there is an isomorphism of Lie groupoids:

$L^{2}\times
J^{k}_{\mathbb{R}}\ni(x_{2},x_{1},\alpha)\mapsto[j^{k}_{x_{2}}(\mathrm{pr}_{2}),\alpha\cdot
j^{k}_{x_{2}}(\mathrm{pr}_{2})]\in\operatorname{Gauge}(P^{k}(\mathcal{F})).$

We also have already a Lie groupoid isomorphism

$G_{\textup{full}}(\mathcal{F})\to(\mathbb{R}^{\ell})^{2}\times
J^{k}_{\mathbb{R}}$

by Theorem 6.15. It remains, therefore, to check that the diagram

${G_{\textup{full}}(\mathcal{F})}$${\operatorname{Gauge}(P^{k}(\mathcal{F}))}$${L^{2}\times
J^{k}_{\mathbb{R}}}$

is commutative. For this purpose, recall the full holonomy atlas
$\\{\widetilde{\Omega}_{\theta}:\theta\in\mathbb{R}\\}$ (Definition 6.9). Fix
$\theta\in\operatorname{Diff}_{0}(\mathbb{R})$;
$x_{1},x_{2}\in\mathbb{R}^{\ell}$; $t\in\mathbb{R}$ and put
$w\coloneqq(x_{2},x_{1},t,0)\in\widetilde{\Omega}_{\theta}$. It is
straightforward to chase $Q_{\widetilde{\Omega}_{\theta}}(w)\in
G_{\textup{full}}(\mathcal{F})$ both ways around the above diagram and confirm
that result is the same. ∎

##  9 Extracting the holonomy groupoid from the full holonomy groupoid

The holonomy groupoid of a singular foliation is contained in the full
holonomy groupoid as an open subgroupoid. To be precise, $G(\mathcal{F})$ is
the $s$-connected component $G_{\textup{full}}(\mathcal{F})$ (c.f. Proposition
2.30 and [2], Theorem 0.1).

######  Definition 9.1.

The $\mathbf{s}$-connected component of a Lie groupoid $G$ is the subgroupoid
$G_{0}\subseteq G$ consisting of all $g\in G$ that can be connected to $s(g)$
by a path in $G_{s(g)}$.

In the preceding section, we obtained a description

$G_{\textup{full}}(\mathcal{F})=(M\setminus
L)^{2}\cup\operatorname{Gauge}(P^{k}(\mathcal{F}))$

for the full holonomy groupoid (Theorem 8.6). As a corollary, we have:

######  Theorem 9.2.

Let $(M,\mathcal{F})$ be a transverse order $k\geq 2$ foliation with singular
leaf $L$. If $M\setminus L$ is connected, then

$G(\mathcal{F})\cong(M\setminus
L)^{2}\cup\operatorname{Gauge}(P^{k}(\mathcal{F}))_{0}.$

If $M\setminus L$ has two connected components $U$ and $V$, then

$G(\mathcal{F})\cong U^{2}\cup
V^{2}\cup\operatorname{Gauge}(P^{k}(\mathcal{F}))_{0}.$

Here, $\operatorname{Gauge}(P^{k}(\mathcal{F}))_{0}$ denotes the $s$-connected
component of $\operatorname{Gauge}(P^{k}(\mathcal{F}))$. The smooth structure
of $G(\mathcal{F})$ is the one inherited from
$G_{\textup{full}}(\mathcal{F})$. ∎

Although the bundle $P^{k}(\mathcal{F})$ is in fact a manifold in our sense
(i.e. is metrizable), it is still very large; both it and its structure group
have continuum-many components. One may therefore desire a more concrete
description of $G(\mathcal{F})_{L}$ which avoids this bundle. The following
proposition, whose proof we omit, shows that we in fact only need to deal with
one connected component of $P_{0}\subseteq P^{k}(\mathcal{F})$, which is
automatically a second-countable manifold.

######  Proposition 9.3.

Let $P\to B$ be a (smooth, left) principal $H$-bundle, where $H$ is a
(possibly disconnected) Lie group.

  1. 1.

If $p\in P$, then the set of $h\in H$ for which $hp$ belongs to the same
component of $P$ as $p$ is a closed and open subgroup $H_{p}\subseteq H$.

  2. 2.

If $p$ and $q$ belong to the same component of $P$, then $H_{p}=H_{q}$.

  3. 3.

Let $P_{0}$ be a component of $P$. Then $P_{0}$ is a principal $H_{0}$-bundle,
where $H_{0}$ is the stabilizer of $P_{0}$ or, equivalently, any particular
point of $P_{0}$.

  4. 4.

The inclusion $P_{0}\times P_{0}\to P\times P$ descends to a Lie groupoid
isomorphism from $\operatorname{Gauge}(P_{0})$ onto the $s$-connected
component $\operatorname{Gauge}(P)_{0}\subseteq\operatorname{Gauge}(P)$. ∎

######  Corollary 9.4.

Let $(M,\mathcal{F})$ be a transverse order $k\geq 2$ foliation with singular
leaf $L$. Then $G(\mathcal{F}^{K})_{L}$ is isomorphic to
$\operatorname{Gauge}(P_{0})$, where $P_{0}$ is any connected component of the
principal $J^{k}$-bundle $P^{k}(\mathcal{F})$.

The following result gives additional insight into the holonomy _groups_ of a
transverse order $k$ foliation at points of its singular leaf.

######  Theorem 9.5.

Let $(M,\mathcal{F})$ be a transverse order $k$ foliation with singular leaf
$L$. Fix $x_{0}\in L$ and $q_{0}\in P^{k-1}(\mathcal{F})_{x_{0}}$. Let
$\gamma:\pi_{1}(L,x_{0})\to J^{k-1}$ be the monodromy homomorphism determined
by $q_{0}$ and let $\Gamma\subseteq J^{k-1}$ be the range of $\gamma$.
Finally, let $\Gamma_{\mathbb{R}}\subseteq J^{k}$ be the preimage of $\Gamma$
under the projection $P^{k}(\mathcal{F})\to P^{k-1}(\mathcal{F})$. Then, the
isotropy group of $G(\mathcal{F})$ at any point of $L$ is isomorphic, as a Lie
group, to the one-dimensional group $\Gamma_{\mathbb{R}}$. For convenience,
the following diagram summarizes the relationship between the various groups
at hand:

${\mathbb{R}}$${J^{k}_{\mathbb{R}}}$${J^{k-1}_{d}}$${\mathbb{R}}$${\Gamma_{\mathbb{R}}}$${\Gamma}$${\pi_{1}(L,x_{0})\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\gamma}$

###### Proof.

Let $Q_{0}$ denote the connected component of $P^{k-1}(\mathcal{F})$
containing $q_{0}$. By standard covering space theory, the subgroup $\Gamma$
equals the stabilizer of $Q_{0}$ so that, by Proposition 9.3, $Q_{0}$ becomes
a principal $\Gamma$-bundle. Let $P_{0}$ be the preimage of $Q_{0}$ under the
projection $P^{k}(\mathcal{F})\to P^{k-1}(\mathcal{F})$. Since the fibers of
this projection are copies of $\mathbb{R}$, it follows that $P_{0}$ is a
connected component of $P^{k}(\mathcal{F})$. Thus, by Proposition 9.3 (4),

$\operatorname{Gauge}(P_{0})\cong\operatorname{Gauge}(P^{k}(\mathcal{F}))_{0}\cong
G(\mathcal{F})_{L}.$

The subgroup of $J^{k}_{\mathbb{R}}$ stabilizing $P_{0}$ is
$\Gamma_{\mathbb{R}}$. By Proposition 9.3 (3), $P_{0}$ is a principal
$\Gamma_{\mathbb{R}}$-bundle, whence every isotropy group of
$\operatorname{Gauge}(P_{0})$ is isomorphic to $\Gamma_{\mathbb{R}}$. ∎

######  Corollary 9.6.

With notation as in Theorem 9.5, we have

$C^{*}(G(\mathcal{F})_{L})\cong C^{*}(\Gamma_{\mathbb{R}})\otimes\mathbb{K},$

where $\mathbb{K}$ denotes the compact operators on the canonical $L^{2}$
space of $L$.

###### Proof.

From [17], Theorem 3.1 it follows that, if $G\rightrightarrows M$ is any
transitive Lie groupoid (hence a gauge groupoid), then $C^{*}(G)\cong
C^{*}(H)\otimes\mathbb{K}(L^{2}(M))$, where $H$ denotes any isotropy group of
$G$. ∎

######  Remark 9.7.

The homomorphisms $\gamma$ in Theorem 9.5 are exactly the same ones used to
define the holonomy invariant $h(\mathcal{F})$ (Definition 7.6), so Corollary
9.6 shows that knowledge of the invariant $h(\mathcal{F})$ is enough to
compute $C^{*}(G(\mathcal{F})_{L})$.

##  10 Transverse order $k$ foliations on line bundles

Throughout this section, $L$ is a connected, smooth manifold. If $E\to L$ is a
smooth line bundle, we always identity $L$ with the zero section of $E$, so
that $L\subseteq E$.

######  Definition 10.1.

Fix a positive integer $r$.

  * •

If $E\to L$ is a line bundle, we write $J^{r}(E,\mathbb{R})\to L$ for the
smooth, left principal $J^{r}$-bundle whose fiber over $x\in L$ consists of
all $r$-jets at $x$ of local diffeomorphisms $E_{x}\to\mathbb{R}$ sending
$x\mapsto 0$.

  * •

If $U\subseteq E$ is open and $p:U\to\mathbb{R}$ is a submersion with
$p^{-1}(0)=U\cap L$, we write $j^{r}(p):U\cap L\to J^{r}(E,\mathbb{R})$ for
the smooth section defined by $j^{r}(p)(x)\coloneqq j_{x}^{r}(p|_{E_{x}\cap
U})$.

  * •

If $\pi_{i}:E_{i}\to L$ is a line bundle for $i=1,2$, we write
$J^{r}(E_{1},E_{2})\to L$ for the smooth fiber bundle whose fiber over $x\in
L$ consists of $r$-jets at $x$ of (local) diffeomorphisms
$(E_{1})_{x}\to(E_{2})_{x}$ sending $x\mapsto x$.

  * •

Let $\theta:U_{1}\to U_{2}$ be a diffeomorphism, where $U_{i}\subseteq E_{i}$
is open for $i=1,2$. We say that $\theta$ is fiberwise if
$\pi_{2}(\theta(e))=\pi_{1}(e)$ for all $e\in U_{1}$ and $\theta(L\cap
U_{1})=L\cap U_{2}$. If $\theta$ is fiberwise, we write $j^{r}(\theta):U\cap
L\to J^{r}(E_{1},E_{2})$ for the smooth section defined by
$j^{r}(\theta)(x)=j^{r}_{x}(\theta|_{E_{x}\cap U})$.

The following proposition shows that all sections of these jet bundles can be
lifted to honest maps.

######  Proposition 10.2.

Let $r$ be a positive integer.

  1. 1.

Let $E$ be a line bundle over $L$. Then, every local section of
$J^{r}(E,\mathbb{R})$ is equal to $j^{r}(p)$ for some local submersion $p:U\to
L$ with $p^{-1}(0)=L\cap U$.

  2. 2.

Let $E_{1}$ and $E_{2}$ be line bundles over $L$. Then, every local section of
$J^{r}(E_{1},E_{2})$ is equal to $j^{r}(\theta)$ for some local fiberwise
diffeomorphism $\theta$.

###### Proof.

Both statements follow from the fact that each $r$-jet between a pair of one-
dimensional vector spaces preserving $0$ is represented by a unique polynomial
mapping of degree $\leq r$. ∎

If $P\to M$ and $Q\to M$ are smooth principal $H$-bundles, we write
$\operatorname{Hom}_{H}(P,Q)$ for the usual smooth fiber bundle over $M$ whose
fiber over $x\in M$ is the the set of $H$-equivariant maps $P_{x}\to Q_{x}$.
We note the following without proof.

######  Lemma 10.3.

If $E_{1}$ and $E_{2}$ are line bundles over $L$, then there is a fiber bundle
isomorphism
$\alpha\mapsto\alpha_{*}:J^{r}(E_{1},E_{2})\to\operatorname{Hom}_{J^{r}}(J^{r}(E_{1},\mathbb{R}),J^{r}(E_{2},\mathbb{R}))$
defined by $\alpha_{*}(\beta)=\beta\circ\alpha^{-1}$. ∎

In particular, if $\theta$ is a fiberwise diffeomorphism from $E_{1}$ to
$E_{2}$ defined on a neighbourhood of $L$, the corresponding global section
$j^{r}(\theta):L\to J^{r}(E_{1},E_{2})$ determines a principal bundle
isomorphism $j^{r}(\theta)_{*}:J^{r}(E_{1},\mathbb{R})\to
J^{r}(E_{2},\mathbb{R})$. Thus, $E\mapsto J^{r}(E,\mathbb{R})$ is functorial
for fiberwise diffeomorphisms. In fact, by Proposition 10.2, _every_
isomorphism $J^{r}(E_{1},\mathbb{R})\to J^{r}(E_{2},\mathbb{R})$ is induced by
a fiberwise diffeomorphism.

The following two results are the main findings of this section.

######  Theorem 10.4.

Let $\pi:E\to L$ be a smooth line bundle with connected base manifold $L$.
There is a unique bijection $\mathcal{F}\mapsto\nabla_{\mathcal{F}}$ from

  1. 1.

the set of transverse order $k$ foliations of $E$ with singular leaf $L$ to

  2. 2.

the set of (smooth) flat connections on $J^{k-1}(E,\mathbb{R})$

such that a submersion $p:U\to\mathbb{R}$, where $U\subseteq E$ is open,
satisfying $p^{-1}(0)=U\cap L$ is a local
$\mathcal{F}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersion if and only if the
associated section $j^{k-1}(p):U\cap L\to J^{k-1}(E,\mathbb{R})$ is parallel
for $\nabla_{\mathcal{F}}$.

###### Proof.

Suppose $U\subseteq E$ is open and $p,q:U\to\mathbb{R}$ are submersions with
$p^{-1}(0)=q^{-1}(0)=U\cap L$. Up to shrinking $U$ around $U\cap L$, we may
assume that $p_{x}\coloneqq p|_{U\cap E_{x}}$ and $q_{x}\coloneqq q|_{U\cap
E_{x}}$ are diffeomorphisms onto their images for all $x\in U\cap L$. Define
$\theta_{x}\coloneqq q_{x}\circ(p_{x})^{-1}$ for all $x\in U\cap L$ so that,
by construction, $\theta_{x}$ is a smooth family of local diffeomorphisms of
$\mathbb{R}$ satisfying $\theta_{\pi(e)}(p(e))=q(e)$ on a neighbourhood of
$U\cap L$. Define $h_{qp}:U\cap L\to J^{k-1}$ by
$h_{qp}(x)=j^{k-1}_{0}(\theta_{x})\in J^{k-1}$. By construction, $h_{qp}$ is
the transition function for the sections $j^{k-1}(p)$ and $j^{k-1}(q)$
(Definition 10.1).

Now suppose $\mathcal{F}$ is a transverse order $k$ foliation of $E$ with
singular leaf $L$ and $p,q:U\to\mathbb{R}$ are local
$\mathcal{F}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersions. We claim that the
transition map $h_{qp}:U\cap L\to J^{k-1}$ is locally constant. By Proposition
5.4, given any point $x_{0}\in U\cap L$, we may choose coordinates so that
$x_{0}$ is the point $(0,0)$ in
$\mathbb{R}^{n}=\mathbb{R}^{\ell}\times\mathbb{R}$ and the foliation is
$\mathcal{F}^{k}_{\mathbb{R}^{n}}$. Moreover, we may perform this coordinate
change in such a way that $p:E\to\mathbb{R}$ and $\pi:E\to L$ are given by
$p(x,y)=y$ and $\pi(x,y)=x$ (the linear structure of the fibers is not likely
to be preserved under this coordinate change, but that is irrelevant). By
Theorem 6.6, there are constants $a_{1},\ldots,a_{k-1}$ and a smooth function
$f$ such that

$\theta_{x}(y)=q(x,y)=a_{1}y+\ldots+a_{k-1}y^{k-1}+f(x,y)y^{k}$

holds on a neighbourhood of $(0,0)$, whence
$h_{qp}(x)=j_{0}^{k-1}(\theta_{x})$ is constant on a neighbourhood of any
$x_{0}\in U\cap L$. Therefore, applying Lemma 7.2, there is a unique
$J^{k-1}_{d}$-bundle structure on $J^{k-1}(E,\mathbb{R})$ for which
$j^{k-1}(p)$ is a smooth section for every local
$\mathcal{F}$-$\mathcal{F}^{k}_{\mathbb{R}}$ submersion $p$. This shows the
map $\mathcal{F}\mapsto\nabla_{\mathcal{F}}$ is well-defined.

Conversely, suppose that $\nabla$ is a flat connection on
$J^{k-1}(E,\mathbb{R})$. For each local submersion $p:U_{p}\to\mathbb{R}$ with
$p^{-1}(0)=L\cap U_{p}$ satisfying $j^{k-1}(p)=U_{p}\cap L$, define
$\mathcal{F}_{p}=p^{-1}(\mathcal{F}^{k}_{\mathbb{R}})$. Thus,
$\mathcal{F}_{p}$ is a foliation of $U_{p}$. We use Proposition 2.25 to glue
these foliations together into a foliation $\mathcal{F}$ of $M$. To this end,
it suffices to consider $p$ and $q$ with $U_{p}=U_{q}=U$. It is easy to see
that $\mathcal{F}_{p}$ and $\mathcal{F}_{q}$ both restrict to the trivial one
leaf foliation on $U\setminus L$. It remains only to check the foliations
agree on a neighbourhood of each point $x_{0}\in L$. Using that $j^{k-1}(p)$
and $j^{k-1}(q)$ are $\nabla$-parallel and shrinking $U$ about $x_{0}$, there
exists $h\in J^{k-1}$ with

$j^{k-1}(q)=h\cdot j^{k-1}(p).$

We may pass again to coordinates such that
$x_{0}=(0,0)\in\mathbb{R}^{\ell}\times\mathbb{R}$ and $\pi(x,y)=x$, $p(x,y)=y$
near $(0,0)$. Writing $h=a_{1}y+\ldots+a_{k-1}y^{k-1}$, the equation above
then says that $\frac{\partial^{r}q}{\partial y^{r}}(x,0)=a_{r}$ holds near
$(0,0)$ for $r=1,\ldots,k-1$. Thus, applying Theorem 6.6 again, we have that,
in these coordinates, $p^{-1}(\mathcal{F}^{k}_{\mathbb{R}})$ and
$q^{-1}(\mathcal{F}^{k}_{\mathbb{R}})$ agree with
$\mathcal{F}^{k}_{\mathbb{R}^{n}}$ in a neighbourhood of $(0,0)$.

It is easy to see that these mappings between foliations and flat connections
are inverses of each other. ∎

Next we show that the correspondence of the above theorem is natural with
respect to fiberwise diffeomorphisms.

######  Theorem 10.5.

Let $L$ be a connected, smooth manifold. For $i=1,2$, let $E_{i}$ be a smooth
line bundle over $L$ equipped with a transverse order $k$ foliation
$\mathcal{F}_{i}$ with singular leaf $L$. Suppose that $\theta:U_{1}\to U_{2}$
is a fiberwise diffeomorphism, where $U_{i}$ is an open subset of $E_{i}$
containing $L_{i}$. Then, the following are equivalent:

  1. 1.

$\theta_{*}((\mathcal{F}_{1})_{U_{1}})=(\mathcal{F}_{2})_{U_{2}}$.

  2. 2.

The induced $J^{k-1}$-bundle isomorphism
$j^{k-1}(\theta)_{*}:J^{k-1}(E_{1},\mathbb{R})\to J^{k-1}(E_{2},\mathbb{R})$
pushes forward $\nabla_{\mathcal{F}_{1}}$ to $\nabla_{\mathcal{F}_{2}}$.

###### Proof.

Assume for convenience that $U_{1}=E_{1}$, $U_{2}=E_{2}$. Let $U\subseteq L$
be open and let $p:U\to L$ be a submersion satisfying $p^{-1}(0)=U\cap L$.
Clearly (1) is equivalent to the assertion: $p$ is a local
$\mathcal{F}_{1}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersion if and only if
$p\circ\theta^{-1}$ is a local
$\mathcal{F}_{2}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersion. Meanwhile
(applying Proposition 10.2 (2)), (2) is equivalent to the assertion:
$j^{k-1}(p)$ is $\nabla_{\mathcal{F}_{1}}$ parallel if and only if
$j^{k-1}(\theta)_{*}(j^{k-1}(p))$ is $\nabla_{\mathcal{F}_{2}}$ parallel.
Since

$j^{k-1}(\theta)_{*}(j^{k-1})(p)=j^{k-1}(p)\circ
j^{k-1}(\theta)^{-1}=j^{k-1}(p\circ\theta^{-1}),$

the desired conclusion is an immediate consequence of Theorem 10.4. ∎

##  11 Completeness of the holonomy invariant

Let $(M,\mathcal{F})$ be a transverse order $k$ foliation with singular leaf
$L$. In this section, we show that the holonomy invariant (Definition 7.6)

$h(\mathcal{F})\in[\pi_{1}(L),J^{k-1}]$

together with the diffeomorphism type of $L$ is a complete invariant for the
structure of $\mathcal{F}$ nearby to $L$.

Given a line bundle equipped with a transverse order $k$ foliation whose
singular leaf is the zero section, there are ostensibly two principal
$J^{k-1}_{d}$-bundles at play: the one from Section 7 and the one from Section
10. As one would probably suspect, these two bundles are in fact the same.

######  Proposition 11.1.

Let $E\to L$ be a smooth line bundle and let $\mathcal{F}$ be transverse order
$k$ foliation of $E$ with singular leaf $L$. There is an isomorphism of
principal $J^{k-1}_{d}$-bundles $P^{k-1}(\mathcal{F})\to
J^{k-1}(E,\mathbb{R})$ such that, if $p:U\to\mathbb{R}$ is a local
$\mathcal{F}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersion and $x\in U\cap L$,
then $j^{k}_{x}(p)\mapsto j^{k}_{x}(p|_{E_{x}\cap U})$. Here,
$J^{k}(E,\mathbb{R})$ is given the structure of a $J^{k-1}_{d}$-bundle using
the flat connection $\nabla_{\mathcal{F}}$ of Theorem 7.

###### Proof.

The map is clearly well-defined and $J^{k-1}$-equivariant. Given a local
$\mathcal{F}$-$\mathcal{F}^{k}_{\mathbb{R}}$-submersion $p:U\to\mathbb{R}$,
the sections

$\displaystyle x\mapsto j^{k-1}_{x}(p):U\cap L\to P^{k-1}(\mathcal{F})$
$\displaystyle x\mapsto j^{k-1}_{x}(p|_{E_{x}\cap U}):U\cap L\to
J^{k-1}(E,\mathbb{R})$

are smooth as part of the definition of those bundles. The first of these
smooth local sections is clearly mapped to the second, and from this it
follows that the map $P^{k-1}(\mathcal{F})\to J^{k-1}(E,\mathbb{R})$ is
smooth. ∎

We now state and prove the main result of the section. One may refer to
Subsection 2.4 and Definition 7.6 for context.

######  Theorem 11.2.

For $i=1,2$, let $(M_{i},\mathcal{F}_{i})$ be transverse order $k$ foliations
with singular leaf $L_{i}$ . Suppose there is a diffeomorphism
$\theta_{0}:L_{1}\to L_{2}$ whose induced map
$[\pi_{1}(L_{1}),J^{k-1}]\to[\pi_{1}(L_{2}),J^{k-1}]$, given by pushing
forward loops, sends $h(\mathcal{F}_{1})$ to $h(\mathcal{F}_{2})$. Then, we
can extend $\theta_{0}$ to a diffeomorphism $\theta:U_{1}\to U_{2}$, where
$U_{i}\subseteq M_{i}$ are open neighbourhoods of $L_{i}$, so as to have
$\theta_{*}(\mathcal{F}_{1}|_{U_{1}})=\mathcal{F}_{2}|_{U_{2}}$.

###### Proof.

Without loss of generality, $L=L_{1}=L_{2}$ and $\theta_{0}=\mathrm{id}_{L}$.
Thus $h(\mathcal{F}_{1})=h(\mathcal{F}_{2})$. By Definition 7.6, this means
exactly that the principal $J^{k-1}_{d}$-bundles $P^{k-1}(\mathcal{F}_{1})$
and $P^{k-1}(\mathcal{F}_{2})$ have the same monodromy invariant so, by
Proposition 2.11, there is an isomorphism of principal $J^{k-1}_{d}$-bundles
$P^{k-1}(\mathcal{F}_{1})\to P^{k-1}(\mathcal{F}_{2})$ covering the identity
map on $L$.

By passing to tubular neighbourhoods, we may assume that $M_{i}$ is the total
space of a line bundle $\pi_{i}:E_{i}\to L$, with $L$ contained in $E_{i}$ as
the zero section. Then, applying Proposition 11.1, there is a $J^{k-1}$-bundle
isomorphism $J^{k-1}(E_{1},\mathbb{R})\to J^{k-1}(E_{2},\mathbb{R})$,
equivalently a section of $J^{k-1}(E_{1},E_{2})$, which moreover maps
$\nabla_{\mathcal{F}_{1}}$ to $\nabla_{\mathcal{F}_{2}}$. By Proposition 10.2,
we can take this section of the form $j^{k-1}(\theta)$ for $\theta$ some
fiberwise diffeomorphism $E_{1}\to E_{2}$ preserving $L$. Then, by Theorem
10.5, $\theta_{*}(\mathcal{F}_{1}|_{U_{1}})=\mathcal{F}_{2}|_{U_{2}}$, as
desired. ∎

As a corollary, when $L$ is simply connected, there is only one “$L$-local”
isomorphism class of transverse order $k$ foliation with singular leaf $L$.

######  Corollary 11.3.

Let $(M,\mathcal{F})$ be a transverse order $k$ foliation with singular leaf
$L$. If $L$ is simply connected, then there exists a diffeomorphism
$\theta:U\to L\times\mathbb{R}$, where $U\subseteq M$ is an open neighbourhood
of $L$, such that $\theta|_{L}=\mathrm{id}_{L}$ and
$\theta_{*}(\mathcal{F}_{U})$ is the product the trivial one-leaf foliation of
$L$ with $\mathcal{F}^{k}_{\mathbb{R}}$.

More generally, the same conclusion holds when there is no nontrivial
homomorphism $\pi_{1}(L)\to J^{k-1}$. For example, this is the case when
$\pi_{1}(L)$ is a simple, nonsolvable group (see Proposition 3.6).

##  12 Range of the holonomy invariant

In this section we show that the range of the holonomy invariant (Definition
7.6) is as large as one could hope it to be. Specifically, we prove the
following:

######  Theorem 12.1.

Let $L$ be a connected, smooth manifold, let $x_{0}\in L$ and let
$\gamma:\pi_{1}(L,x_{0})\to J^{k-1}$ be a group homomorphism. There exists a
line bundle $E\to L$ and a transverse order $k$ foliation $\mathcal{F}$ on $E$
with singular leaf $L$ (identified with the zero section of $E$) whose
holonomy invariant $h(\mathcal{F})$ is equal to the class of $\gamma$ in
$[\pi_{1}(L),J^{k-1}]$.

Together, Theorems 11.2 and 12.1 imply that the problem of enumerating the
“$L$-local” isomorphism classes of transverse order $k$-foliations with a
fixed singular leaf $L$ is equivalent to the problem of enumerating
homomorphisms $\gamma:\pi_{1}(L)\to J^{k-1}$ modulo a certain equivalence
relation. Specifically:

$\gamma\sim\alpha\circ\gamma\circ\beta$

as $\alpha$ ranges over inner automorphisms of $J^{k-1}$ and $\beta$ ranges
over automorphisms of $\pi_{1}(L)$ that can be induced by a diffeomorphism of
$L$. Under a variety of assumptions on $L$, one is assured of being able to
implement any automorphism of $\pi_{1}(L)$ with a diffeomorphism of $L$. For
example, if $L$ is a surface, then the homomorphism from the mapping class
group to the group of outer automorphisms of $\pi_{1}(L)$ is an isomorphism
(the Dehn-Nielsen-Baer theorem). Another example: for $L$ a hyperbolic
3-manifold of dimension $\geq 3$, one has that
$\mathrm{Isom}(L)\to\mathrm{Out}(\pi_{1}(L))$ is an isomorphism by the Mostow
rigidity theorem. In cases such as these, the $L$-local isomorphism classes of
transverse order $k$ foliations with singular leaf $L$ can be put into
correspondence with the quotient set

$\mathrm{Inn}(J^{k-1})\setminus\operatorname{Hom}(\pi_{1}(L),J^{k-1})/\operatorname{Aut}(\pi_{1}(L)).$

The proof of Theorem 12.1 will be an exercise in manipulating principal
bundles and will have little to do with foliations per se. Let us first recall
the associated principal bundle construction:

######  Definition 12.2.

If $\varphi:G\to H$ is a homomorphism of Lie groups and $P$ is a principal
$G$-bundle, then $H\times_{\varphi}P$ denotes the principal $H$-bundle whose
total space is the quotient of $H\times P$ by the equivalence relation defined
by $(h,p)\sim(hg,g^{-1}p)$ for all $g\in G$ and equipped with the $H$-action
defined by $h[h^{\prime},p]=[hh^{\prime},p]$. Here we denote the class of
$(h,p)$ by $[h,p]$.

See for example [11], Chapter 5. The following instances of this construction
will be relevant:

######  Example 12.3.

If $1\to N\to G\overset{\pi}{\to}H\to 1$ is an exact sequence of Lie groups
and if $P$ is a principal $G$-bundle, then there is a principal $H$-bundle
isomorphism $P/N\to H\times_{\pi}P$ sending $[p]\mapsto[1,p]$.

######  Definition 12.4.

Given a line bundle $E$, the coframe bundle $F^{*}(E)$ of $E$ is the principal
$\operatorname{GL}(1)\coloneqq\operatorname{GL}(1,\mathbb{R})$-bundle whose
fiber at $x$ is the collection of nonzero linear functionals
$E_{x}\to\mathbb{R}$.

######  Example 12.5.

Let $E$ be a smooth line bundle and $r$ a positive integer. Denote by $\iota$
the natural inclusion $a\mapsto ay:\operatorname{GL}(1)\to J^{r}$. Then there
is a principal $J^{r}$-bundle isomorphism $J^{r}\times_{\iota}F^{*}(E)\to
J^{r}(E,\mathbb{R})$ sending $[j^{r}_{0}(\theta),\varphi]\mapsto
j^{r}_{x}(\theta\circ\varphi)$ for all $j^{r}_{0}(\theta)\in J^{r}$ and all
$\varphi:E_{x}\to\mathbb{R}$.

######  Example 12.6.

Let $M$ be a connected smooth manifold with basepoint $x_{0}$. Let
$\widetilde{M}$ be the universal cover of $M$ with corresponding basepoint
$\widetilde{x}_{0}$. Let $\Gamma$ be a discrete group and
$\varphi:\pi_{1}(M,x_{0})\to\Gamma$ a homomorphism. View $\widetilde{M}$ as a
principal $\pi_{1}(M,x_{0})$-bundle in the usual way using deck
transformations. Put $P=\Gamma\times_{\varphi}\widetilde{M}$ and
$p_{0}=[1,\widetilde{x}_{0}]$. Then, $P$ is a principal $\Gamma$-bundle over
$M$ for which the monodromy homomorphism $\pi_{1}(M,x_{0})\to\Gamma$
determined by the point $p_{0}$ is exactly $\varphi$.

Suppose we have an exact sequence of Lie groups $1\to N\to
G\overset{\pi}{\to}H\to 1$ that is split by a homomorphism $\iota:H\to G$. In
other words, suppose $G$ is presented as a semidirect product $N\rtimes H$.
Then, using the fact that $\pi\circ\iota=\mathrm{id}_{H}$, there is a natural
isomorphism

$\displaystyle P\cong H\times_{\pi}(G\times_{\iota}P)$ (5)

for every principal $H$-bundle $P$. On the other hand,
$\iota\circ\pi\neq\mathrm{id}_{G}$ (unless $\pi$ is an isomorphism), so there
is no reason to expect the opposite relation

$\displaystyle Q\cong G\times_{\iota}(H\times_{\pi}Q)$ (6)

to hold for every principal $G$-bundle $Q$. With some knowledge of classifying
space theory , however, one may prove the following.

######  Proposition 12.7.

Let $1\to N\to G\overset{\pi}{\to}H\to 1$ be an exact sequence of Lie groups
and let $\iota:H\to G$ satisfy $\pi\circ\iota=\mathrm{id}_{H}$. If either
$\iota$ or $\pi$ is a homotopy equivalence, then $Q\cong
G\times_{\iota}(H\times_{\pi}Q)$ for every principal $G$-bundle $Q$.

Note that, unlike the isomorphism (5), the isomorphism given by the above
proposition is not natural.

###### Proof sketch.

A homomorphism between $G$ and $H$ which is also a homotopy equivalence
induces a homotopy equivalence at the level of classifying spaces. It follows
that the associated bundle construction for such a homomorphism induces a
bijection between isomorphism classes of $G$-bundles and isomorphism classes
of $H$-bundles. The desired claim then follows from the principle that a one-
sided inverse for an invertible map is the two-sided inverse. We refer the
reader to [12], Chapter 4 or [23], Part II for relevant aspects of classifying
space theory. ∎

In the following two lemmas, we give an alternative proof of a special case of
the above proposition which avoids the technology of classifying space theory.

######  Lemma 12.8.

Let $1\to N\to G\to H\to 1$ be short exact exact sequence of Lie groups. Let
$P\to B$ and $Q\to B$ be smooth principal $G$-bundles. If $P/N\cong Q/N$ as
principal $H$-bundles and $N$ is diffeomorphic to a finite-dimensional
Euclidean space, then $P\cong Q$ as principal $G$-bundles.

###### Proof.

Recall $\operatorname{Hom}_{G}(P,Q)$ is defined as the fiber bundle over $B$
whose fiber at $x\in B$ is the set of $G$-equivariant maps from $P_{x}$ to
$Q_{x}$. Reformulated slightly, the statement to be proven is that
$\operatorname{Hom}_{G}(P,Q)\to B$ admits a (smooth) global section. We are
given that the fiber bundle $\operatorname{Hom}_{H}(P/N,Q/N)\to B$ has a
global section. We may instead view $\operatorname{Hom}_{G}(P,Q)$ as a fiber
bundle over $\operatorname{Hom}_{H}(P/N,Q/N)$ with typical fiber $N$. Since
$N$ is locally Euclidean, it follows that the latter fiber bundle has a global
section. Indeed, the Tietze extension theorem can be used to show that every
fiber bundle over a manifold with Euclidean space fibers has a continuous
section (see [23], Theorem 12.2) and routine smoothing arguments can then be
applied. Composing the section $B\to\operatorname{Hom}_{H}(P/N,Q/N)$ with the
section $\operatorname{Hom}_{H}(P/N,Q/N)\to\operatorname{Hom}_{G}(P,Q)$ gives
the result. ∎

######  Lemma 12.9.

Let $1\to N\to G\to H\to 1$ be a short exact exact sequence of Lie groups and
let $\iota:H\to G$ satisfy $\pi\circ\iota=\mathrm{id}_{H}$. Let $P\to B$ be a
smooth principal $G$ bundle. If is $N$ diffeomorphic to a finite-dimensional
Euclidean space, then there is a smooth isomorphism of principal $G$-bundles
from $P$ to $G\times_{\iota}(H\times_{\pi}P)$.

###### Proof.

Let $Q\coloneqq H\times_{\pi}P$. Since $\pi\circ\iota=\mathrm{id}_{H}$, we
have $Q\cong H\times_{\pi}(G\times_{\iota}Q)$ as principal $H$-bundles. Noting
that $Q\cong P/N$ and
$H\times_{\pi}(G\times_{\iota}Q)\cong(G\times_{\iota}Q)/N$ (see Example 12.3),
we have $P/N\cong(G\times_{\iota}Q)/N$. Thus, applying Lemma 12.8, $P$ is
isomorphic to $G\times_{\iota}Q$, as desired. ∎

In particular, the above lemma applies to the extension

$1\to N\to J^{r}\overset{\pi}{\to}\operatorname{GL}(1)\to 1,$

where the quotient map is $a_{1}y+\ldots+a_{r}y^{r}\mapsto a_{1}$, $N$ is its
kernel and $\iota(a)=ay$ provides a splitting on the right. From this we
obtain:

######  Proposition 12.10.

Let $P\to B$ be any principal $J^{r}$ bundle, where $r$ is a positive integer.
Then, there exists a line bundle $E\to B$ such that $P\cong
J^{r}(E,\mathbb{R})$ as principal $J^{r}$-bundles.

###### Proof.

Applying Lemma 12.9 to $P$ with the extension described above, we obtain
$P\cong J^{r}\times_{\iota}Q$ where
$Q\coloneqq\operatorname{GL}(1)\times_{\pi}P$. Any
$\operatorname{GL}(1)$-bundle is isomorphic to the coframe bundle of its
associated vector bundle, so we have $Q\cong F^{*}(E)$ where
$E\coloneqq\mathbb{R}\times_{\operatorname{GL}(1)}Q$. Thus, $P\cong
J^{r}\times_{\iota}F^{*}(E)\cong J^{r}(E,\mathbb{R})$ (see Example 12.5). ∎

We conclude with the proof of the main result of this section.

###### Proof of Theorem 12.1.

Let $\gamma:\pi_{1}(L,x_{0})\to J^{k-1}$ be any homomorphism. Let $P\to L$ be
any principal $J^{k-1}_{d}$-bundle such that the monodromy mapping associated
to some point $p_{0}\in P_{x_{0}}$ is $\gamma$. For instance, we may use
$P=J^{k-1}\times_{\gamma}\widetilde{L}$, as in Example 12.6. We may
equivalently view $P$ as a principal $J^{k-1}$-bundle equipped with a flat
connection. According to the preceding lemma, there exists a line bundle $E\to
L$ and an isomorphism $P\to J^{k-1}(E,\mathbb{R})$ of principal
$J^{k-1}$-bundles. Push forward the flat connection on $P$ to
$J^{k-1}(E,\mathbb{R})$ through this isomorphism, and then use Theorem 10.4 to
produce a corresponding transverse order $k$ foliation $\mathcal{F}$ of $E$
with singular leaf $L$. By Proposition 11.1, $J^{k-1}(E,\mathbb{R})$ and
$P^{k-1}(\mathcal{F})$ are isomorphic as $J^{k-1}_{d}$-bundles, so the
holonomy invariant of $\mathcal{F}$ is as desired. ∎

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