Gbe a
Lie group.
Definition 2.20. Aprincipal G-bundle over Gconsists of a principal G-bundle π:P(0) →
G(0) over G(0) together with an action σ:P:= G ×s,π P(0) → P(0) that commutes with the right
action of Gon P(0). We will often refer to Pas a principal G-groupoid over G.
Let us for the rest of this subsection fix a principal G-bundle π:P(0) → G(0) over G. We
will primarily consider the associated action groupoid P:= G ×s,π P(0) , and the resulting spaces
P(k)of composable k-tuples of elements of P. In order to make notation less cumbersome, for
(u1, . . . , uk)∈ G(k)and p∈ P (0)
s(uk)we will denote the composable k-tuple
(u1,(u2· · · uk)·p),(u2,(u3· · · uk)·p),...,(uk, p)∈ P(k)
by simply
(u1, . . . , uk)·p.
The next result tells us that the P(k)fibre over the G(k)as principal G-bundles, and is a
straightforward consequence of the fact that P(0) → G(0) is a principal G-bundle, together with
the fact that the action of Gcommutes with that of G.
Lemma 2.21. Let π:P(0) → G(0) be a principal G-bundle over G. Then for each k∈Nand
(u1, . . . , uk)·p∈ P(k), the formula
π(k)((u1, . . . , uk)·p) := (u1, . . . , uk)
defines a principal G-bundle π(k):P(k)→ G(k).
Remark 2.22. Note that the definition and properties of the exterior derivative don a manifold
Ydepend only on the local structure of the manifold. Consequently, the differential forms
(Ω∗(Y), d) on Yare a differential graded algebra even if Yis only locally Hausdorff. We will
use this fact freely and without further comment in what follows.
We now have the following immediate consequence of Lemma 2.21 and Example 2.6.
Corollary 2.23. For all k∈N, the differential forms Ω∗(P(k))on P(k)form a G-differential
graded algebra. If K⊂Gis a Lie subgroup, then the K-basic elements of Ω∗(P(k))identify with
the differential forms Ω∗(P(k)/K)on the quotient of P(k)by the right action of K.
Let us now recall the definition of the de Rham cohomology of Ptogether with its relative
versions. Observe that there exist face maps k
i:P(k)→ P(k−1) defined for all k > 1 and
0≤i≤kby the formulae
k
0(u1, u2, . . . , uk)·p:= (u2, . . . , uk)·p,
12

k
k(u1, . . . , uk)·p:= (u1, . . . , uk−1)·(uk·p),
and
k
i(u1, . . . , uk)·p:= (u1, . . . , uiui+1, . . . , uk)·p
for 1 ≤i≤k−1. For k= 1, we obtain the range and source maps 1
0:= s:P → P (0) and
1
1:= r:P → P (0).
Remark 2.24. The collection {P(k)}k≥0, taken together with the face maps k
i:P(k)→ P(k−1),
is known in the literature as the nerve NPof P. The nerve NPis an example of a (semi)
simplicial manifold (the terminology used depends on the author). It is in the general setting of
simplicial manifolds that most of the technology in this section was developed by Bott-Shulman-
Stasheff [5] and Dupont [17, 18].
The next result is an immediate consequence of the fact that the action of Gon P(0) com-
mutes with the right action of Gon P(0).
Lemma 2.25. The face maps k
i:P(k)→ P(k−1) commute with the actions of Gon P(k)and
P(k−1).
Since each P(k)is a manifold, the exterior derivative d: Ω∗(P(k))→Ω∗+1(P(k)) is defined for
all k≥0 and satisfies the usual property d2= 0. The exterior derivative will form the vertical
differential of the double complex from which we will construct the de Rham cohomology of P.
To obtain the horizontal differential, notice that the face maps k
i:P(k)→ P(k−1) allow us to
define a natural map ∂: Ω∗(P(k−1))→Ω∗(P(k)) via an alternating sum of pullbacks
∂ω :=
k
X
i=0
(−1)i(k
i)∗ω.
A routine calculation shows that ∂2= 0. By Lemma 2.25 the coboundary mapping ∂:
Ω∗(P(k−1))→Ω∗(P(k)) preserves K-basic elements for any Lie subgroup Kof G, and we
therefore obtain the double complex
.
.
..
.
..
.
.
Ω1(P(0) /K) Ω1(P(1) /K) Ω1(P(2) /K )· · ·
Ω0(P(0) /K) Ω0(P(1) /K) Ω0(P(2) /K )· · ·
d
d
d d
d d
∂∂∂
∂∂∂
Definition 2.26. Let Kbe a Lie subgroup of G. The double complex (Ω∗(P(∗)/K), d, ∂)is
called the K-basic de Rham complex of the principal G-groupoid P. The associated total
complex is given by
Tot∗Ω(P/K) = M
n+m=∗
Ωn(P(m)/K), δ|Ωn(P(m)/K):= (−1)md+∂.
13

The cohomology of (Tot∗Ω(P/K), δ)is denoted by H∗
dR(P/K)and is called the K-basic de
Rham cohomology of the principal G-groupoid P.
Remark 2.27. The de Rham complex of Definition 2.26 is frequently referred to in the literature
as the Bott-Shulman-Stasheff complex associated to the simplicial manifold N(P/K), named
after its originators [5].
In the same way that the exterior product of differential forms induces a multiplication in
de Rham cohomology groups of any manifold, the exterior product of differential forms also
induces a multiplication in the de Rham cohomology of G.
Definition 2.28. Let Kbe a Lie subgroup of G. Given ω1∈Ωk(P(m)/K)and ω2∈Ωl(P(n)/K)
we define the cup product ω1∨ω2∈Ωk+l(P(m+n)/K)of ω1and ω2by
(ω1∨ω2)(u1,...,um+n):= (−1)kn(p∗
1ω1)(u1,...,um+n)∧(p∗
2ω2)(u1,...,um+n),
where p1:P(m+n)/K → P(m)/K is given by
p1(u1, . . . , um+n) = 








(u1, . . . , um)if m≥1
r(u1)if m= 0, n ≥1
id if m=n= 0
and p2:P(m+n)/K → P(n)/K is given by
p2(u1, . . . , um+n) = 








(um+1, . . . , um+n)if n≥1
s(um)if n= 0, m ≥1
id if m=n= 0
Proposition 2.29. The cup product descends to give a well-defined multiplication on the co-
homology H∗
dR(P/K), gifting H∗
dR(P/K)the structure of a graded ring.
Proof. The cup product respects the bi-grading of Ω∗(P(∗)/K) by definition. Well-definedness
on H∗
dR(P/K) follows from the Liebniz rule for the exterior derivative, and from noting that
we may rewrite p1:P(m+n)/K → P(m)/K as
p1=m+1
m+1 ◦ · · · ◦ m+i
m+i◦ · · · ◦ m+n
m+n
and p2:P(m+n)/K → P(n)/K as
p2=n+1
0◦ · · · ◦ n+i
0◦ · · · ◦ n+m
0,
where the k
i:P(k)→ P(k−1) are the face maps.
For our characteristic map we will also need a particular differential graded algebra which
encodes the face maps of the P(k)into those of the standard simplices. For k∈N, let ∆kdenote
14

the standard k-simplex
∆k:= (t0, t1, . . . , tk)∈[0,1]k+1 :
k
X
i=1
ti= 1.(4)
We have face maps ˜k
i: ∆k−1→∆kdefined for all k > 1 and 1 ≤i≤kby the formulae
˜k
i(t0, . . . , tk−1) := (t0, . . . , ti−1,0, ti, . . . , tk−1).
and for i= 0 by simply
˜k
0(t0, . . . , tk−1) := (0, t0, . . . , tk−1).
Definition 2.30. For l∈N, a simplicial l-form on Pis a sequence ω={ω(k)}k∈Nof
differential l-forms ω(k)∈Ωl(∆k× P(k))such that
(˜k
i×id)∗ω(k)= (id ×k
i)∗ω(k−1) ∈Ωl(∆(k−1) × P(k))
for all i= 0, . . . , k and for all k∈N. We denote the space of all simplicial l-forms on Pby
Ωl
∆(P).
Remark 2.31. One identifies (t, k
i(v)) ∈∆k−1× P(k−1) with (˜k
i(t), v)∈∆k× P(k)for all
k > 0, and defines the fat realisation of kNP k of NPto be the space
kNP k := G
k≥0∆k× P(k)/∼.
The fat realisation is a geometric realisation of the classifying space BPof the groupoid P
[32], and is not generally a manifold even though each of its “layers” ∆k× P(k)is. Simplicial
differential forms were defined by Dupont [18] so as to descend to “forms on BP”. We will see
shortly that together with the usual de Rham differential, simplicial differential forms define a
differential graded algebra whose cohomology can be taken as the definition of the cohomology
of the classifying space.
Importantly, simplicial differential forms on Pdetermine a differential graded algebra which
will be instrumental in the construction of our characteristic map. The next result follows from
routine verification.
Proposition 2.32. The wedge product and exterior derivative of differential forms on the man-
ifolds ∆k× P(k)together with the action of Gon the principal G-bundle P(k), make the space
Ω∗
∆(P)of all simplicial differential forms on Pinto a G-differential graded algebra. If Kis
any Lie subgroup of G, then the subcomplex (Ω∗
∆(P)K−basic, d)of (Ω∗
∆(P), d)coincides with the
complex (Ω∗
∆(P/K), d)of simplicial differential forms on the groupoid P/K.
In order to relate simplicial differential forms to the de Rham cohomology of P, we make use
of the integration over the fibres map. The next result is proved in [18, Theorem 2.3, Theorem
2.14].
15

Proposition 2.33. Let Kbe a Lie subgroup of G. Then the map I: Ω∗
∆(P/K)→Tot∗Ω(P/K)
defined by
I(ω) := X
l∈NZ∆l
ω(l)
is a map of cochain complexes. Moreover the map determined by Ion cohomology is a homo-
morphism of rings, where the ring structure on H∗(Ω∗
∆(P/K)) is induced by the wedge product
and where the ring structure on H∗
dR(P/K)is induced by the cup product.
Remark 2.34. When Gis Hausdorff, the map I: Ω∗
∆(P/K)→Tot∗Ω(P/K) descends to
an isomorphism on cohomology [18, Theorem 2.3]. Thus for Hausdorff Gthe double complex
Ω∗(G(∗)) computes the cohomology of the classifying space BG.
Before we give the characteristic map, we need to show that a connection form on P(0)
induces a connection on the differential graded algebra Ω∗
∆(P). The universal property of the
Weil algebra W(g) (Theorem 2.11) will then guarantee a homomorphism from W(g) to Ω∗
∆(P)
which, composed with the cochain map I, will give us our characteristic map.
Construction 2.35. For each 0 ≤i≤k, define pk
i:P(k)→ P(0) by
pk
0((u1, . . . , uk)·p) := (u1· · · uk)·p,
pk
k((u1, . . . , uk)·p) := p,
and
pk
i((u1, . . . , uk)·p) := (ui+1 · · · uk)·p.
for all 1 ≤i≤k−1. Since the range and source maps are G-equivariant, so too are the maps
pk
i. Given a connection form α∈Ω1(P(0);g), for each k∈Nwe define a differential form
α(k)∈Ω1(∆k× P(k)) by the formula
α(k)
(t0,...,tk;(u1,...,uk)·p):=
k
X
i=0
ti((pk
i)∗α)(u1,...,uk)·p.(5)
The next lemma now follows from routine calculations (see also [27, Proposition 5.3]).
Lemma 2.36. The sequence ˜α:= {α(k)}k∈Nof 1-forms α(k)∈Ω1(∆k× P(k))⊗gdetermines a
connection on the differential graded algebra Ω∗
∆(P).
Finally we can give the characteristic map as in [27].
Theorem 2.37. A choice of connection form α∈Ω1(P(0);g)determines, for any Lie subgroup
Kof G, a homomorphism
ΨG
α:W(g, K)→Ω∗
∆(P/K)
of differential graded algebras, hence a cochain map
ψG
α=I◦ΨG
α:W(g, K)→Tot∗Ω(P/K)
of total complexes. The induced map on cohomology is a homomorphism of graded rings and
does not depend on the connection chosen.
16

Proof. The existence of ΨG
αfollows from Lemma 2.36 together with Theorem 2.11. That ψG
αis a
cochain map is true by Proposition 2.33, while the cohomological independence of the choice of
connection follows from Theorem 2.11. That ψG
αdescends to a homomorphism H∗(W(g, K)) →
H∗
dR(P/K) of graded rings follows from Proposition 2.33 together with the fact that ΨG
α:
W(g, K)→Ω∗
∆(P/K) is a homomorphism of differential graded algebras.
3 Characteristic map for foliated manifolds
We will now use the background material of Section 2 to prove new results on the secondary
characteristic classes for the transverse frame holonomy groupoid of a foliation. Let us consider
a transversely orientable foliated manifold (M, F) of codimension qwith holonomy groupoid
G. As we have already discussed in Subsection 2.2, the principal GL+(q, R)-bundle πFr+(N):
Fr+(N)→Mof positively oriented frames for Ncarries a left action of the holonomy groupoid
Gthat commutes with the canonical right action of GL+(q, R).
Definition 3.1. We will denote by G1=GnFr+(N)the principal GL+(q, R)-groupoid over G
corresponding to the foliated principal GL+(q, R)-bundle πFr+(N): Fr+(N)→M.
Given a connection form α∈Ω1(Fr+(N); gl(q, R)), the characteristic map ψG
αof Theo-
rem 2.37 composes with the inclusion W Oq→W(gl(q, R),SO(q, R)) to give a cochain map
ψG
α:W Oq→Ω∗(G(∗)
1/SO(q, R)), whose induced map on cohomology does not depend on
the connection α. The image of ψG
αin the first-column subcomplex Ω∗(Fr+(N)/SO(q, R)) of
Ω∗(G(∗)
1/SO(q, R)) coincides with that of the characteristic map ψαof Theorem 2.18. More gen-
erally the image of ψG
αin H∗
dR(G1/SO(q, R)) consists of the Pontryagin classes of the groupoid
G1/SO(q, R), accessed in the same manner as in [27]. In order to construct secondary charac-
teristic classes for the groupoid G1/SO(q, R), we must prove an analogue of Bott’s vanishing
theorem - that is, we must prove that the Pontryagin classes of G1/SO(q, R) vanish in total
degree greater than 2q. To this end, we present the following generalisation of Bott’s vanish-
ing theorem, which is the non-´etale analogue of [16, Theorem 2 (iv)]. Regard elements of the
subalgebra I∗
q(R) := R[c1, . . . , cq]⊂W Oqas invariant polynomials as in Equation (2).
Theorem 3.2 (Bott’s vanishing theorem for G1).Let α[∈Ω1(Fr+(N); gl(q, R)) be a Bott
connection form. If P∈I∗
q(R)is an invariant polynomial of degree deg(P)> q (so that its
degree in I∗
q(R)is greater than 2q), then ψα[(P)=0∈Ω∗(G(∗)
1/SO(q, R)).
Proof. For each k∈N, let (R[)(k)=d(α[)(k)+ (α[)(k)∧(α[)(k)denote the curvature of the
connection form (α[)(k)on ∆k× G(k)
1obtained as in Lemma 2.36. Let P∈I∗
q(R). The cochain
ψα[(P) in Ω∗(G(∗)
1/SO(q, R)) identifies in the same manner as in Example 2.6 with the SO(q, R)-
basic cochain X
kZ∆k
P((R[)(k)),(6)
in Ω∗(G(∗)
1). Thus it suffices to show that the cochain in Equation (6) is zero.
The form (R[)(k)is by construction of degree at most 1 in the ∆kvariables due to the d(α[)(k),
and therefore P((R[)(k)) is of degree at most deg(P) in the ∆kvariables. Thus R∆kP((R[)(k))
vanishes when deg(P)< k, implying that ψα(P) vanishes in Ω∗(G(k)
1) for k > deg(P).
17

Let us assume therefore that k≤deg(P). We will show that R∆kP((R[)(k)) = 0 as a
differential form on Ω2 deg(P)−k(G(k)
1). On ∆k× G(k)
1, using Equation (5), we compute
(R[)(k)=
k
X
i=0
dti∧(pk
i)∗α[+
k
X
i=0
ti(pk
i)∗dα[+k
X
i=0
ti(pk
i)∗α[∧k
X
i=0
ti(pk
i)∗α[,(7)
with the pk
i:G(k)
1→Fr+(N) defined as in Construction 2.35. To proceed further, we must
consider a local coordinate picture.
About a point (u1, . . . , uk)·φ∈ G(k)
1, consider a local coordinate chart for G(k)
1of the form
(xj
1)dim(F)
j=1 ;. . . ; (xj
k)dim(F)
j=1 ; (zj)q
j=1;g∈B1× · · · × Bk×V×GL+(q, R),
where the B1, . . . , Bkare open balls in Rdim(F)corresponding to plaques in foliated charts
U1, . . . , Ukin (M, F), and where Vis an open ball in Rqsuch that Bk×V∼
=Uk. For ˜uj+1 =
uj+1 · · · uk∈ G we let (h˜uj+1 )l:V→Rdenote the lth component function of some holonomy
transformation h˜uj+1 representing ˜uj+1. Then in these coordinates the maps pk
i:G(k)
1→Fr+(N)
take the form
pk
i(xj
1)dim(F)
j=1 ;. . . ; (xj
k)dim(F)
j=1 ; (zj)q
j=1;g:= (xj
i+1)dim(F)
j=1 ;(h˜ui+1 )j(z1, . . . , z q)q
j=1;g.
To write (R[)(k)in these local coordinates, consider the chart Ui×GL+(q, R) of Fr+(N). In
the foliated chart Uiwe have the local connection form αi∈Ω1(Ui;gl(q, R)) corresponding
to ∇[, which by Proposition 2.15 vanishes on plaquewise1tangent vectors. Letting π1:Ui×
GL+(q, R)→Uiand π2:Ui×GL+(q, R)→GL+(q, R) denote the projections, over Ui×
GL+(q, R) the form α[can be written
α[
(x,g)= Adg−1π∗
1αi(x,g)+π∗
2ωM C (x,g),(x, g)∈Ui×GL+(q, R)
where ωMC is the Maurer-Cartan form on GL+(q, R) [29, Section 2.4 (b)]. For simplicity, let
us abuse notation in letting αidenote the form Ad−1π∗
1αion Ui×GL+(q, R). Then by
Proposition 2.15 the matrix components of αican all be written in terms of the differentials of
the transverse coordinates zjin Ui. Consequently, in coordinates we can write
(pk
i)∗α[= (pk
i)∗αi+π∗
kωM C ,
where πk:B1× · · · × Bk×V×GL+(q, R)→GL+(q, R) is the projection and where (pk
i)∗αiis
agl(q, R)-valued 1-form in the coordinate differentials (dzj)q
j=1.
Let us now rewrite the expression (7) for (R[)(k)in coordinates. The first term on the right
hand side can be written
k
X
i=0
dti∧(pk
i)∗α[=
k
X
i=0
dti∧(pk
i)∗αi+k
X
i=0
dti∧π∗
kωMC =
k
X
i=0
dti∧(pk
i)∗αi(8)
1i.e. locally leafwise
18

since Pk
i=0 ti= 1. The middle term on the right hand side of (7) can be written
k
X
i=0
ti(pk
i)∗dα[=
k
X
i=0
ti(pk
i)αi+k
X
i=0
tiπ∗
kdωM C =
k
X
i=0
ti(pk
i)αi+π∗
kdωM C ,(9)
while the last term on the right hand side of (7) can be written
k
X
i=0
ti(pk
i)∗α[∧k
X
i=0
ti(pk
i)∗α[=k
X
i=0
ti(pk
i)∗αi∧k
X
i=0
ti(pk
i)∗αi
+k
X
i=0
ti(pk
i)∗αi∧π∗
kωMC
+π∗
kωMC ∧k
X
i=0
ti(pk
i)∗αi+π∗
k(ωMC ∧ωM C ).(10)
Adding the expressions (8), (9) and (10) and using the fact that the Maurer-Cartan form satisfies
dωM C +ωM C ∧ωM C = 0 [29, Equation 2.46], we find that
(R[)(k)=
k
X
i=0
dti∧(pk
i)∗αi+
k
X
i=0
ti(pk
i)∗dαi+k
X
i=0
ti(pk
i)∗αi∧k
X
i=0
ti(pk
i)∗αi
+k
X
i=0
ti(pk
i)∗αi∧π∗
kωMC +π∗
kωMC ∧k
X
i=0
ti(pk
i)∗αi.(11)
For a summand of R∆kP((R[)(k)) to be nonzero, it must contain precisely kfactors of the
first term appearing in (11). Thus, in our coordinates, due to the (pk
i)∗αiappearing in this first
term of (11) each summand of R∆kP((R[)(k)) contains a string of wedge products of at least
kof the dzi. This consideration accounts for 2kof the coordinate differentials that appear in
each summand of R∆kP((R[)(k)), and we must concern ourselves now with the 2deg(P)−2k
coordinate differentials that remain.
Now each of the final four terms in (11) is a matrix of 2-forms, and contains either an αi
or a dαias a factor. Consequently, all the components of each such matrix must contain at
least one dzias a factor. Therefore, of the remaining 2 deg(P)−2kcoordinate differentials in
each summand of R∆kP((R[)(k)), at least deg(P)−kmore must be dzi’s. Thus in our local
coordinate system for G(k)
1, each summand in R∆kP((R[)(k)) contains a string of wedge products
of at least k+ (deg(P)−k) = deg(P)> q of the dzi, and must therefore be zero by dimension
count.
Bott’s vanishing theorem at the level of the holonomy groupoid enables us to refine our
characteristic map in a way entirely analogous to the classical case.
Theorem 3.3. If α[∈Ω1(Fr+(N); gl(q, R)) is a Bott connection on N, the cochain map ψG
α[:
W Oq→Ω∗(G(∗)
1/SO(q, R)) descends to a cochain map
φG
α[:W Oq→Ω∗(G(∗)
1/SO(q, R))
19

whose induced map on cohomology is independent of the Bott connection chosen.
Recall now the characteristic map φα[:W Oq→Ω∗(Fr+(N)/SO(q, R)) of Theorem 2.18,
and for any b∈Ω∗(G(∗)
1/SO(q, R)) let b0denote its component in Ω∗(Fr+(N)/SO(q, R)). Then
by construction we have
φG
α[(a)0=φα[(a),for all a∈W Oq.
Thus φα[should be thought of as encoding the “static” transverse geometric information that
can be accessed via classical Chern-Weil theory, while the “larger” characteristic map φG
α[en-
codes both the static and dynamic information pertaining to the relationship of the groupoid
action with transverse geometry. As discussed in Remark 2.19, one can pull back the static
information encoded by φα[to Ω∗(M) through the choice of a Euclidean structure for N, so
one might hope that it is also possible to pull back all the dynamical information encoded by
φG
α[to the double complex Ω∗(G(∗)) in the same way.
Indeed it is claimed, with some vagueness, by Crainic and Moerdijk in [16, Section 3.4] (who
work with an ´etalified, ˇ
Cech version of the double complex Ω∗(G(∗))) that the contractibility of
the fibres of Fr+(N)/SO(q, R) allows one to pull all of φG
α[(W Oq) down to Ω∗(G(∗)) “as in” the
static case. While it is unclear exactly what Crainic and Moerdijk mean by this, let us point out
here that one is prevented from na¨ıvely extending the cochain map σ∗: Ω∗(Fr+(N)/SO(q, R)) →
Ω∗(M) to a cochain map Ω∗(G(∗)
1/SO(q, R)) →Ω∗(G(∗)) precisely by the lack of invariance of
the Euclidean structure on Ndefining σunder the action of G. More precisely, we have the
following proposition.
Proposition 3.4. Define σ(k):G(k)→ G(k)
1/SO(q, R)by the formula
σ(k)(u1, . . . , uk) := (u1, . . . , uk)·σ(s(uk)),(u1, . . . , uk)∈ G(k).
Pulling back by the σ(k)defines a cochain map Ω∗(G1/SO(q, R)) →Ω∗(G(∗))if and only if σis
invariant under the action of G.
Proof. The σ(k)define a cochain map if and only if σ(k−1) ◦k
i=k
i◦σ(k)for all i≤k. However,
we see that
k
kσ(k)(u1, . . . , uk)= (u1, . . . , uk)·uk·σ(s(uk))
while
σ(k−1)k
k(u1, . . . , uk)= (u1, . . . , uk−1)·σ(s(uk−1))
for all (u1, . . . , uk)∈ G(k). Consequently we have k
k◦σ(k)=σ(k−1) ◦k
kif and only if u·σ(s(u)) =
σ(r(u)) for all u∈ G, which occurs if and only if the Euclidean structure σon Nis preserved
by the action of G.
An invariant section σ:M→Fr+(N)/SO(q, R) is the same thing as a G-invariant Eu-
clidean structure on N, which is not always guaranteed to exist. Moreover in any situation
where such an invariant Euclidean structure does exist, it induces via its determinant an in-
variant transverse volume form. In this case, well-known results [26, Theorem 2] state that all
20

generalised Godbillon-Vey classes (that is, those classes determined by cocycles h1hIcJ∈W Oq,
for multi-indices Iand Jwith deg(cJ)=2qand with hI= 1 permitted) vanish in H∗
dR(M). In
particular, whenever (M, F) has nonvanishing Godbillon-Vey invariant, in order to probe the
algebraic topology of Gusing the characteristic map φG
α[we need a more sophisticated method
of getting from Ω∗(G(∗)
1/SO(q, R)) to Ω∗(G(∗)) which takes into account the lack of invariance
of Euclidean structures on Nunder the action of G. Giving such a construction constitutes an
interesting research question, which we leave to a future paper.
4 The codimension 1 Godbillon-Vey cyclic cocycle
Connes and Moscovici [11, Section 4] use the ´etale picture of a foliation groupoid to obtain
an analogue of Theorem 3.3. More specifically, they replace G1with the groupoid F X oΓX
of germs of local diffeomorphisms of an q-manifold X, lifted to the frame bundle F X of X.
Then they obtain a characteristic map from H∗(W Oq) to the cyclic cohomology of the algebra
C∞
c(F X )oΓX. While unfortunately the lack of an easily-defined “transverse exterior derivative”
prevents a complete replication of the Connes-Moscovici construction in the non-´etale case, we
can use Theorem 3.3 to give, in codimension 1, a cyclic cocycle for the Godbillon-Vey invariant
on the algebra C∞
c(G1; Ω 1
2) (recall from [8] that C∞
c(G1; Ω 1
2) is the convolution algebra spanned
by leafwise half-densities that are smooth with compact support in some Hausdorff open subset
of the locally Hausdorff Lie groupoid G1).
Let us begin with a preliminary calculation. Suppose that (M, F) is of codimension 1 (in
which case SO(1,R) is the trivial group so we need not concern ourselves with basic elements),
and let α∈Ω1(Fr+(N)) correspond to a Bott connection on N(we have dropped the super-
script for notational simplicity). We obtain the corresponding connection forms α(0) =αon
G(0)
1= Fr+(N) and α(1) on ∆1× G(1)
1defined by
α(1)
(t;u):= t(p1
0)∗α+ (1 −t)(p1
1)∗α=t r∗α+ (1 −t)s∗α
for (t;u)∈∆1× G(1)
1. For simplicity let us denote (p1
i)∗αby simply αi,i= 0,1. Then since
q= 1, the curvature of α(1) is given simply by
R(1)
(t;u):= dt ∧(α0−α1) + tdα0+ (1 −t)dα1.
Now in W O1the Godbillon-Vey invariant is given by the cocycle h1c1, which is mapped via the
φG
αof Theorem 3.3 to the simplicial differential form
α(1) ∧R(1) = (tα0+ (1 −t)α1)∧(dt ∧(α0−α1) + tdα0+ (1 −t)dα1)
=−dt ∧(tα0+ (1 −t)α1)∧(α0−α1)+(tα0+ (1 −t)α1)∧(tdα0+ (1 −t)dα1)
21

on ∆1× G(1)
1. Integration over ∆1then produces the form
Z1
0
α(1) ∧R(1) =−Z1
0
tdt ∧α0∧(α0−α1)−Z1
0
(1 −t)dt ∧α1∧(α0−α1)
=−1
2(α0+α1)∧(α0−α1) (12)
on G1. Equation (12) is geometrically opaque, and our immediate task now is to elucidate its
geometric content. First, we will prove that the factor α0−α1has an interpretation as a path
integral of the Bott curvature form R.
Proposition 4.1. Let (M, F)be codimension q, and let α∈Ω1(Fr+(N); gl(q, R)) correspond
to a Bott connection on Nwith associated curvature R∈Ω2(Fr+(N); gl(q, R)). For u∈ G1,
let γ: [0,1] →Fr+(N)be any smooth path in a leaf of FFr+(N)that represents u. Letting
p:TFr+(N)→NFr+(N)denote the projection, for any X∈TuG1choose a smooth vector field
˜
X∈Γ∞(γ([0,1]); TFr+(N)) along γfor which
1. dsuX=˜
Xγ(0) and druX=˜
Xγ(1), and
2. the projection Z=p˜
X∈Γ∞(γ([0,1]); NFr+(N))of ˜
Xto a normal vector field is parallel
along γwith respect to the Bott connection ∇Fr+(N)(see Proposition 2.15) for the foliation
FFr+(N)determined by α.
Then
(α0−α1)u(X) = Zγ
R( ˙γ, ˜
X).(13)
In particular, the integral on the right hand side does not depend on the choices of γand ˜
X.
Proof. That such a vector field ˜
Xcan be chosen is a consequence of the surjectivity of the
projection ptogether with the definition of the parallel transport map for NFr+(N)along γ.
More precisely, since Gacts on Nby parallel transport with respect to ∇[, the projections of
Xuto a normal vector on Nvia the range and source are mapped to one another by parallel
transport along any path representing u. We compute
R( ˙γ, ˜
X) = dα( ˙γ, ˜
X)+(α∧α)( ˙γ, ˜
X) = ˙γα(˜
X)−˜
Xα( ˙γ)−α([ ˙γ, ˜
X]) + (α∧α)( ˙γ, ˜
X).
Since αis a Bott connection, all the leafwise tangent vectors ˙γlie in the kernel of αand so we
can simplify to
R( ˙γ, ˜
X) = ˙γα(˜
X)−α([ ˙γ, ˜
X]).
By definition of a Bott connection we moreover have
[ ˙γ, ˜
X] = ∇Fr+(N)
˙γ(Z) + [ ˙γ, ˜
XTFFr+(N)] = [˙γ, ˜
XTFFr+(N)]
since Zis parallel along γ, and where ˜
XTFFr+(N)is the leafwise component of ˜
X. Since TFFr+(N)
is closed under brackets, [˙γ, ˜
X] is also annihilated by αand we have
R( ˙γ, ˜
X) = ˙γα(˜
X).
22

Therefore by Stokes’ theorem
Zγ
R( ˙γ, ˜
X) = αγ(1)(˜
Xγ(1))−αγ(0)(˜
Xγ(0))=(α0−α1)u(X)
as claimed.
Next we show that the number (α0−α1)u(X) defined for X∈TuG1depends only on the
projection of Xto a vector in the normal bundle Nof (M, F), obtained via the range or source.
Proposition 4.2. Let πG1:G1→ G be the projection induced by πFr+(N): Fr+(N)→M, and
let TrG1and TsG1denote the tangent bundles to the range and source fibres of G1respectively,
so that the differentials of r◦πG1and s◦πG1define fibrewise isomorphisms
N1:= TG1/TrG1⊕TsG1⊕ker(dπG1)→N.
Then the formula in Equation (13) depends only on the class [X]∈(N1)udetermined by X,
and not on the choices of γand ˜
X.
Proof. To see that (α0−α1)u(X) depends only on the class of Xin N1we consider a perturbation
X0=X+Yof Xwhere Y∈ker(dπG1)u. Identify ker(dπG1) with Gn(ker(dπFr+(N))) =
Gn(Fr+(N)×gl(q, R)). By commutativity of the action of Gon Fr+(N) with that of GL+(q, R),
the action of Gon the gl(q, R) factor is by the identity, and we have dru(Y) = dsu(Y) = Y.
Since αis a connection form we have α(Y) = Yand therefore
(α0−α1)u(X+Y)=(α0−α1)u(X) + Y−Y= (α0−α1)u(X).
Now suppose that X0=X+Z, where Z∈T(Gr(u)
1)⊕T((G1)s(u)). Then both druZand dsuZ
are contained in TFFr+(N)and therefore are annihilated by α. Hence
(α0−α1)u(X+Z) = (α0−α1)u(X)
as required.
In the paper [27], the differential form
∂α = (p1
0)∗α−(p1
1)∗α= (1
0)∗α−(1
1)∗α=r∗α−s∗α∈Ω1(G1;gl(q, R))
determined by a connection form αappears as a measure of the failure of the connection form α
to be invariant under the action of G1. In light of Proposition 4.1 we give any such differential
form arising from a Bott connection a special name.
Definition 4.3. Given a Bott connection form α∈Ω1(Fr+(N); gl(q, R)), we refer to the 1-form
RG:= ∂α = (r∗α−s∗α)∈Ω1(G1;gl(q, R))
as the integrated curvature of α.
23

Remark 4.4. Path integrals of differential forms such as in Proposition 4.1 are already of great
use in determining de Rham representatives for loop space cohomology [6, 7, 19, 33]. Since
the holonomy groupoid is really a coarse sort of “path space”, in light of Proposition 4.1 one
expects it to be possible to obtain a characteristic map for the holonomy groupoid defined in
terms of iterated path integrals. This would provide an exciting new geometric window into the
existing theory of foliations, and has the potential to open up links with loop space theory. We
leave this question to a future paper.
Let us now come back to the Godbillon-Vey invariant of a codimension 1 foliated manifold
(M, F). Denoting αG:= −1
2(r∗α+s∗α) for notational simplicity, the differential form in (12)
can now be written Z1
0
α(1) ∧R(1) =αG∧RG∈Ω2(G(1)
1).
Thus we have reconciled the Chern-Weil description of the Godbillon-Vey invariant, as “Bott
connection wedge curvature”, with the image of the Godbillon-Vey invariant arising from the
characteristic map of Theorem 3.3. Proposition 4.2 now allows us to integrate against αG∧RG
in the following way.
Lemma 4.5. Let (M, F)be a transversely orientable foliated n-manifold of codimension 1, and
let a0, a1∈C∞
c(G1; Ω 1
2). Then, setting x=πFr+(N)(φ)for φ∈Fr+(N), the formula
gv(a0, a1)φ:= Zu∈Gx
a0(u−1, u ·φ)a1(u, φ)(αG∧RG)(u,φ)
defines a compactly supported 1-density gv(a0, a1)∈Γ(|Fr+(N)|).
Proof. Fix φ∈Fr+(N). For each (u, φ)∈ G1, we have
a0(u−1, u ·φ)a1(u, φ)∈ | F Fr+(N)|φ⊗ | F Fr+(N)|u·φ,
while (αG∧RG)(u,φ)∈Λ2(T∗
(u,φ)G1), which we now describe using coordinates.
Consider a chart B1×B2×V×R∗
+for G1about (u, φ), where B1, B2are open balls in Rdim(F)
and Van open ball in Rsuch that B2×V∼
=U2⊂Mis a foliated chart about πFr+(N)(φ)
and B1×hu(V)∼
=U1is a foliated chart about πFr+(N)(u·φ), where hu:V→hu(V)⊂R
is a holonomy diffeomorphism representing u. By Proposition 4.2, in the local coordinates
((xi
1)dim F
i=1 ; (xi
2)dim(F)
i=1 ;z;t)∈B1×B2×V×R∗
+we have that RG=f1dz for some f1defined on
B1×B2×V×R∗
+. Moreover, with t−1dt the Maurer-Cartan form on R∗
+,αGis of the form
f2dz +t−1dt for some smooth f2defined on B1×B2×V. Consequently, αG∧RGis of the form
ft−1dt ∧dz for some smooth function fdefined on B1×B2×V×R∗
+.
Since the coordinate differentials dz and dt span T∗
φFr+(N)T∗
φFFr+(N), at each point
(u, φ)∈ G1we have that
a0(u−1, u ·φ)a1(u, φ)(αG∧RG)(u,φ)∈ | Fr+(N)|φ⊗ |TFFr+(N)|u·φ.
24

Then by compact support of a0and a1, the integral
gv(a0, a1)φ=Zu∈Gx
a0(u−1, u ·φ)a1(u, φ)(αG∧RG)(u,φ)∈ | Fr+(N)|φ
is well-defined and gv(a0, a1) is a compactly supported density on Fr+(N).
Theorem 4.6. Let (M, F)be a transversely orientable foliated n-manifold of codimension 1.
Then for a0, a1∈C∞
c(G1; Ω 1
2)the formula
ϕgv (a0, a1) := Zφ∈Fr+(N)
gv(a0, a1)φ=Z(u,φ)∈G1
a0(u−1, u ·φ)a1(u, φ)(αG∧RG)(u,φ)
defines a cyclic 1-cocycle ϕgv on the convolution algebra C∞
c(G1; Ω 1
2).
Proof. We work with Connes’ λ-complex [10]. For notational simplicity we denote elements
of G1by vi, and we use the notation Rv0v1v2∈Fr+(N)to mean the iterated integral over all
triples (v0, v1, v2)∈ G(3)
1for which v0v1v2∈Fr+(N), followed by an integral over Fr+(N).
For a0, a1, a2∈C∞
c(G1; Ω 1
2), we calculate
ϕgv (a0a1, a2) = Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2)(αG∧RG)v2
=Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2) (2
0)∗(αG∧RG)(v1,v2),
ϕgv (a0, a1a2) = Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2)(αG∧RG)v1v2
=Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2) (2
1)∗(αG∧RG)(v1,v2)
and
ϕgv (a2a0, a1) = Zv2v0v1∈Fr+(N)
a2(v2)a0(v0)a1(v1)(αG∧RG)v1
=Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2) (2
2)∗(αG∧RG)(v1,v2).
Because h1c1∈W O1is closed under d, the component αG∧RG∈Ω2(G(1)
1) of its image under
the cochain map ψα:W O1→Ω∗(G(∗)
1) of Theorem 3.3 is closed under ∂: Ω2(G(1)
1)→Ω2(G(2)
1).
Thus
bϕgv (a0, a1, a2) =ϕgv (a0a1, a2)−ϕgv (a0, a1a2) + ϕgv (a2a0, a1)
=Zv0v1v2∈Fr+(N)
a0(v0)a1(v1)a2(v2)∂(αG∧RG)(v1,v2)
=0
making ϕgv a Hochschild cocycle.
25

It remains only to check that ϕgv(a0, a1) = −ϕgv (a1, a0). For this, we observe that by
definition RG
φ=αφ−αφ= 0 for any unit φ∈ G1, hence
0 = ∂(αG∧RG)(v−1,v)= (αG∧RG)v−(αG∧RG)v−1v+ (αG∧RG)v−1= (αG∧RG)v−1+ (αG∧RG)v.
Therefore
ϕgv (a0, a1) = Zv∈G1
a0(v−1)a1(v)(αG∧RG)v=−Zv−1∈G1
a1(v)a0(v−1)(αG∧RG)v−1
=−ϕgv (a1, a0)
making ϕgv a cyclic cocycle.
Definition 4.7. We refer to the cyclic cocycle ϕgv on C∞
c(G1; Ω 1
2)given in Theorem 4.6 as the
Godbillon-Vey cyclic cocycle.
Remark 4.8. The Godbillon-Vey cyclic cocycle for C∞
c(G1; Ω 1
2) is the analogue of the Connes-
Moscovici formula [12, Proposition 19] for the crossed product of a manifold by a discrete group
action. Note that in contrast with the ´etale setting of Connes and Moscovici, the differential
form αG∧RGon G1with respect to which ϕgv is defined has, by Proposition 4.1, an explicit
interpretation in terms of the integral of the Bott curvature along paths representing elements
in G1. Such a geometric interpretation is novel, and is completely lost in the ´etale setting that
has been almost exclusively used in studying the secondary characteristic classes of foliations
using noncommutative geometry.
One has the following immediate corollary of Proposition 4.1 which, while completely un-
surprising, is novel due to our non-´etale perspective that incorporates the global transverse
geometry of (M, F).
Corollary 4.9. If (M , F)is a codimension 1, transversely orientable foliated manifold with a
flat Bott connection, then the Godbillon-Vey cyclic cocycle vanishes.
Our final task is to demonstrate that the Godbillon-Vey cyclic cocycle coincides with the
cocycle obtained from the local index formula for the semifinite spectral triple considered in
[28, Section 4.3]. For this purpose, it will be convenient to have a formula for the Godbillon-
Vey cyclic cocycle in terms of a transverse volume form ω∈Ω1(M) (that is, a form ωthat
is nonvanishing and is identically zero on leafwise tangent vectors). By the final statement of
Proposition 4.2, we know that we can write
RG=δ(s◦π(1))∗ω(14)
for some smooth function δ:G1→R, where s:G → Mis the source and where π(1) :G1→ G
is the projection. Since r∗ωand s∗ωboth annihilate the tangents to the range and source fibres
we can formulate the following definition.
Definition 4.10. Given a transverse volume form ω∈Ω1(M), the smooth homomorphism
∆ : G → R∗
+defined by the equation
r∗ω= ∆ s∗ω
26

is called the modular function or Radon-Nikodym derivative associated to ω.
Note that the face maps 2
i:G(2) → G of Gsatisfy
s◦2
0(u1, u2) = s◦2
1(u1, u2) = s(u2), s ◦2
2(u1, u2) = r◦2
0(u1, u2) = r(u2)
for all (u1, u2)∈ G(2) . Via a mild abuse of notation let us also denote the face maps of G1by j
i.
Then the fact that RG=∂α gives (2
0)∗RG−(2
1)∗RG+ (2
2)∗RG=∂2α= 0. Therefore, letting
π(2) :G(2)
1→ G(2) denote the projection, we have
0 = δ(u2)(s◦π(1) ◦2
0)∗ω(u1,u2)−δ(u1u2)(s◦π(1) ◦2
1)∗ω(u1,u2)
+δ(u1)(s◦π(1) ◦2
2)∗ω(u1,u2)
=δ(u2)(s◦2
0◦π(2))∗ω(u1,u2)−δ(u1u2)(s◦2
1◦π(2))∗ω(u1,u2)
+δ(u1)(s◦2
2◦π(2))∗ω(u1,u2)
=δ(u1)−δ(u1u2)(s◦2
0◦π(2))∗ω(u1,u2)+δ(u1)(r◦2
0◦π(2))∗ω(u1,u2)
=δ(u2)−δ(u1u2) + δ(u1)∆(u2)(s◦2
0◦π(2))∗ω(u1,u2)
for all (u1, u2)∈ G(2) . Hence
δ(u1u2) = δ(u2) + δ(u1)∆(u2),(u1, u2)∈ G(2) .(15)
Now the choice of ωdetermines a trivialisation Fr+(N)∼
=M×R∗
+in which we can write
elements of G1as (u, x, t)∈ G n(M×R∗
+). By the arguments of the second paragraph in the
proof of Lemma 4.5 we can now write
(αG∧RG)(u,x,t)=δ(u)
tdt ∧ωx,(u, x, t)∈ G n(M×R∗
+)
so that our Godbillon-Vey cyclic cocycle becomes
ϕgv (a0, a1) = −Z(x,t)∈M×R∗
+Zu∈Gx
a0(u−1,∆(u)t)a1(u, t)δ(u)
tωx∧dt
for a0, a1∈C∞
c(G1; Ω 1
2). In order to compare our formula with that in [28], we will want to
assume that G1is of the form G1= Fr+(N)oGrather than GnFr+(N). Now these groupoids
are of course isomorphic via the map Fr+(N)oG 3 (φ, u)7→ (u, u−1·φ)∈ G nFr+(N), and
a function a∈C∞
c(GnFr+(N); Ω 1
2) identifies under this map with ˜a∈C∞
c(Fr+(N)oG; Ω 1
2)
given by
˜a(φ, u) := a(u, u−1·φ),(φ, u)∈Fr+(N)oG.
With these identifications, and using the notational convention
au(φ) := a(φ, u) (φ, u)∈Fr+(N)oG
for a∈C∞
c(Fr+(N)oG; Ω 1
2), we see that the Godbillon-Vey cyclic cocycle is defined for a0, a1∈
27

C∞
c(Fr+(N)oG; Ω 1
2) by the formula
ϕgv (a0, a1) = −Z(x,t)∈M×R∗
+Zu∈Gxa0
u(x, t)a1
u−1(u−1·x, ∆(u−1)t)δ(u−1)
tωx∧dt. (16)
Let us now recall the cocycle φ1obtained via the local index formula in [28, Section 4.3]. In the
coordinates we have chosen in this paper, φ1is given by the equation
φ1(a0, a1) = −(2πi)1
2Z(x,t)∈M×R∗
+Zu∈Gxa0
u(x, t)a1
u−1(u−1·x, ∆(u−1)t)∂log ∆(u−1)
tωx∧dt (17)
for a0, a1∈C∞
c(G1; Ω 1
2). Thus in order to conclude that the index formula φ1of [28] and the
cyclic cocycle ϕgv of Equation (16) coincide (up to the constant multiple (2πi)1
2), we need only
show that δ=∂log ∆. This will be a consequence of the following fact, which holds for foliations
of arbitrary codimension and gives a geometric interpretation (in the non-´etale setting) for the
off-diagonal term appearing in the triangular structures considered by Connes [9, Lemma 5.2]
and Connes-Moscovici [13, Part I].
Proposition 4.11. Let (M, F)be a transversely orientable foliated manifold of codimension q,
and let α[∈Ω1(Fr+(N); gl(q, R)) be a Bott connection form. Let H:= ker(α[)/T FFr+(N)be
the horizontal normal bundle determined by α[, let V:= ker(dπFr+(N))be the vertical tangent
bundle, and let
NFr+(N)=V⊕H∼
=Fr+(N)×(gl(q, R)⊕Rq)
be the corresponding decomposition of NFr+(N), with Vand Htrivialised as in the proof of
Proposition 2.15. With respect to this decomposition, for u∈ G and for any φ∈Fr+(N)s(u)the
action uFr+(N)
∗: (NFr+(N))φ→(NFr+(N))u·φcan be written
uFr+(N)
∗= idgl(q,R)RG
u
0 idRq!.(18)
Proof. That the top left corner is idgl(q,R)follows from the commutativity of the left action of
Gon Fr+(N) with the right action of GL+(q, R), and the bottom left entry is zero for the same
reason. For the bottom right corner, we note that equivariance of the map πFr+(N): Fr+(N)→
Mwith respect to the action of Gimplies that the induced fibrewise isomorphisms πφ:Hφ→
NπFr+(N)(φ)are also equivariant. Thus if [vφ]∈Hφ, denoting u22 ·[vφ] := projHu∗[vφ]and
letting u∗:Ns(u)→Nr(u)denote the action of uon N, we have
(u·φ)−1◦πu·φ(u22 ·[vφ]) = φ−1◦u−1
∗◦u∗◦πφ([vφ]) = φ−1◦πφ(vφ)
giving the bottom right entry of (18).
Finally we come to the top right entry. Since a connection form maps vertical vectors to
themselves, the top right entry of (18) is the map which sends [vφ]∈Hto α[(uFr+(N)
∗vφ), where
uFr+(N)
∗vφis any element of TFr+(N) representing uFr+(N)
∗[vφ]. Since vφis contained in ker(α[),
28

we can equally regard the top right entry as the map which sends [vφ]∈(ker(α[)/T FFr+(N)) to
α[(uFr+(N)
∗vφ)−α[(vφ),
which by (13) in Proposition 4.1 coincides with RG
u(vφ), the well-definedness of which is due to
Proposition 4.2.
Coming back to our codimension 1 foliation (M, F), recall [28, p. 23] that the function
∂log ∆ on Gis by definition the top right corner of the matrix in Equation (18) that gives the
action of Gon NFr+(N). Therefore
δ=∂log ∆
as required, proving the following result.
Theorem 4.12. The Godbillon-Vey cyclic cocycle of Equation (16) coincides with the local
index formula cocycle in Equation (17) for the semifinite spectral triple considered in [28, Section
4.3].
References
[1] M. T. Benameur, Triangulations and the stability theorem for foliations, Pacific J. Math.
179 (1997), 221–239.
[2] R. Bott, On a topological obstruction to integrability, Proc. Sympos. Pure Math. 16 (1970),
127–131.
[3] , Lectures on characteristic classes and foliations, Lectures on Algebraic and Dif-
ferential Topology, vol. 279, Springer, 1972.
[4] , On characteristic classes in the framework of Gelfand-Fuks cohomology, Ast´erique
32–33 (1976), 113–139.
[5] R. Bott, H. Shulman, and J. Stasheff, On the de Rham theory of certain classifying spaces,
Adv. Math. 20 (1976), 43–56.
[6] K. T. Chen, Iterated integrals of differential forms and loop space homology, Ann. Math.
97 (1973), 217–246.
[7] , Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831–879.
[8] A. Connes, A survey of foliations and operator algebras, Proc. Sympos. Pure, 1982, pp. 38–
521.
[9] , Cyclic cohomology and the transverse fundamental class of a foliation, Geometric
methods in operator algebras (Kyoto, 1983) (H. Araki and E. G. Effros, eds.), Longman
Sci. Tech., Harlow, 1986, pp. 52–144.
[10] , Noncommutative Geometry, Academic Press, 1994.
29

[11] A. Connes and H. Moscovici, Differentiable cyclic cohomology and Hopf algebraic structures
in transverse geometry, Essays on geometry and related topics, Vol. 1, 2, vol. 38, pp. 217–
255, Monogr. Enseign. Math., Geneva, 2001.
[12] , Background independent geometry and Hopf cyclic cohomology,
arXiv:math/0505475v1, 2015.
[13] A. Connes and M. Moscovici, The local index formula in noncommutative geometry, GAFA
5(1995), 174 – 243.
[14] , Hopf algebras, cyclic cohomology and transverse index theory, Comm. Math. Phys.
198 (1998), 199 – 246.
[15] M. Crainic and I. Moerdijk, Foliation groupoids and their cyclic homology, Adv. Math. 157
(2001), 177–197.
[16] , Cech-De Rham theory for leaf spaces of foliations, Math. Ann. 328 (2004), 59–85.
[17] J. Dupont, Curvature and Characteristic Classes, Lecture Notes in Mathematics, Springer-
Verlag, 1973.
[18] , Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology
15 (1976), 233–245.
[19] E. Getzler, J. D. S. Jones, and S. Petrack, Differential forms on loop spaces and the cyclic
bar complex, Topology 30 (1991), 339–371.
[20] A. Gorokhovsky, Characters of cycles, equivariant characteristic classes and Fredholm mod-
ules, Comm. Math. Phys. 208 (1999), 1–23.
[21] , Secondary characteristic classes and cyclic cohomology of Hopf algebras, Topology
41 (2002), 993–1016.
[22] S. Guelorget, Alg`ebre de Weil du groupe lin´eaire, Application aux classes caract´eristiques
d’un feuilletage, Springer Lecture Notes in Math., vol. 484, pp. 179–191, Springer, Berlin,
Heidelberg, 1975.
[23] V. W. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory,
Springer, 1999.
[24] J. Heitsch and S. Hurder, Secondary classes, Weil measures and the geometry of foliations,
J. Differential Geom. 20 (1984), 291–309.
[25] M. Hilsum and G. Skandalis, Stabilit´e des C∗-alg`ebres de feuilletages, Ann. Inst. Fourier
(Grenoble) 33 (1983), 201–208.
[26] S. Hurder and A. Katok, Secondary classes and transverse measure theory of a foliation,
Bull. Amer. Math. Soc. 11 (1984), 347–350.
30

[27] C. Laurent-Gengoux, J.-L. Tu, and P. Xu, Chern-Weil map for principal bundles over
groupoids, Math. Z. 255 (2007), 451–491.
[28] L. MacDonald and A. Rennie, The Godbillon-Vey invariant in equivariant KK-theory,
arXiv:1811.04603, 2018.
[29] S. Morita, Geometry of Differential Forms (Translations of Mathematical Monographs),
American Mathematical Society, 2001.
[30] H. Moscovici, Geometric construction of Hopf cyclic characteristic classes, Adv. Math. 274
(2015), 651–680.
[31] H. Moscovici and B. Rangipour, Cyclic cohomology of hopf algebras of transverse symme-
tries in codimension 1, Adv. Math. 210 (2007), 323–374.
[32] G. Segal, Classifying spaces and spectral sequences, Inst. Hautes ´
Etudes Sci. Publ. Math.
34 (1968), 105–112.
[33] T. Tradler, S. Wilson, and M. Zeinalian, Loop differential K-theory, Ann. Math. Blaise
Pascal 22 (2015), 121–163.
[34] H. E. Winkelnkemper, The graph of a foliation, Ann. Global Anal. Geom. 1(1983), no. 3,
51–75.
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