# Positive arborealization of polarized Weinstein manifolds

Daniel Álvarez-Gavela  Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA, 02139<EMAIL_ADDRESS>,  Yakov Eliashberg  Department of
Mathematics
Stanford University
Stanford, CA 94305<EMAIL_ADDRESS>and  David Nadler  Department of
Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a Weinstein manifold. We show that the existence of a global field
of Lagrangian planes in $TX$ is equivalent to the existence of a positive
arboreal skeleton for the Weinstein homotopy class of $X$.

DA was partially supported by NSF grant DMS-1638352 and the Simons Foundation

YE was partially supported by NSF grant DMS-1807270.

DN was partially supported by NSF grant DMS-1802373.

###### Contents

  1. 1 Introduction
  2. 2 Wc-manifolds and cotangent buildings
  3. 3 Symplectic neighborhoods of arboreal Lagrangians
  4. 4 Positive cotangent buildings
  5. 5 Ridgification of Lagrangians
  6. 6 Arborealization of skeleta

##  1\. Introduction

###  1.1. Arborealization program

The symplectic topology of the cotangent bundle $T^{*}M$ of a smooth manifold
is determined by the smooth topology of its Lagrangian zero-section $M$. A
general Weinstein symplectic manifold $X$ (see precise definitions below),
which is the symplectic counterpart of a Stein complex manifold, can be viewed
as a symplectic thickening of its skeleton, which is a singular isotropic
subcomplex $\mathrm{Skel}\,X\subset X$. By analogy with the special case
$X=T^{*}M$, $\mathrm{Skel}\,X=M$, one would like to view $X$ as the “cotangent
bundle” of $\mathrm{Skel}\,X$, and characterize the smooth symplectic topology
of $X$ in terms of the differential topology of $\mathrm{Skel}\,X$. However,
in general the singularities of $\mathrm{Skel}\,X$ are too complicated to be
amenable to a differential topological treatment.

In the paper [N13], the third author introduced a class of Lagrangian
singularities, called arboreal. For arboreal singularities, the smooth
topology of the singularity determines the symplectic topology of its
neighborhood. In particular, it is possible to calculate the local symplectic
invariants of a neighborhood of an arboreal singularity in terms of the
singularity. This was shown in [N13] for microlocal sheaves; for Fukaya
categories, one can apply Lefschetz fibration calculations [Se08] to the
plumbing characterization of [Sh18]. Going further, if a Weinstein manifold
$W$ has a skeleton with arboreal singularities, then its global symplectic
invariants can be effectively computed knowing the smooth topology of the
skeleton, see [N16] for microlocal sheaves and [GPS17] for Fukaya categories.

The paper [N13] initiated a program to determine which Weinstein manifolds
admit arboreal skeleta, or as we say, which Weinstein manifolds could be
arborealized. As evidence that this program might achieve the arborealization
of a large class of Weinstein manifolds, it was shown in [N15] that germs of
Whitney stratified Lagrangians can always be deformed to arboreal Lagrangians
in a non-characteristic fashion, i.e. without changing their microlocal
invariants.

The question of whether a Weinstein manifold can be arborealized via a
homotopy of its Weinstein structure is more subtle. In two dimensions, the
story is classical: generic ribbon graphs provide arboreal skeleta. In four
dimensions, Starkston proved that arboreal skeleta always exist [St18]. In
this paper, we establish that any Weinstein manifold admitting a polarization,
i.e. a Lagrangian plane field, or equivalently a reduction of structure group
of the tangent bundle from $Sp(2n)$ to $GL(n)$, can be arborealized, see
Theorem 1.5 below. Moreover, our proof yields skeleta with the more specific
class of positive arboreal singularities. Conversely, Weinstein manifolds with
positive arboreal skeleta admit polarizations. On the other hand it turns out
not all Weinstein manifolds can be arborealized, see the discussion in Section
1.3 below. Perhaps this is not surprising: there are homotopical obstructions
to defining many symplectic invariants, and so any combinatorial route to
realizing them must also encounter these obstructions.

###  1.2. Main results

Let $(W,\lambda)$ be a $2n$-dimensional Liouville domain. We recall the
definition: $W$ is a compact $2n$-manifold with boundary; $\lambda$ is a
1-form, called the Liouville form, such that $\omega=d\lambda$ is a symplectic
form; and the corresponding vector field $Z=\omega^{-1}(\lambda)$, called the
Liouville vector field, is outward transverse to $\partial W$.

By definition, the skeleton of a Liouville domain $(W,\lambda)$ is the
attractor of the negative flow of the Liouville vector field:

$\mathrm{Skel}(W,\lambda)=\bigcap_{t>0}Z^{-t}(W).$

While the $2n$-dimensional Lebesgue measure of $\mathrm{Skel}(W,\lambda)$ is
equal to $0$, in general $\mathrm{Skel}(W,\lambda)$ can be quite large if no
additional conditions are imposed.

A Weinstein domain is a Liouville domain $(W,\lambda)$ which admits a Morse
Lyapunov function $\phi:W\to{\mathbb{R}}$, i.e. the critical points of $\phi$
are non-degenerate and $Z$ is gradient-like for $\phi$. Sometimes it is
convenient to relax the Morse condition to generalized Morse or Morse-Bott. We
do not consider the Lyapunov function as part of the defining data of a
Weinstein domain, we merely require its existence.

The skeleton of a Weinstein domain $(W,\lambda)$ is known to be the union of
stable manifolds

$\textstyle{\mathrm{Skel}(W,\lambda)=\bigcup_{\lambda_{p}=0}S_{p},}$$\textstyle{S_{p}=\\{x\in
W\,|\,\lim_{t\to\infty}Z^{t}(x)=p\\}.}$

Each stable manifold is $\lambda$-isotropic, hence $\omega$-isotropic and so
at most half-dimensional. A result of F. Laudenbach [L92] states that if $Z$
is Morse-Smale, i.e. its stable and unstable manifolds intersect transversely,
and if moreover $Z$ is the Euclidean gradient near critical points with
respect to the coordinates which give the Morse normal form, then
$\mathrm{Skel}(W,\lambda)$ can be Whitney stratified.

However, even if $Z$ satisfies the above conditions, in general the
singularities of $\mathrm{Skel}(W,\lambda)$ are quite complicated, and their
smooth topology does not determine the symplectic topology of their
neighborhoods. For example, in the simplest case when $W$ is obtained by
attaching a handle to a ball along a Legendrian sphere, the skeleton has a
unique conical singular point and the symplectic topology depends on the
Legendrian isotopy class of the link, not its smooth isotopy class (which is
always trivial for manifolds of dimension $>4$). However, for the class of
arboreal singularities described in Definition 1.1 below the situation is
different.

First, we introduce some auxilliary notions. A closed subset of a symplectic
or contact manifold is called isotropic if it is stratified by isotropic
submanifolds. It is called Lagrangian or Legendrian if it is isotropic and
purely of the maximal possible dimension. The germ at the origin of a locally
simply-connected isotropic subset $L\subset T^{*}{\mathbb{R}}^{n}$ of the
cotangent bundle with its standard Liouville structure $\lambda=pdq$ admits a
unique lift to an isotropic germ at the origin $\widehat{L}\subset
J^{1}{\mathbb{R}}^{n}=T^{*}{\mathbb{R}}^{n}\times{\mathbb{R}}$ of the 1-jet
bundle. Given an isotropic subset $\Lambda\subset S^{*}{\mathbb{R}}^{n}$ of
the cosphere bundle, its Liouville cone $C(\Lambda)\subset T^{*}{\mathbb{R}}$,
i.e. the closure of its saturation by trajectories of the Liouville vector
field $Z=p\frac{\partial}{\partial p}$, is an isotropic subset.

######  Definition 1.1.

Arboreal Lagrangian (resp. Legendrian) singularities form the smallest class
$\mathrm{Arb}^{\mathit{symp}}_{n}$ (resp. $\mathrm{Arb}^{\mathit{cont}}_{n}$)
of germs of closed isotropic subsets in $2n$-dimensional symplectic (resp.
$(2n+1)$-dimensional contact) manifolds such that the following properties are
satisfied:

  1. (i)

(Invariance) $\mathrm{Arb}^{\mathit{symp}}_{n}$ is invariant with respect to
symplectomorphisms and $\mathrm{Arb}^{\mathit{cont}}_{n}$ is invariant with
respect to contactomorphisms.

  2. (ii)

(Base case) $\mathrm{Arb}^{\mathit{symp}}_{0}$ contains
$pt={\mathbb{R}}^{0}\subset T^{*}{\mathbb{R}}^{0}=pt$.

  3. (iii)

(Stabilizations) If $L\subset(X,\omega)$ is in
$\mathrm{Arb}^{\mathit{symp}}_{n}$, then the product
$L\times{\mathbb{R}}\subset(X\times T^{*}{\mathbb{R}},\omega+dp\wedge dq)$ is
in $Arb^{\mathit{symp}}_{n+1}$.

  4. (iv)

(Legendrian lifts) If $L\subset T^{*}{\mathbb{R}}^{n}$ is in
$\mathrm{Arb}^{\mathit{symp}}_{n}$, then its Legendrian lift
$\widehat{L}\subset J^{1}{\mathbb{R}}^{n}$ is in
$\mathrm{Arb}^{\mathit{cont}}_{n}$.

  5. (v)

(Liouville cones) Let $\Lambda_{1},\dots,\Lambda_{k}\subset
S^{*}{\mathbb{R}}^{n}$ be a finite disjoint union of arboreal Legendrian germs
from $Arb^{\mathit{cont}}_{n-1}$ centered at points $z_{1},\dots,z_{k}\in
S^{*}{\mathbb{R}}^{n}$. Let $\pi:S^{*}{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be
the front projection. Suppose

     * -

$\pi(z_{1})=\dots=\pi(z_{k})$.

     * -

For any $i$, and smooth submanifold $Y\subset\Lambda_{i}$, the restriction
$\pi|_{Y}:Y\to{\mathbb{R}}^{n}$ is an embedding (or equivalently, an
immersion, since we only consider germs).

     * -

For any distinct $i_{1},\ldots,i_{\ell}$, and any smooth submanifolds
$Y_{i_{1}}\subset\Lambda_{i_{1}},\dots,Y_{i_{\ell}}\subset\Lambda_{i_{\ell}}$,
the restriction $\pi|_{Y_{i_{1}}\cup\dots\cup
Y_{i_{\ell}}}:Y_{i_{1}}\cup\dots\cup Y_{i_{\ell}}\to{\mathbb{R}}^{n}$ is self-
transverse.

Then the union ${\mathbb{R}}^{n}\cup C(\Lambda_{1})\cup\dots\cup
C(\Lambda_{k})$ of the Liouville cones with the zero-section form an arboreal
Lagrangian germ from $\mathrm{Arb}^{\mathit{symp}}_{n}$.

With the above classes defined, we can also allow boundary by additionally
taking the product $L\times{\mathbb{R}}_{\geq 0}\subset(X\times
T^{*}{\mathbb{R}},\omega+dp\land dq)$ for any arboreal Lagrangian
$L\subset(X,\omega)$, and similarly for arboreal Legendrians.

Figure 1.1. The simplest non-smooth arboreal singularity is the zero section
union the positive conormal of a smooth co-oriented hypersurface. Figure 1.2.
This arboreal singularity consists of the zero section union the positive
conormal of two smoothly intersecting co-oriented hypersurfaces. Figure 1.3.
This arboreal singularity consists of the zero section union the positive
conormal of a singular co-oriented hypersurface (namely the front of the
arboreal singularity of Figure 1.1).

In our paper [AGEN20a] we proved that for fixed dimension $n$ the class of
arboreal singularities contains only finitely many local models up to ambient
symplectomorphism or contactomorphism. More precisely, to each member of the
class $\mathrm{Arb}^{\mathit{symp}}_{n}$ one can assign a signed rooted tree
$\mathcal{T}=(T,\rho,\varepsilon)$ with at most $n+1$ vertices. Here $T$ is a
finite acyclic graph, $\rho$ is a distinguished root vertex, and $\varepsilon$
is a sign function on the edges of $T$ not adjacent to $\rho$. This discrete
data completely determines the germ:

######  Theorem 1.2 ( [AGEN20a]).

If two arboreal singularities $L\subset(X,\omega)$,
$L^{\prime}\subset(X^{\prime},\omega^{\prime})$ have the same dimension and
signed rooted tree $\mathcal{T}$, then there is (the germ of) a
symplectomorphism $(X,\omega)\simeq(X^{\prime},\omega^{\prime})$ identifying
$L$ and $L^{\prime}$.

Within all arboreal singularities, there is the distinguished class with
positive sign $\varepsilon\equiv+1$.

Figure 1.4. One can obtain a new arboreal singularity from the arboreal
singularity of Figure 1.1 in two different ways depending on how it is
embedded in $S^{*}M$ relative to the vertical distribution $\nu=\ker(d\pi)$.
One yields a positive arboreal singularity and the other does not.

######  Definition 1.3.

An arboreal Lagrangian $L$ (with boundary) in a symplectic manifold
$(X,\omega)$ is a piecewise smooth Lagrangian with arboreal singularities
(with boundary), i.e. locally modeled on the above class (with boundary). When
$L$ is an arboreal Lagrangian whose boundary $\partial L$ is a smooth
manifold, we will say that $L$ has smooth boundary.

A positive arboreal Lagrangian $L$ (with boundary) in a symplectic manifold
$(X,\omega)$ is a Lagrangian with positive arboreal singularities (with
boundary), i.e. locally modeled on the distinguished class with positive sign
$\varepsilon\equiv+1$.

Given any positive arboreal Lagrangian $L\subset(X,\omega)$, possibly with
boundary, its neighborhood $U\subset X$ admits a canonical, up to homotopy,
Lagrangian plane field $\xi\subset TU$. This homotopy class admits a
representative $\xi$ which is transverse to $L$, i.e. transverse to the
closure of each smooth piece.

######  Definition 1.4.

A polarization of a symplectic manifold $(X,\omega)$ is the choice of a
Lagrangian plane field $\xi\subset TX$.

If $(W,\lambda)$ is a Weinstein domain with positive arboreal skeleton, then
$(W,\lambda)$, which retracts onto an arbitrarily small neighborhood of its
skeleton, admits a polarization. The main result of this paper is that the
converse also holds:

Figure 1.5. The canonical polarization in the neighborhood of an $A_{2}$
singularity, where the positivity condition is vacuous. One could also turn
the tangent field to the zero section counter-clockwise instead of clockwise.
The choice is determined by what we call an orientation structure, which in
this case is equivalent to a co-orientation of the origin inside the zero-
section. Figure 1.6. The canonical polarization in the neighborhood of a
positive $A_{3}$ singularity, where the orientation structure is given by the
co-orientation of the $A_{3}$ front.

######  Theorem 1.5.

A Weinstein domain $(W,\lambda)$ is homotopic to a Weinstein domain whose
skeleton is positive arboreal with smooth boundary if and only if
$(W,\lambda)$ admits a polarization.

######  Remark 1.6.

The conclusion of Theorem 1.5 can be refined in several ways, some of which we
sketch below. For precise formulations see Theorem 6.16 and its corollaries,
which use the language of Wc-manifolds, see Definition 2.24.

  1. (i)

In fact, for $\xi$ a polarization of $(W,\lambda)$ we can arrange it so that
$\xi$ is the canonical polarization in the neighborhood of the positive
arboreal skeleton, so in particular $\xi$ is transverse to the skeleton.

  2. (ii)

If so desired, the smooth boundary of the positive arboreal skeleton can be
pushed out to the boundary of $W$, in which case it serves as a skeleton for a
Weinstein pair, see Section 2.4. The Weinstein hypersurface in the pair is
thus the ribbon of a smooth Legendrian, and can also be thought of as a stop.

  3. (iii)

Theorem 1.5 also holds with input a Weinstein pair $(W,A)$ instead of a
Weinstein domain. In this case the conclusion is that the Weinstein
hypersurface $A$ can be enlarged by the ribbon of a smooth Legendrian so that
the resulting Weinstein pair has a positive arboreal skeleton.

  4. (iv)

Moreover, the version of Theorem 1.5 for Weinstein pairs also holds in
relative form if the Weinstein hypersurface in the Weinstein pair already has
a positive arboreal skeleton and the polarization restricts on it to the
canonical polarization. In this case the Weinstein structure can be kept fixed
near the Weinstein hypersurface.

  5. (v)

Theorem 1.5 and its refinements hold more generally for arbitrary Weinstein
manifolds, not necessarily of finite type (i.e. not completions of Weinstein
domains), and the result is proved in the same way since the inductive
argument stabilizes. We restrict the discussion to the finite type case for
simplicity of exposition.

Existence of a polarization for a symplectic manifold $(X^{2n},\omega)$ is
equivalent to asking that the classifying map $X\to BU_{n}$ of the tangent
bundle lifts to $BO_{n}$, where we endow $TX$ with the homotopically unique
almost complex structure compatible with $\omega$.

######  Definition 1.7.

A stable polarization of a symplectic manifold $(X,\omega)$ is the choice of a
Lagrangian plane field $\xi\subset TX\oplus{\mathbb{C}}^{d}$ for some $d\geq
1$.

The existence of a stable polarization is equivalent to asking that the
classifying map $W\to BU$ of the stable tangent bundle lifts to $BO$. Recall
that a Weinstein domain $(W,\lambda)$ has the homotopy type of a half-
dimensional CW complex. Recall also that $BO_{k}\to BO_{k+1}$ is $k$-connected
and $BU_{k}\to BU_{k+1}$ is $(2k+1)$-connected. Therefore, it follows that for
Weinstein domains the existence of a polarization is equivalent to the
existence of a stable polarization. From the above, one can check that many
Weinstein domains admit polarizations, notably smooth complete intersections
in complex affine space. Indeed, if $X\subset{\mathbb{C}}^{N}$ is a smooth
complete intersection, then its normal bundle is trivial, hence its tangent
bundle is stably trivial and in particular $X$ admits a stable polarization.
This proves:

######  Corollary 1.8.

Let $X\subset{\mathbb{C}}^{N}$ be a complete intersection. Then $X$ is
Weinstein homotopic to a Weinstein manifold whose skeleton is positive
arboreal with smooth boundary.

An arboreal skeleton with boundary comes with a natural additional orientation
structure, induced from the ambient symplectic structure. We formalize this in
the notion of an arboreal space and prove the following counterpart to Theorem
1.5.

######  Theorem 1.9.

Any compact arboreal space with boundary arises as the skeleton of a Weinstein
domain $(W,\lambda)$, unique up to Weinstein homotopy.

Figure 1.7. The simplest example of two non-equivalent arboreal spaces with
diffeomorphic underlying arboreal Lagrangians. These are skeleta for the pair
of pants and the once-punctured torus repsectively. The arboreal spaces are
distinguished by the orientation structure, which is additional data, and in
this dimension reduces to the usual cyclic structure on ribbon graphs. We
thank A. Oancea for pointing out this example.

Hence, up to Weinstein homotopy, those Weinstein domains which admit
polarizations are precisely the Weinstein thickenings of positive arboreal
spaces. Moreover, the relative form of Theorem 1.5 implies a uniqueness result
for positive arboreal skeleta. To state it we need the notion of a Weinstein
concordance, which is best defined in the language of Wc-manifolds, see
Definition 2.24. Briefly, if $\lambda_{1}$ and $\lambda_{2}$ are two Weinstein
structures on the same symplectic domain $(W,\omega)$ with Liouville fields
$Z_{1},Z_{2}$, respectively, then a Weinstein concordance between them is a
Weinstein-type primitive $\lambda$ for $\omega+du\land dt$ on $W\times
T^{*}[0,1]$ such that the Liouville field $Z$ of $\lambda$ is tangent to the
boundary $W\times T_{0}^{*}[0,1]\cup W\times T_{1}^{*}[0,1]$ and restricts to
$Z_{1}+u\partial/\partial u$ and $Z_{2}+u\partial/\partial u$ over $W\times
T^{*}_{0}[0,1]$ and $W\times T_{1}^{*}[0,1]$ respectively.

######  Definition 1.10.

A positive arboreal concordance is a Weinstein concordance $(W\times
T^{*}[0,1],\lambda)$ whose skeleton is a positive arboreal Lagrangian with
boundary. We assume that the boundary of the skeleton is smooth in the
interior of $W\times T^{*}[0,1]$ but we allow simple corners with one face
contained in $W\times T^{*}_{0}[0,1]$ or $W\times T^{*}_{1}[0,1]$ and the
other face transverse to it.

######  Remark 1.11.

We must allow simple corners so that for example if $(W,\lambda)$ is a
Weinstein manifold whose skeleton is a positive arboreal with nonempty smooth
boundary, then the trivial product $(W\times T^{*}[0,1],\lambda+udt)$ is a
positive arboreal concordance.

The uniqueness statement then reads as follows:

######  Theorem 1.12.

Suppose that $\lambda_{1}$ and $\lambda_{2}$ are two homotopic Weinstein
structures on $W$ whose skeleta are positive arboreal Lagrangians with smooth
boundary, transverse to polarizations $\xi_{1}$ and $\xi_{2}$ respectively.
Then $\xi_{1}$ and $\xi_{2}$ are stably homotopic if and only if there is a
positive arboreal concordance between $\lambda_{1}$ and $\lambda_{2}$.

We remark that the classification of polarizations on a Weinstein manifold up
to stable homotopy is in general weaker than the classification up to
homotopy.

###  1.3.  Further development of the arborealization program

Work in progress of the authors extends the results of this paper in two
directions. First, in our forthcoming work [AGEN21] we consider a
$1$-parametric version of the problem, by introducing a suitable notion of
“arboreal homotopy of positive arboreal spaces”, i.e. the minimal sequence of
combinatoral Reidemeister type moves necessary to connect two homotopic
Weinstein structures with positive arboreal skeleta that have homotopic
canonical polarizations. Note that substantial progress in this direction
appeared in Zorn’s thesis [Z18], where local mutations of arboreal skeleta are
classified.

The second direction is an extension of the arborealization program to a more
general class of Weinstein manifolds. While most Weinstein manifolds
considered in applications such as mirror symmetry are complete intersections
and hence polarized, not all Weinstein manifolds admit polarizations. For
example a cohomological obstruction to existence of a polarization is that all
odd Chern classes are 2-torsion. However, this cohomological condition does
not obstruct existence of a not necessarily positive arboreal skeleton.

But even allowing for non-positive arboreal skeleta, there are still
obstructions to their existence. Indeed, while existence of a polarization is
not a necessary condition for existence of a non-positive arboreal skeleton,
there is a closely related necessary condition: existence of an
$(n,n-1)$-polarization. This is, roughly speaking, a polarization which is
allowed to degenerate to an $(n-1)$-dimensional isotropic subspace. More
precisely, it is defined as follows.

Given a symplectic vector space $(E,\omega)$ let $L_{n}(E)$ denote its
Lagrangian Grassmanian of non-oriented Lagrangian planes, and $I_{n-1}(E)$
denote the Grassmanian of non-oriented isotropic (n-1)-dimensional subspaces.
We further denote by $F_{n,n-1}(E)$ the flag manifold
$\\{(\lambda,\mu);\lambda\in L_{n}(E),\mu\in I_{n-1}(E),\mu\subset\lambda\\}$.
Let $\pi_{L}$ and $\pi_{I}$ be the tautological projections
$\pi_{L}:F_{n,n-1}(E)\to L_{n}(E)$ and $\pi_{I}:F_{n,n-1}(E)\to I_{n-1}(E)$.
Consider the double cone $C_{n}(E):=(F_{n,n-1}\times[0,1]\cup L_{n}(E)\cup
I_{n-1}(E))/\\{f\times 0\sim\pi_{L}(f),f\times 1\sim\pi_{I}(f)\\}.$ Given any
symplectic vector bundle $E$, we will use the notation $C_{n}(E)$ for the
associated cone bundle. An $(n,n-1)$-polarization on a symplectic manifold $V$
is by definition a section of the cone bundle $C_{n}(T(V))$.

One can show that for $n\leq 5$ any $2n$-dimensional Weinstein manifold admits
a $(n,n-1)$-polarization, while for $n=6$ there are obstructions (e.g. one
must have $c_{3}(V)=c_{1}(V)c_{2}(V)$).111We thank John Pardon for suggesting
to us the notion of a $(n,n-1)$-polarization, and to Søren Galatius for
computations of obstructions. It is possible that this condition is also
sufficient for existence of an arboreal skeleton. Note in particular that a
Weinstein manifold of dimension four always admits an $(n,n-1)$-polarization,
so this discussion is consistent with Starkston’s result [St18] that four-
dimensional Weinstein manifolds always admit arboreal skeleta.

For completely arbitrary Weinstein manifolds it is not clear whether one
should expect there to be a simple class of singularities which the skeleton
can always be arranged to have, but if it exists such a class would have to be
more general than the arboreal class. In any case, even the most optimistic
hopes of constructing a reasonable skeleton for a Weinstein manifold will in
all likelihood require the existence of “Maslov data”, i.e. the homotopical
trivialization required to define Fukaya categories or microlocal sheaves.

###  1.4. Flexibility of caustics and the ridgification theorem

We discuss a toy example to illustrate our strategy of proof. Consider a
Weinstein domain $(W^{2n},\lambda)$ which is obtained from the standard
Darboux ball $B\subset({\mathbb{R}}^{2n},pdq-qdp)$ by attaching an index $n$
handle, i.e. the neighborhood of a Lagrangian disk, along a Legendrian sphere
$\Lambda\subset\partial B=(S^{2n-1},\xi_{\text{std}})$. The skeleton of $W$ is
homeomorphic to the $n$-sphere: it consists of the Lagrangian disk union the
Liouville cone on $\Lambda$ with respect to the radial Liouville field on
${\mathbb{R}}^{2n}$, union the origin.

Replace the radial Liouville structure $pdq-qdp$ on ${\mathbb{R}}^{2n}$ with
the Morse-Bott canonical Liouville structure $pdq$ on the cotangent bundle
$T^{*}{\mathbb{R}}^{n}$, or rather on $T^{*}D$ for $D\subset B$ a Lagrangian
$n$-disk bounding a Legendrian unknot $\partial D\subset S^{2n-1}$ which we
assume disjoint from $\Lambda$. Then the radial Liouville cone on
$\Lambda\subset S^{2n-1}$ in $B$ gets replaced with the fibrewise Liouville
cone on $\Lambda\subset S^{*}D$ in $T^{*}D$, where $S^{*}D$ denotes the
cosphere bundle, see Figure 1.8.

Observe that as a result, the singularity gets spread out over the disk, i.e.
instead of a cone on a point it is now a fibrewise cone over the disk. The
skeleton of the new Weinstein structure on $W$ consists of the Lagrangian
handle, union the Liouville cone on $\Lambda$ with respect to the canonical
Liouville structure on $T^{*}D$, union the zero section $D$. The singularities
of this new Lagrangian skeleton are thus related to the singularities of the
map $\pi|_{\Lambda}:\Lambda\to D$, where $\pi:S^{*}D\to D$ is the front
projection. These singularities correspond to the tangencies of $\Lambda$ with
the distribution $\nu=\ker(d\pi)\subset T(S^{*}D)$ and are also known as
caustics in the literature. For example, if $\pi|_{\Lambda}:\Lambda\to D$ has
no caustics, then it is an immersion and so by axiom (v) of Definition 1.1,
after a generic perturbation (to ensure self-transversality), the skeleton of
$W$ is arboreal.

Figure 1.8. The blow-up procedure for a Weinstein manifold $W^{2n}$ with two
critical points: one of index 0 and one of index $n$.

The idea of blowing up conical Lagrangian singularities was already present at
the inception of the arborealization program, but for our purposes it is
necessary to perform the blowing up procedure globally, at the level of
Weinstein structures. This strategy was successfully implemented by Starkston
in [St18] for the $4$-dimensional case. Starkston moreover showed how to
explicitly arborealize the semi-cubical cusp singularities which generically
appear in the front projections of 1-dimensional Legendrians, thus leading to
her result that 4-dimensional Weinstein manifolds always admit arboreal
skeleta. In higher dimensions this strategy encounters the difficulty that
there is no classification theorem for the generic singularities of front
projections, nor can anything remotely close to a satisfactory classification
be hoped for.

Nevertheless, there holds an h-principle for the simplification of caustics.
The h-principle states that if there is no homotopy theoretic obstruction to
the simplification of caustics, then the simplification can be achieved by
means of a Legendrian isotopy (or for Lagrangians, by means of a Hamiltonian
isotopy). Results in this direction first appeared in work of Entov [En99],
and the full h-principle was established in work of the first author [AG18a],
[AG18b]. Returning to our toy example, if we know that $W$ admits a
polarization as in the hypothesis of our main Theorem 1.5 and we further
assume that the dimension of $W$ is congruent to $2$ modulo $4$, then it
follows from the h-principle that there exists a Legendrian isotopy which
deforms $\Lambda$ to a Legendrian whose front projection
$\pi|_{\Lambda}:\Lambda\to D$ only has cusp singularities. In this case we can
proceed as in [St18] to conclude that the skeleton of $W$ can be arborealized
by a Weinstein homotopy. When the dimension of $W$ is congruent to $0$ modulo
$4$ there is an additional homotopical subtlety to consider, even in this toy
example, but we do not pursue this further since one also encounters
additional and more serious difficulties when attempting to implement the
above strategy on a Weinstein manifold with a more complicated handlebody
presentation. For example, one would need to know that the relevant
homotopical obstruction to applying the h-principle vanishes for every handle,
yet it is unclear how to arrange for this even under the global assumption of
existence of a polarization.

So instead of attempting a direct application of the h-principle, we will
exploit the fact that for the purposes of arborealization we can allow
deformations which are more general than a Hamiltonian isotopy, namely we can
deform the skeleton of a Weinstein manifold by any homotopy of the Weinstein
structure. This allows the skeleton to develop genuine singularities (i.e.
non-smooth points) along which its field of tangent planes jumps
discontinuously. By introducing these jumps we destroy the homotopical
obstruction to simplifying the singularities with the distribution
$\nu=\ker(d\pi)$, and in fact we can always completely eliminate these
tangencies.

This strategy was made precise in [AGEN19], where we introduced a class of
singular Lagrangians, called ridgy, and proved an h-principle without pre-
conditions, which allows one to deform any Lagrangian so that it becomes
transverse to any given Lagrangian distribution, at the expense of developing
ridgy singularities. This result, which we refer to as the ridgification
theorem, uses the h-principle for the simplification of caustics as one of the
key ingredients in its proof, but is better suited for the task at hand since
there are no hypotheses needed for its application.

Figure 1.9. The ridge singularity in one dimension. Figure 1.10. The
ridgification theorem in action: the zero section $M\subset T^{*}M$ gets
deformed to a ridgy Lagrangian $L\subset T^{*}M$ which is transverse to the
red distribution. Of course in this simple example a single ridge point would
suffice, but the proof always produces a ridge locus which is the union of
homologically trivial hypersurfaces, in this case two ridge points.

Ridgy singularities are extremely simple: they consist of the corner
$\\{p=|q|\\}\subset{\mathbb{R}}^{2}$ together with its products and
stabilizations. Moreover, ridgy singularities can be explicitly arborealized
while maintaining transversality to any given Lagrangian distribution, see
Figure 1.11. So our revised strategy to arborealize a Weinstein manifold $W$
is the following: (1) blow up the singularities as before to replace Liouville
cones on a point with Liouville cones on Lagrangian disks, (2) apply the
ridgification theorem to get rid of all singularities of the resulting front
projections, and finally (3) arborealize the ridgy Lagrangians produced by the
ridgification theorem, while maintaining transversality to the relevant
distributions, to obtain an arboreal skeleton for $W$.

Figure 1.11. Arborealization of an order 1 ridge. Figure 1.12. Arborealization
of an order 2 ridge.

Finally, a word on how we deal with the issue of compatibility at the
interaction of three or more strata of the skeleton: this is where the notion
of positivity comes up in an essential way. The key fact is that if a
Lagrangian distribution $\eta$ on $T^{*}{\mathbb{R}}^{n}$ corresponds to a
family of positive definite quadratic forms, then it is transverse to the
conormal $T^{*}_{H}{\mathbb{R}}^{n}\subset T^{*}{\mathbb{R}}^{n}$ of any
immersed hypersurface $H\subset{\mathbb{R}}^{n}$ in a neighborhood of the zero
section. For the interaction of say three strata, we use the ridgification
theorem to achieve transversality for two strata at a time and then conclude
that transversality also holds for the third stratum for free using the above
observation on positivity. This yields the desired compatibility. The
existence of a global polarization for $W$ will play a key role in guiding the
positivity condition globally, thus enabling the successful implementation of
the above strategy.

###  1.5. Structure of the article

The paper is organized as follows:

In Section 2 we give the necessary preliminaries regarding Weinstein
manifolds. This includes a review of notions in the literature as well as the
introduction of a new framework for working with Weinstein structures. The key
notions are that of a Weinstein manifold with boundary and corners, which we
call a Wc-manifold, and that of a cotangent building, which provides a way of
presenting a Wc-manifold in terms of successive attachments of cotangent
bundles. The concept of a cotangent building gives an alternative to the
standard Weinstein handlebody presentation for Weinstein manifolds, and is
better suited for our purposes.

In Section 3 we prove the existence and uniqueness of symplectic neighborhoods
of arboreal spaces. The existence part is an easy consequence of the stability
theorem for arboreal singularities, which we proved in [AGEN20a]. The
uniqueness part consists of a Darboux-Weinstein theorem for the symplectic
neighborhoods of arboreal Lagrangians.

In Section 4 we discuss the notion of positivity, which is the key ingredient
that allows us to control the interaction of three or more strata in the
Lagrangian skeleta under consideration. We first discuss the notion of
positivity for ordered tuples of Lagrangian planes, then discuss positivity
for cotangent buildings and finally we relate this to the notion of positivity
for arboreal spaces. We show that existence of a positive arboreal skeleton
for the homotopy class of a Weinstein structure implies the existence of a
polarization, proving one half of our main theorem 1.5.

In Section 5 we recall the notion of a ridgy Lagrangian from our previous work
[AGEN19] and state the main result of that paper, which we call the
ridgification theorem, as well as its non-integrable counterpart, which we
call the formal ridgification theorem. The formal ridgification theorem also
plays an important role since it holds in extension form. The main point is
that by allowing for ridgy singularities we can deform any Lagrangian
submanifold so that it becomes transverse to any given Lagrangian
distribution, and moreover the deformation can be achieved by means of a
Weinstein homotopy.

Finally, in Section 6 we show how the ridgy singularities produced by the
ridgification theorem can be further deformed into arboreal singularities.
Furthermore, the deformation is realized by a Weinstein homotopy and the end
result is a positive arboreal Lagrangian which serves as skeleton for the
deformed Weinstein structure. This proves the other half of our main theorem
1.5.

###  1.6. Acknowledgements

We are very grateful to Laura Starkston who collaborated with us on the
initial stages of this project. We are also grateful to John Pardon who helped
us to crystalize the notion of an $(n,n-1)$-polarization and to Søren Galatius
who explained to us how to compute obstructions to its existence.

The first author thanks the Institute for Advanced Study and Princeton
University for a great research environment and thanks CRM Montreal for its
hospitality. The second author thanks RIMS Kyoto and ITS ETH Zurich for their
hospitality. The third author thanks MSRI for its hospitality.

Finally, we are very grateful for the support of the American Institute of
Mathematics, which hosted a workshop on the arborealization program in 2018
from which this project has greatly benefited.

##  2\. Wc-manifolds and cotangent buildings

In this section we recall the basic definitions concerning Liouville and
Weinstein manifolds, and introduce the language of Wc-manifolds and cotangent
buildings, which will be used throughout the text.

###  2.1. Liouville manifolds

####  2.1.1. Liouville domains

An exact symplectic manifold $(W,\lambda)$ is a compact manifold with boundary
equipped with an exact symplectic form $\omega$, together with a choice of
primitive $\lambda$, i.e. $\omega=d\lambda$. Since $\omega$ is non-degenerate,
the 1-form $\lambda$ corresponds to a vector field $Z$ under the contraction
$\iota_{Z}\omega=\lambda$.

######  Definition 2.1.

A Liouville domain is an exact symplectic manifold $(W,\lambda)$ such that $Z$
is outwards transverse to $\partial W$.

Equivalently, $(W,\lambda)$ is Liouville if $\lambda|_{\partial W}$ is a
contact form and the orientation defined on $\partial W$ by the volume form
$\lambda\wedge(d\lambda)^{n-1}|_{\partial W}$ coincides with the orientation
of $\partial W$ as the boundary of $W$, where $W$ is itself oriented by the
volume form $\omega^{\land n}$.

We call $\lambda$ the Liouville form and $Z$ the Liouville field. By Cartan’s
formula for the Lie derivative, the condition $\omega=d\lambda$ is equivalent
to $\mathcal{L}_{Z}\omega=\omega$. This means that the vector field $Z$ is
symplectically conformally expanding, i.e. $(Z^{t})_{*}\omega=e^{t}\omega$,
$t\geq 0$, for $Z^{t}$ the flow of $Z$. Equivalently, the vector field $-Z$ is
symplectically conformally contracting and this viewpoint will sometimes be
more natural in what follows. Note that we also have
$(Z^{t})_{*}\lambda=e^{t}\lambda$. The object of interest in this paper is the
following.

######  Definition 2.2.

The skeleton of a Liouville domain $(W,\lambda)$ is the subset

$\mathrm{Skel}(W,\lambda)=\bigcap\limits_{t>0}Z^{-t}(W).$

In other words, $\mathrm{Skel}(W,\lambda)$ is the attractor of the positive
flow of the contracting field $-Z$. We will be going back and forth between
Liouville domains and the closely related notion of a Liouville manifold,
whose definition we recall next.

####  2.1.2. Liouville manifolds

Recall that the symplectization $S(Y,\xi)$ of a contact manifold $(Y,\xi)$
equipped with a contact form $\alpha$ for $\xi$ is the exact symplectic
manifold $Y\times{\mathbb{R}}$ equipped with the primitive
$\lambda=e^{t}\alpha$, where $t\in{\mathbb{R}}$. We can equivalently consider
$Y\times{\mathbb{R}}^{+}$ with $\lambda=s\alpha$ for
$s=e^{t}\in{\mathbb{R}}^{+}$ the multiplicative coordinate.

The symplectization of a contact manifold $(Y,\xi)$ can be defined invariantly
without choosing a contact form. Assuming that $Y$ is connected and $\xi$ is
co-oriented, we define $S(Y,\xi)$ as follows. Let $N(\xi)\subset T^{*}Y$ be
the total space of the conormal line bundle to $\xi$ and $N_{+}(\xi)$ the
component of $N(\xi)\setminus Y$ consisting of $1$-forms defining the given
co-orientation of $\xi$. Then $\lambda_{\xi}=pdq|_{N_{+}(\xi)}$ is a primitive
of the exact symplectic form $\omega=d\lambda_{\xi}$ and we define $S(Y,\xi)$
as $N_{+}(\xi)$ endowed with this structure. A choice of a contact form
$\alpha$ for $\xi$ identifies $N_{+}(\xi)$ with $Y\times{\mathbb{R}}^{+}$ and
$\lambda_{\xi}$ with $s\alpha$. For example, the symplectization of the
cosphere bundle $S^{*}M$ is the complement of the zero section in $T^{*}M$.

In the other direction, the contactization of an exact symplectic manifold
$(N,\mu)$ is the manifold $M\times{\mathbb{R}}$ equipped with the contact form
$\mu-du$, where $u\in{\mathbb{R}}$. For example, the contactization of the
cotangent bundle $T^{*}M$ is the 1-jet bundle $J^{1}M$.

######  Definition 2.3.

A finite type Liouville manifold $(X,\lambda)$ consists of a (non-compact)
boundaryless exact symplectic manifold such that the following conditions
hold.

  * (L1)

The vector field $Z$ which is $\omega$-dual to $\lambda$ is complete for
$t\to\pm\infty$.

  * (L2)

There exists a compact domain $W\subset X$ such that $Z$ is outward transverse
to the boundary $\partial W$, i.e. $(W,\lambda|_{W})$ is a Liouville domain,
and such that the union of forward trajectories of $Z$ starting at $\partial
W$ is equal to $X\setminus{\rm Int\,}W$.

We call Liouville domains as in (L2) defining.

######  Definition 2.4.

The skeleton of a finite type Liouville manifold $(X,\lambda)$ is defined to
be

$\mathrm{Skel}(X,\lambda)=\bigcup\limits_{C}\mathop{\bigcap}\limits_{t>0}Z^{-t}(C),\qquad\text{$C\subset
X$ is compact}.$

Clearly, $\mathrm{Skel}(X,\lambda)=\mathrm{Skel}(W,\lambda)$ for any defining
Liouville domain $W\subset X$. Hence the skeleton of a finite type Liouville
manifold is compact. Note that for any such $W$ we may use the Liouville flow
starting at $\partial W$ to parametrize $X\setminus\text{int}W$ as $\partial
W\times[0,\infty)$. Since the Liouville field conformally expands $\lambda$ we
have $\lambda|_{\partial W\times[0,\infty)}=e^{t}(\lambda|_{\partial W})$ for
$t\in[0,\infty)$ the coordinate parametrizing the Liouville flow, so
$X\setminus\text{int}W$ is the positive part of the symplectization of the
contact manifold $(\partial W,\lambda|_{\partial W})$.

Conversely, any Liouville domain $(W,\lambda)$ can be completed to a Liouville
manifold by attaching to it the positive part of the symplectization of
$(\partial W,\lambda|_{\partial W})$. Explicitly:

$X=W\mathop{\bigcup}\limits_{\partial W\sim\partial W\times 0}\partial
W\times[0,\infty),$

where the Liouville form $\lambda$ on $W$ extends to to $\partial
W\times[0,\infty)$ as $e^{t}\left(\lambda|_{\partial W}\right)$.

######  Definition 2.5.

The ideal boundary of a finite type Liouville manifold $(X,\lambda)$ is
defined to be
$\partial_{\infty}X=\big{(}X\setminus\mathrm{Skel}(X,\lambda)\big{)}/{\mathbb{R}}^{+}$,
where the ${\mathbb{R}}^{+}$ action is given by the positive flow of $Z$.

Since $\lambda(Z)=0$, the form $\lambda$ descends to $\partial_{\infty}X$ as a
contact structure $\xi_{\infty}$. A choice of a contact form $\alpha$ for
$\xi_{\infty}$ yields an isomorphism
$j_{\alpha}:S(\partial_{\infty}X,\xi_{\infty})=(\partial_{\infty}X\times{\mathbb{R}},e^{t}\alpha)\to(X\setminus\mathrm{Skel}(X,\lambda),\lambda)$
which maps $\partial_{\infty}X\times 0$ contactomorphically onto the boundary
of a defining Liouville domain. We denote by
$\pi_{\infty}:X\setminus\mathrm{Skel}(X,\lambda)\to\partial_{\infty}X$ the
projection along the trajectories of $Z$.

####  2.1.3. Deformations of Liouville structures

######  Definition 2.6.

A homotopy of finite type Liouville structures $(X,\lambda_{t})$ is a family
$\lambda_{t}$, $t\in[0,1]$, of finite type Liouville structures on a manifold
$X$ admitting a smooth family $W_{t}\subset X$ of defining Liouville domains.

Given such a homotopy $(X,\lambda_{t})$ there exists an isotopy $\phi_{t}:\
X\to X$ such that $\phi_{t}^{*}\omega_{t}=\omega_{0}$, where
$\omega_{t}=d\lambda_{t}$. Moreover, one can always arrange that
$\phi_{t}^{*}\lambda_{t}=\lambda_{0}+dH_{t}$ for uniformly compactly supported
functions $H_{t}$, see [CE12], Sections 11.1 and 11.2. In particular, it is
always sufficient to consider homotopies fixing the symplectic form, and
moreover changing the Liouville form by adding a compactly supported exact
form. Note that for any fixed $\omega$ the space of Liouville structures on
$(X,\omega)$ which are fixed at infinity is convex, hence any two such
Liouville structures are canonically homotopic.

######  Remark 2.7.

Unless explicitly stated otherwise, all Liouville structures under
consideration in this paper are of finite type and hence we will stop
mentioning this below.

We will often abuse terminology and say that two Liouville manifolds
$(X_{1},\lambda_{1})$ and $(X_{2},\lambda_{2})$ are Liouville homotopic if
there exists a symplectomorphism
$F:(X_{1},d\lambda_{1})\to(X_{2},d\lambda_{2})$ such that
$(X_{1},F^{*}\lambda_{2})$ is a Liouville manifold which is Liouville
homotopic to $(X_{1},\lambda_{1})$. In all cases where such terminology will
be abused, the symplectomorphism will be obvious and unique up to contractible
choice.

If $F^{*}\lambda_{2}=\lambda_{1}$ on the nose, we say that
$(X_{1},\lambda_{1})$ and $(X_{2},\lambda_{2})$ are Liouville isomorphic. For
$(X,\lambda)$ and $(Y,\mu)$ Liouville domains or manifolds we call a smooth,
proper embedding $F:(X,\lambda)\to(Y,\mu)$ a Liouville embedding if
$F^{*}\mu=\lambda$.

####  2.1.4. Liouville germs

It will often be convenient to express our constructions in the language of
germs.

######  Definition 2.8.

A Liouville germ  $(\mathscr{X},\lambda)$ is the equivalence class of defining
Liouville domains for a fixed Liouville manifold $(X,\lambda)$.

######  Definition 2.9.

A Liouville embedding of a Liouville germ $(\mathscr{X},\lambda)$ into a
Liouville domain or manifold $(Y,\mu)$ is an equivalence class of Liouville
embeddings $F_{W}:(W,\lambda|_{W})\to(Y,\mu)$, where $W\subset X$ is a
defining domain for $(X,\lambda)$ and we identify $F_{W_{0}}$ with $F_{W_{1}}$
if they agree on a defining domain $W_{2}$ for $(X,\lambda)$ contained in both
$W_{0}$ and $W_{1}$.

######  Example 2.10.

The cotangent bundle $X=T^{*}M$ of a closed manfiold $M$ is a Liouville
manifold with $\lambda=pdq$ the canonical 1-form. The Liouville field is the
fibrewise radial vector field $Z=p\partial_{p}$ and the skeleton is the zero
section $M\subset T^{*}M$. As a Liouville domain we can take the unit disk
bundle $W=\\{\|p\|\leq 1\\}$ relative to any Riemannian metric on $M$. The
ideal boundary is the cosphere bundle $S^{*}M$, i.e the (positively)
projectivized cotangent bundle, which is contactomorphic to the unit sphere
bundle $\partial W=\\{\|p\|=1\\}$ for any choice of Riemannian metric. We will
use the notation $\mathscr{T}^{*}M$ for the germ of the cotangent bundle
$T^{*}M$.

###  2.2. Weinstein manifolds

The notion of Weinstein manifold was first introduced in [EG91] building on
[E90] and [W91].

####  2.2.1. Lyapunov functions

Let $(X,\lambda)$ be a Liouville manifold. While $\mathrm{Skel}(X,\lambda)$
always has its $2n$-dimensional Lebesgue measure equal to $0$, its dimension
can nonetheless be quite large if no extra conditions are imposed on the
Liouville structure. For example, McDuff constructed in [MD91] a Liouville
structure on $T^{*}\Sigma_{g}\setminus\Sigma_{g}$ for $\Sigma_{g}$ a closed
surface of genus $g>1$ whose skeleton has codimension 1. To tame the dynamics
of the Liouville flow we require existence of a Lyapunov function.

######  Definition 2.11.

We say that $\phi:X\to{\mathbb{R}}$ is a Lyapunov function for the Liouville
manifold $(X,\lambda)$ if $Z$ is gradient-like for $\phi$, i.e. if there holds

  * (W1)

$d\phi(Z)\geq\delta(\|Z\|^{2}+\|d\phi\|^{2})$ for some Riemannian metric on
$X$ and some constant $\delta>0$.

Note that the space of Lyapunov functions for a given Liouville structure is
convex, hence contractible as soon as it is nonempty. Consider the set of
critical points $\text{Crit}(\phi)=\\{d\phi=0\\}$ of the Lyapunov function,
which by (W1) is the same as the zero set $\\{\lambda=0\\}$ of the Liouville
form. The conditions (L1) and (W1) imply that $\mathrm{Skel}(X,\lambda)$ is
the union of the $Z$-stable manifolds of the critical points of $\phi$, i.e.
points converging to $\text{Crit}(\phi)$ in forward time. However, as far as
we know, we have to add some further constraints on the Lyapunov function
$\phi$ in order to deduce any meaningful properties.

######  Definition 2.12.

A Morse-Weinstein manifold is a Liouville manifold $(X,\lambda)$ for which
there exists a Morse Lyapunov function $\phi:X\to{\mathbb{R}}$.

Under this assumption it was shown in [CE12], see also [EG91, E95], that

  * (W2)

$\mathrm{Skel}(X,\lambda)$ is the union of submanifolds which are isotropic
for $\lambda$, and hence for $\omega$.

Figure 2.1. The skeleton of an open Riemann surface with Morse Lypapunov
function.

######  Remark 2.13.

Note that $\lambda$-isotropic is equivalent to $\omega$-isotropic and
$Z$-conic.

Without any assumptions on $\phi$, is not known whether a triple
$(X,\lambda,\phi)$ which satisfies (L1), (L2) and (W1) is homotopic in the
class of such triples to one with a Morse Lyapunov function. However,
condition (W2) holds for a larger class of taming functions, for example when
$\phi$ is generalized Morse, see [CE12]. Moreover, it is often most natural to
consider a Lyapunov function which is not Morse or even generalized Morse,
such as the Morse-Bott function $\phi=\|p\|^{2}$ on $(T^{*}M,pdq)$.

####  2.2.2. Morse-Bott Lyapunov functions

In this paper we will consider a version of Morse-Bott Weinstein structures
which allows the critical set of the Lyapnuov function to have nonempty
boundary, and even corners. Our definition is a slight variation of
Starkston’s definition in [St18] of a Weinstein structure which is Morse-Bott
with boundary.

Let $E^{+}\oplus E^{0}\oplus E^{-}$ be the decomposition of the tangent space
to $X$ at a zero of $Z$ into generalized eigenspaces of the differential $dZ$
with eigenvalue having positive, zero or negative real part respectively.
Suppose first that a connected component $C$ of $\text{Crit}(\phi)$ is a
smooth, proper submanifold of $X$ without boundary. Then the Morse-Bott
condition on a triple $(X,\lambda,\phi)$ along the component $C$ is the
following:

  * (MB1)

$Z$ is non-degenerate in the directions normal to $C$, i.e. $TC=E^{0}|_{C}$.

When $\partial C\neq\varnothing$ we moreover demand that in a neighborhood of
$\partial C$ the triple $(X,\lambda,\phi)$ takes a special form, which we now
describe. Recall the Weinstein normal form for Morse critical points in a
Weinstein manifold of dimension $2$. There are two possibilities depending on
the index: $k=0$ or $k=1$.

  * ($k=1$)

$\lambda_{1}=2pdq+qdp$, $\phi_{1}=p^{2}-\frac{1}{2}q^{2}$.

  * $(k=0$)

$\lambda_{0}=\frac{1}{2}(pdq-qdp)$, $\phi_{0}=\frac{1}{4}(p^{2}+q^{2})$.

We also recall the standard Morse-Bott normal form corresponding to the
canonical Liouville structure on $T^{*}{\mathbb{R}}$.

  * (std)

$\lambda_{\text{std}}=pdq$, $\phi=\frac{1}{2}p^{2}$.

We construct local models for Morse-Bott with boundary Weinstein structures on
$T^{*}{\mathbb{R}}$ with the half line $\\{p=0,q\geq 0\\}$ as the critical
set. The construction consists of patching up the above Morse normal forms on
the half plane $\\{q\leq 0\\}$ with the standard Morse-Bott normal form on the
half plane $\\{q\geq 0\\}$. Take any smooth function
$\psi:{\mathbb{R}}\to[0,1]$ satisfying the following properties:

  * (i)

$\psi=0$ on $[0,\infty)$

  * (ii)

$\psi=1$ on $(-\infty,-\varepsilon]$ for some $\varepsilon>0$.

  * (iii)

$\psi<1$ and $\psi^{\prime}<0$ on $(-\varepsilon,0)$ for that same
$\varepsilon>0$.

In the case $k=1$ we set $f_{1}(q,p)=\psi(q)pq$ and in the case $k=0$ we set
$f_{0}(q,p)=-\frac{1}{2}\psi(q)pq$. In both cases the patched up model is then
given by $\lambda_{k}^{\text{MB}}=\lambda_{\text{std}}+df_{k}$, which has
Liouville field $Z_{k}^{\text{MB}}=Z_{\text{std}}+X_{f_{k}}$ for $X_{f_{k}}$
the Hamiltonian vector field of $f_{k}$. Explicitly, when $k=1$ we have

$Z_{1}^{\text{MB}}=\left(1+\psi(q)+\psi^{\prime}(q)q\right)p\partial_{p}-\psi(q)q\partial_{q}.$

Since the coefficient which multiplies $p\partial_{p}$ is positive, we deduce
that $Z_{1}^{\text{MB}}$ admits a Lyapunov function. For example, if we set
$\phi=p^{2}-h(q)$ for

$h(q)=\int_{0}^{q}\psi(t)tdt,$

then $Z_{1}^{\text{MB}}$ is the Euclidean gradient of $\phi$ along $p=0$ and
we have $d\phi(Z_{1}^{\text{MB}})>0$ on all of $T^{*}{\mathbb{R}}$, from which
the claim follows. Similarly, one verifies that $Z_{0}^{\text{MB}}$ also
admits a Lyapunov function.

######  Definition 2.14.

We call $(T^{*}{\mathbb{R}},\lambda_{k}^{\text{MB}})$ the Weinstein normal
form for Morse-Bott boundary of index $k=0,1$.

Figure 2.2. The Weinstein normal form for Morse-Bott boundary of index 1 and 0
respectively.

The space of permissible $\psi$ is convex, hence the normal form is well
defined up to a contractible choice. Note that the behavior of the Liouville
flow along the half line $\\{p=0,q\leq 0\\}$, which shares its boundary point
$\\{p=0,q=0\\}$ with the critical set $\\{p=0,q\geq 0\\}$, can be either
repelling or attracting depending on whether $k=0$ or $k=1$. We also consider
products of these normal forms. Thus we obtain Weinstein structures on
$T^{*}{\mathbb{R}}^{n}$ where the critical set is the positive quadrant
$\\{p_{j}=0,q_{j}\geq 0\\}$, which not only has boundary but also has corners
of all orders $\leq n$.

######  Definition 2.15.

The Weinstein normal form for Morse-Bott $n$-fold corners
$(T^{*}{\mathbb{R}}^{n},\lambda_{r}^{\text{MB}})$ of index $0\leq r\leq n$ is
the product of $n-r$ copies of $(T^{*}{\mathbb{R}},\lambda_{0}^{\text{MB}})$
and $r$ copies of $(T^{*}{\mathbb{R}},\lambda_{1}^{\text{MB}})$.

Going back to a triple $(X,\lambda,\phi)$ as before, consider a component
$C\subset X$ of the critical set of $\phi$ such that $C$ is a smooth, proper
submanifold of $X$ with nonempty boundary $\partial C$ and possibly with
higher order corners. By definition, if $x\in\partial_{k}C\subset\partial C$
is a corner of order $k$, then there is a collar neighborhood
$U\times\mathcal{I}^{k}$ of $x$ in $C$, where $\mathcal{I}$ is the germ of the
interval $[0,1)$ at $0$ and $U\subset\partial_{k}C$ is a neighborhood of $x$
in the order $k$ corner locus. For such a collar neighborhood we have a local
symplectomorphism near $x$ between $X$ and $T^{*}U\times
T^{*}{\mathbb{R}}^{k}\times{\mathbb{C}}^{m}$, with $C$ corresponding to
$U\times\mathcal{I}^{k}\times 0$. The condition we require is that for each
such $k$-fold point $x\in\partial_{k}C$ there exists a collar neighborhood
$U\times\mathcal{I}^{k}$ as above such that with respect to the splitting
$T^{*}U\times T^{*}{\mathbb{R}}^{k}\times{\mathbb{C}}^{m}$ there holds:

  * (MB2)

$\lambda$ splits as a direct sum with the summand corresponding to $T^{*}U$
given by the canonical Liouville form and the summand corresponding to
$T^{*}{\mathbb{R}}^{k}$ given by the Weinstein normal form for Morse-Bott
$k$-fold corners $\lambda_{r}^{\text{MB}}$ for some $r\leq k$.

######  Definition 2.16.

A generalized Morse-Bott Weinstein manifold is a Liouville manifold
$(X,\lambda)$ which admits a Lyapnuov function $\phi$ whose critical set is a
union of smooth, proper submanifolds with corners $C\subset X$ satisfying
(MB1) and (MB2).

######  Remark 2.17.

It was shown by Starkston [St18] that in the boundary repelling Morse-Bott
case the intersection of the $Z$-stable manifold of a component $C$ of the
critical set of $\phi$ with a neighborhood of $C$ is a smoothly embedded
isotropic submanifold of $X$, even without requiring standard local models. To
check that this property also holds near the boundary attracting or mixed
behavior points of the Morse-Bott structures considered above is
straightforward, since it suffices to consider the explicit local model.

######  Lemma 2.18.

Suppose that $(X,\lambda_{0},\phi_{0})$ is a generalized Morse-Bott Weinstein
manifold. Then there exists a homotopy $(\lambda_{t},\phi_{t})$, fixed outside
of a compact subset, such that $(\lambda_{t},\phi_{t})$ is Morse-Weinstein for
$t>0$.

###### Proof.

We give the proof in the case where each component $C\subset\Sigma$ of the
critical set of $\phi$ is of the critical dimension $n=\frac{1}{2}\dim(X)$,
the general case is similar and in any case will not be needed for our
purposes. Suppose first that $C\subset\Sigma$ has empty boundary. Since $C$ is
Lagrangian, by the Weinstein neighborhood theorem it has a neighborhood in $X$
symplectomorphic to a neighborhood of the zero section in
$(T^{*}C,\lambda_{\text{std}}=pdq)$. Therefore we may from the onset replace
$X$ by $T^{*}C$ with the pulled back Liouville form which we still denote by
$\lambda_{0}$.

Let $f:C\to{\mathbb{R}}$ be a Morse function and let $\eta:T^{*}C\to[0,1]$ be
a cutoff function such that $\eta=1$ on ${\mathcal{O}}{\it p}\,(C)$ and
$\eta=0$ outside of a slightly bigger neighborhood. Fix an auxiliary
Riemannian metric on $C$ and put $h=\eta\lambda_{\text{std}}(\nabla f)$, a
function on $T^{*}C$. The restriction of the Hamiltonian vector field of the
function $h$ to $C$ is $\nabla f$. Hence $\phi_{0}+\eta f$ is a Lyapunov
function for $\lambda_{0}+dh$ in a neighborhood of $C$ and therefore on all of
$T^{*}C$ if $f$ is taken sufficiently $C^{1}$ small. Moreover, for such $h$
the Liouville form $\lambda_{0}+tdh$ is Morse-Weinstein for any $t>0$.

If $\partial C$ is non-empty, then we need to modify the construction
somewhat. Consider a slight enlargement $\widehat{C}$ of $C$ in $X$ obtained
by attaching an isotropic collar along $\partial C$ so that the Liouville
field $Z_{0}$ is tangent to $\widehat{C}$. Then we should require that our
function $f:\widehat{C}\to{\mathbb{R}}$ is Morse on $C$, compactly supported
in $\widehat{C}$, has no critical points on $\partial C$ and moreover has the
following behavior on $\partial C$. Let $P$ be the closure of a component of
the order 1 corner locus $\partial_{1}C=\partial
C\setminus\bigcup_{k>1}\partial_{k}C$. If the Liouville structure is boundary
attracting $(k=1)$ along $P$, then we demand that $df(v)>0$ for any inwards
pointing vector $v\in TC|_{\partial C}$. If the Liouville structure is
boundary repelling ($k=0$) along $P$, then we demand that $df(v)>0$ for any
outwards pointing vector $v\in TC|_{\partial C}$. With this restriction on $f$
the proof proceeds just as before. ∎

######  Remark 2.19.

As we will see, if a Morse-Bott Weinstein manifold has arboreal skeleton, then
one can choose the Morsifying homotopy such that
$\mathrm{Skel}(X,\lambda_{t})=\mathrm{Skel}(X,\lambda_{0})$ for all $t\geq 0$.

####  2.2.3. Weinstein manifolds

It is easy to check that Lemma 2.18 also holds for the generalized Morse
condition instead of Morse-Bott, see Proposition 9.13 in [CE12]. Therefore we
will take the following as our working definition, which includes Morse,
generalized Morse and Morse-Bott Weinstein structures.

######  Definition 2.20.

A Liouville manifold $(X,\lambda_{0})$ is called Weinstein if there exists a
Liouville homotopy $\lambda_{t}$ and a family of Lyapunov functions $\phi_{t}$
for $Z_{t}$ such that $(X,\lambda_{t},\phi_{t})$ is Morse-Weinstein for $t>0$.

The notions of a Weinstein domain, defining Weinstein domain for a Weinstein
manifold and a Weinstein germ are entirely analogous to the corresponding
notions in the Liouville case. We emphasize that the Lyapunov function $\phi$
is not part of the structure of a Weinstein manifold, we merely require its
existence. For example, Weinstein embeddings are just Liouville embeddings of
Weinstein type Liouville manifolds (or domains, or germs). We do not require
the embedding to pull back one Lyapunov function to the other. Similarly, a
Weinstein isomorphism (resp. homotopy) is a Liouville isomorphism (resp.
homotopy) of Liouville manifolds of Weinstein type.

######  Example 2.21.

The standard Liouville structure $\lambda=pdq$ on the cotangent bundle
$T^{*}M$ of a closed manifold $M$ is Weinstein. Given a Riemannian metric on
$M$, the function $\phi=\|p\|^{2}$ is a Morse-Bott Lyapunov function for
$\lambda$. By the procedure explained in Lemma 2.18, we can Morsify the
Lyapunov function $\phi$ using a Morse function on $M$. By inspection, this
procedure does not change the skeleton of $T^{*}M$, which is the zero section.

######  Remark 2.22.

Although Definition 2.20 allows for rather degenerate Weinstein structures, in
the Weinstein homotopies constructed in this paper for the purposes of
arborealization of a Morse-Weinstein manifold the Weinstein structures are
always of Morse-Bott type except for a finite number of times at which there
is a birth/death of Morse-Bott critical components.

###  2.3. Wc-manifolds

####  2.3.1. Manifolds with boundary and corners

Recall that an $n$-dimensional smooth manifold $M$ has a corner of order
$k\leq n$ at $x\in M$ if there is a neighborhood of $x$ in $M$ diffeomorphic
to a neighborhood of the origin in $\mathcal{I}^{k}\times{\mathbb{R}}^{n-k}$,
where $\mathcal{I}$ denotes the germ of the interval $[0,1)$ at $0$. We denote
the locus of order $k$ corners by $\partial_{k}M$. The closure $P$ of a
connected component of $\partial_{k}M$ is called a boundary $k$-face. For
$k=1$ we will more simply call $P$ a boundary face. Sometimes we will call a
manifold with boundary and corners (of any order) a bc-manifold for short.

We always assume that each component of $\partial_{k}M$ is regularly embedded,
so that each $k$-face is itself a manifold with corners. Under this
assumption, if $Q$ is a $k$-face, then there is an embedded collar
neighborhood $Q\times\mathcal{I}^{k}\subset M$. We consider the germs of these
collars as part of the structure. In particular, near each point
$x\in\partial_{k}M$ we have canonical collar coordinates $x=(y,t)$, where
$y\in\partial_{k}M$ and $t=(t_{1},\ldots,t_{k})\in\mathcal{I}^{k}$. Note that
in a neighborhood of $x$ we have $\partial_{k}M$ cut out by
$t_{1}=\cdots=t_{k}=0$. More generally, for $j\leq k$ the components of
$\partial_{j}M$ whose closure contains $x$ are given by setting exactly $j$ of
the coordinates $t_{i}$ equal to zero.

We demand compatibility of these collars in the sense that the remaining $n-j$
coordinates $t_{i}$ give the collar structure for
$\partial_{j}M\times\mathcal{I}^{n-j}$ near $x$. We call such a compatible
system of collar germs a corner structure on $M$. Henceforth all manifolds
with corners will be implicitly equipped with a fixed but otherwise arbitrary
corner structure.

Figure 2.3. A corner structure on $M$.

Let us choose a partial ordering of the boundary faces of a manifold $M$ with
corners such that the faces at all corners are ordered in a compatible way,
i.e. the order is that induced by the order of the coordinates $t_{i}$ in the
collar structure. We will assume this order to be part of the structure of a
manifold with corners.

Given manifolds with corners $W$ and $X$, we consider embeddings of $W$ into
$X$ which satisfy the property that each $j$-face of $W$ is properly embedded
into some $k$-face of $X$, where $k\geq j$.

####  2.3.2. Liouville manifolds with boundary and corners

We now give an inductive definition of the notion of a Liouville manifold with
corners of order $k\leq\frac{1}{2}\dim X$, starting with the base case $k=0$
which is just our previous definition.

######  Definition 2.23.

A _Liouville manifold with corners of order $k$_ consists of an exact
symplectic manifold $(X^{2n},\lambda)$ with smooth corners of order $k\leq n$
such that the following properties hold.

Figure 2.4. A Liouville manifold with coners $W$ near one of its faces $P$
with nucleus $N$.

  * (L1)

The vector field $Z$ which is $\omega$-dual to $\lambda$ is complete for
$t\to\pm\infty$.

  * (L2)

There exists a compact domain with corners $W\subset X$ such that such that
$Z$ is outwards transverse to its vertical boundary $\partial_{v}W$, which is
defined to be the closure of $\partial W\cap\text{int}X$, and such that the
union of forward trajectories of $Z$ starting at $\partial_{v}W$ is equal to
$\partial_{v}W\cup(X\setminus W)$.

  * (L3)

For $r=1,\ldots,k$, each $r$-face $P$ of the horizontal boundary
$\partial_{h}W$, which is defined to be $\partial W\cap\partial X$, contains a
Liouville submanifold with corners $N\subset P$, called the nucleus of $P$,
such that the collar neighborhood of $P$ in $X$ is of the form $N\times
T^{*}\mathcal{I}^{r}$, where $P$ is the product of $N$ with the cotangent
fibre over $0\in\mathcal{I}^{r}$, and in this neighborhood the form $\lambda$
is the direct sum of a Liouville form on $N$ and the canonical 1-form on
$T^{*}\mathcal{I}^{r}$.

It follows from the definition that the Liouville field $Z$ is tangent to
every component of $\partial_{k}X$, $k=1,\ldots,n$. A Liouville manifold with
corners has a horizontal boundary $\partial_{h}X=\partial X$ and an ideal
vertical boundary
$\partial_{\infty}X=\big{(}X\setminus\mathrm{Skel}(X,\lambda)\big{)}/{\mathbb{R}}^{+}$,
where just as before the action of ${\mathbb{R}}^{+}$ is defined by the flow
of $Z$ and the skeleton $\mathrm{Skel}(X,\lambda)$ is defined to be the
attractor of the negative flow of $Z$.

We call $(W,\lambda|_{W})$ as in (L2) a defining Liouville domain with corners
for the Liouville manifold with corners $(X,\lambda)$. Note that the
horizontal boundary of $W$ is $\partial_{h}W=W\cap\partial_{h}X$ and its
vertical boundary $\partial_{v}W$, the closure of $\partial
W\setminus\partial_{h}W$, is a contact manifold with corners which is
naturally contactomorphic to $\partial_{\infty}X$ under the projection
$\pi_{\infty}:\partial_{v}W\to\partial_{\infty}X$. Just as before, the
equivalence class of defining domains for Liouville manifold with corners is
called the Liouville germ of $(X,\lambda)$ and is denoted by
$(\mathscr{X},\lambda)$. Notions of isomorphism and homotopy of Liouville
manifolds with corners are defined exactly the same way as without corners.

####  2.3.3. Wc-manifolds

We now introduce Weinstein structures for Liouville manifolds with corners,
which we call Wc-manifolds. We restrict our discussion to the Morse-Bott
category since this is the framework in which we will work throughout this
paper.

######  Definition 2.24.

A Wc-manifold is a Liouville manifold with corners $(X,\lambda)$ whose
Liouville vector field admits a Morse-Bott Lyapunov function
$\phi:X\to{\mathbb{R}}$, i.e. satisfying (MB1) and (MB2) in the interior of
$X$, and such that on the collars $N\times T^{*}\mathcal{I}^{k}$ of (L3)
$\phi$ has the form $\phi_{N}+\sum_{i=1}^{k}u_{i}^{2}$ for $\phi_{N}$ a Morse-
Bott Lyapunov function on $(N,\lambda|_{N})$ and $u_{1},\ldots,u_{k}$ the
momentum coordinates on $T^{*}\mathcal{I}^{k}$.

Figure 2.5. The prototypical example of a Wc-manifold is the cotangent bundle
of a manifold $M$ with nonempty boundary, or even corners of any order.

Defining Liouville domains for Wc-manifolds will be called defining Wc-
domains, and the equivalence class of such will be called a Wc-germ. Notions
of Wc-isomorphism, embedding and homotopy are the same as the corresponding
notions for Liouville manifolds with corners, but demanding that all objects
in sight are of Weinstein type.

######  Example 2.25.

Let $M$ be a smooth $n$-dimensional manifold with corners. Then $T^{*}M$ is a
Liouville manifold with corners, where the structure (L3) is induced by a
corner structure on $M$. We call the germ $\mathscr{T}^{*}M$ of $T^{*}M$ a
cotangent block. The corresponding Lagrangian distribution tangent to
cotangent fibers will be denoted by $\nu_{M}$. We denote by $\lambda_{M}$ and
$Z_{M}$ the corresponding Liouville form and the Liouville vector field. The
notation $S^{*}M$ will be used for the positively projectivized cotangent
bundle of $M$, which is the ideal boundary of $\mathscr{T}^{*}M$. For $Q$ a
$k$-face of $M$ we have $P=T^{*}M|_{Q}\subset\partial_{k}T^{*}M$ a $k$-face of
$T^{*}M$ and $N=T^{*}Q$ is the nucleus of $P$. The collar coordinates near $Q$
yield the decomposition ${\mathcal{O}}{\it p}\,P=N\times
T^{*}\mathcal{I}^{k}$. A choice of Riemannian metric on $M$ yields defining
Liouville domain with corners $W=\\{\|p\|\leq\varepsilon\\}$ and a Morse-Bott
Lyapunov function $\phi=\|p\|^{2}$.

The rest of this section can be equally discussed in the Liouville and
Weinstein categories. We restrict the discussion only to the Weinstein case as
this is the one needed for our further applications.

###  2.4. Wc-hypersurfaces

####  2.4.1. Weinstein hypersurfaces

We now turn our attention to Weinstein hypersurfaces with corners, or Wc-
hypersurfaces. They were introduced (without corners) in [E18]. These are
special cases of the Liouville hypersurfaces introduced by Avdek in [A12].
This notion is similar to the “stops” of Sylvan, [Sy16] and the Liouville
sectors of Ganatra-Pardon-Shende, [GPS17]. Related constructions are also
considered in Ekholm-Lekili’s paper [EL17]. We first recall the definition
without corners.

######  Definition 2.26.

A Weinstein hypersurface in a Weinstein manifold $(X^{2n},\lambda)$ is a
Weinstein embedding
$(A^{2n-2},\lambda^{\prime}=\lambda|_{A})\hookrightarrow(X\setminus\mathrm{Skel}(X,\lambda),\lambda)$
such that $\pi_{\infty}|_{A}:A\to\partial_{\infty}X$ is an embedding.

Note that any Weinstein hypersurface is contained in a boundary $\partial W$
of some defining domain $W$. Note also that the Reeb vector field of the
contact form $\lambda|_{\partial W}$ is transverse to $A$. Note also that the
boundary $\partial A$ is a codimension 2 contact submanifold of $(\partial
W,\lambda|_{\partial W})$. By the assumption that $(A,\lambda|_{A})$ is
Weinstein, the skeleton $\mathrm{Skel}(A,\lambda|_{A})$ can be decomposed as a
union of isotropic submanifolds, which in the top dimension $n-1$ are
Legendrian for the contact structure $\xi=\ker(\lambda|_{\partial W})$.

######  Lemma 2.27.

The skeleton $\mathrm{Skel}(A,\lambda|_{A})$ depends only on the projection
$\pi_{\infty}(A)\subset\partial_{\infty}X$.

###### Proof.

Indeed, the Liouville fields corresponding to the Liouville structures
$\lambda_{A}$ and $f\lambda|_{A}$ for a positive function $f$ are
proportional. In fact, as computed in Lemma 12.1 of [CE12] the form
$d(f\lambda)|_{A}$ is symplectic if and only if $k:=\inf(f+df(Z))>0$, where
$Z$ is the expanding field for $\lambda|_{A}$, and in that case the
restriction of the form $f\lambda$ to $A$ is automatically Liouville, with the
expanding vector field for $f\lambda|_{A}$ equal to $\frac{1}{k}Z$. Therefore
$\mathrm{Skel}(A,\lambda|_{A})$ only depends on the contact structure
$\xi_{\infty}$. ∎

Note that the space of functions $f>0$ for which $f\lambda$ is Liouville (and
hence in the considered case Weinstein) is convex, therefore contractible.

######  Definition 2.28.

We say that two Weinstein hypersurfaces $A_{0},A_{1}$ in a Weinstein manifold
$(X,\lambda)$ are related by a translation if $A_{1}=Z^{f}(A_{0})$ for some
function $f:A_{0}\to{\mathbb{R}}$.

Note that given a defining domain $W\subset X$, any Weinstein hypersurface $A$
of $X$ can be translated to a Weinstein hypersurface contained in $W$.
However, it may not be possible to translate it to a Weinstein hypersurface
contained in $\partial W$. Indeed, this is equivalent to the condition
$\text{inf}(f+df(Z))>0$ for $f$ the conformal factor between the contact forms
corresponding to $A$ and $\partial W$.

######  Definition 2.29.

A Weinstein hypersurface germ $\mathscr{A}\subset
X\setminus\mathrm{Skel}(X,\lambda)$ is an equivalence class of Weinstein
hypersurfaces
$(A,\lambda^{\prime}=\lambda|_{A})\subset(X\setminus\mathrm{Skel}(X,\lambda),\lambda)$
for $(A,\lambda^{\prime})$ defining domains for a fixed Liouville manifold,
where two embeddings are equivalent if they agree on a smaller defining
domain, possibly after translation.

One can think of $\mathscr{A}$ as living in the ideal boundary
$\partial_{\infty}X$, or one can think of $\mathscr{A}$ as the germ of a
Weinstein embedding
$\mathscr{A}\to\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$. The
two viewpoints are equivalent and we will go back and forth between them.

######  Example 2.30.

An important example of a Weinstein hypersurface is the ribbon of a Legendrian
submanifold. Let $\Lambda$ be a Legendrian submanifold in the contact boundary
of a Liouville domain $(W,\lambda)$. Then it admits a Darboux neighborhood
$U(\Lambda)\subset\partial W$ contactomorphic to
$\\{\|p\|^{2}+z^{2}\leq\varepsilon^{2}\\}\subset J^{1}(\Lambda)$ for some
Riemannian metric on $\Lambda$. The strip
$\Sigma(\Lambda):=U(\Lambda)\cap\\{z=0\\}$ is a Weinstein hypersurface
symplectomorphic to the cotangent disk bundle of $\Lambda$, whose skeleton is
precisely $\Lambda$. The space of all ribbon extensions of a given Legendrian
$\Lambda$ is contractible. An important example to keep in mind is the case
where$\mathscr{X}=\mathscr{T}^{*}M$ for $M$ a smooth manifold, $W=\\{\|p\|\leq
1\\}$ with respect to any Riemannian metric on $M$ and $\Lambda$ is the unit
conormal to a co-oriented hypersurface $H\subset M$.

Figure 2.6. A Legendrian $\Lambda\subset S^{*}M$ and its ribbon $U(\Lambda)$.

####  2.4.2. Weinstein pairs

######  Definition 2.31.

A Weinstein pair $(X,A,\lambda)$ consists of a Weinstein manifold
$(X,\lambda)$ together a Weinstein hypersurface $A\subset
X\setminus\mathrm{Skel}(X,\lambda)$.

We will also call $(X,\mathscr{A},\lambda)$ a Weinstein pair when
$\mathscr{A}$ is the germ of a Weinstein hypersurface $\mathscr{A}\subset
X\setminus\mathrm{Skel}(X,\lambda)$.

######  Definition 2.32.

Let $(X,\lambda)$ be a Liouville manifold and $Y\subset
X\setminus\mathrm{Skel}(X,\lambda)$ any subset. The Liouville cone on $Y$ is
the saturation of $A$ by the Liouville trajectories:

$\mathrm{Cone}(Y,\lambda)=\bigcup_{t\in{\mathbb{R}}}Z^{-t}(Y)\subset X.$

The positive Liouville cone on $A$ is the saturation of $A$ by the backwards
Liouville trajectories:

$\mathrm{Cone}^{+}(Y,\lambda)=\bigcup_{t\geq 0}Z^{-t}(Y)\subset X.$

Note that if $Y$ is compact, then $\text{Cone}^{+}(Y,\lambda)$ has compact
closure while $\text{Cone}(Y,\lambda)$ does not. The skeleton of a Weinstein
pair $(X,A,\lambda)$ is defined as the compact subset:

$\mathrm{Skel}(X,A,\lambda)=\mathrm{Skel}(X,\lambda)\cup\mathrm{Cone}^{+}\left(\mathrm{Skel}(A,\lambda|_{A}),\lambda\right).$

Note that $\mathrm{Skel}(X,A,\lambda)\subset W$ for any defining domain $W$
such that $A\subset\partial W$, moreover $\partial
W\cap\mathrm{Skel}(X,A,\lambda)=\mathrm{Skel}(A,\lambda|_{A})$. The notion of
a pair at the level of germs is defined as follows.

######  Definition 2.33.

A Weinstein (germ) pair $(\mathscr{X},\mathscr{A},\lambda)$ consists of a
Weinstein germ $(\mathscr{X},\lambda)$ together with a Weinstein hypersurface
germ
$\mathscr{A}\subset\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$.

Note that for any representative $(W,\lambda)$ of $(\mathscr{X},\lambda)$ we
may take a representative $A$ for $\mathscr{A}$ contained in $W$. The skeleton
of a Weinstein pair $(\mathscr{X},\mathscr{A},\lambda)$ is defined as the non-
compact subset:

$\mathrm{Skel}(X,\mathscr{A},\lambda)=\mathrm{Skel}(X,\lambda)\cup\mathrm{Cone}\left(\mathrm{Skel}(\mathscr{A},\lambda|_{\mathscr{A}}),\lambda\right).$

Equivalently in terms of germs, we can think of
$\mathrm{Skel}(X,\mathscr{A},\lambda)$ as the equivalence class of the compact
subsets $\mathrm{Skel}(X,A,\lambda)\subset W$ for all defining domains
$(W,\lambda|_{W})$ and representatives $A$ for $\mathscr{A}$ such that
$A\subset W$.

####  2.4.3. Wc-hypersurfaces

Finally, we extend the above to include boundary and corners.

######  Definition 2.34.

A Wc-hypersurface in a Wc-manifold $(X^{2n},\lambda)$ is an equivalence class
of embeddings of Wc-germs
$(\mathscr{A}^{2n-2},\lambda^{\prime}=\lambda|_{\mathscr{A}})\hookrightarrow(X\setminus\mathrm{Skel}(X,\lambda),\lambda)$
such that $\pi_{\infty}|_{\mathscr{A}};\mathscr{A}\to\partial X_{\infty}$ is
an embedding, with the equivalence given by translation.

All the previous discussion carried over word by word to the Wc-setting. The
notion of a Weinstein pair in the Wc-category is called a Wc-pair.

######  Definition 2.35.

A Wc-pair $(\mathscr{X},\mathscr{A},\lambda)$ consists of a Wc-germ
$(\mathscr{X},\lambda)$ together with a Wc-hypersurface germ
$\mathscr{A}\subset\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$.

###  2.5. Conversion between Wc-hypersurfaces and face nuclei

####  2.5.1. Nucleus-to-hypersurface conversion

We start with a Wc-manifold $(X,\lambda)$. Let $P$ be a boundary face of $X$
and $N\subset P$ its nucleus. Let $U$ be a collar neighborhood of $P$ in $X$
as in the definition of a Wc-manifold. So $U$ is of the form $N\times
T^{*}\mathcal{I}$, with $\lambda=\lambda|_{N}+udt$ where $t$ is the defining
coordinate for $P$. Attach to $N\times T^{*}\mathcal{I}$ the half-infinite
collar $N\times T^{*}(-\infty,0]$ along their common intersection and on the
union consider the Weinstein structure given by the direct sum of
$\lambda|_{N}$ and the Weinstein normal form for Morse-Bott boundary of index
$k=0$, which agrees with $\lambda_{N}+udt$ on $N\times T^{*}\mathcal{I}$. The
existence of a Morse-Bott Lyapnuov function is immediate from the local model.
Hence the result is a Wc-manifold $(X^{N},\lambda^{N})$. Let $\mathscr{N}$ be
the germ of $N$. For $\varepsilon>0$ note that
$\mathscr{N}\times\\{t=-\varepsilon\\}\subset\mathscr{N}\times
T^{*}(-\infty,0]$ is a Wc-hypersurface for $\lambda^{N}$. Any two differ by a
translation and all are canonically identified with $\mathscr{N}$. We will
denote this Wc-hypersurface by $\mathscr{A}^{N}\subset
X^{N}\setminus\mathrm{Skel}(X^{N},\lambda^{N})$.

Note that by construction we have
$\mathrm{Skel}(X^{N},\mathscr{A}^{N},\lambda^{N})=\mathrm{Skel}(X,\lambda)$.
The operation $(X,N)\mapsto(X^{N},\mathscr{A}^{N})$ only depends on the
contractible choice of a collar neighborhood and the contractible choice of
the Weinstein normal form for Morse-Bott boundary. Hence it produces a Wc-germ
$(\mathscr{X}^{N},\lambda^{N})$ which is well defined up to Wc-homotopy.

######  Definition 2.36.

The Wc-pair $(\mathscr{X}^{N},\mathscr{A}^{N},\lambda^{N})$ is called the
nucleus-to-hypersurface conversion of the Wc-germ $(\mathscr{X},\lambda)$ and
the boundary nucleus $N$.

Figure 2.7. Converting a boundary nucleus to a Wc-hypersurface.

####  2.5.2. Underlying Weinstein manifold of a Wc-manifold

Let $X$ be a Wc-manifold and let $P$ be a boundary face. The resulting of
converting the nucleus $N$ of $P$ to a Wc-hypersurface is a Wc-manifold which
has one less boundary face than $X$. We can continue this process inductively
over all faces. After forgetting the resulting Wc-hypersurfaces, we end up
with a Weinstein manifold without boundary which we denote $\widehat{X}$. It
is clear that the Weinstein homotopy class of the Weinstein manifold
$\widehat{X}$ does not depend on the order in which the boundary faces are
chosen. If one wishes to, one can further deform the Weinstein structure on
$\widehat{X}$, which at this point is generalized Morse-Bott, so that it
becomes Morse-Weinstein.

######  Definition 2.37.

The Weinstein manifold $\widehat{X}$ obtained from a Wc-manifold $X$ by
converting all of its boundary nuclei to Wc-hypersurfaces is called the
underlying Weinstein manifold.

######  Example 2.38.

the underlying Weinstein manifold of the Wc-manifold $(T^{*}[0,1]^{n},pdq)$ is
Weinstein homotopic to $({\mathbb{R}}^{2n},pdq-qdp)$.

######  Remark 2.39.

A Wc-manifold may be thought of as a Weinstein manifold with ‘stops’. From
this viewpoint, taking the underlying Weinstein manifold of a Wc-manifold is
simply removing the stops.

####  2.5.3. Hypersurface-to-nucleus conversion

The converse operation to the nucleus-to-hypersurface conversion is the
hypersurface-to-nucleus conversion, which we discuss next. Here and below we
denote $U_{a,b}=\\{0<u\leq a,\,-b<t<b\\}\subset T^{*}{\mathbb{R}}$.

######  Lemma 2.40.

The inclusion $(A,\lambda^{\prime}=\lambda|_{A})\hookrightarrow(X,\lambda)$ of
a Wc-hypersurface extends, uniquely up to contractible choice, to a Liouville
embedding $(A\times U_{1,\varepsilon},\lambda^{\prime}+udt)\to(X,\lambda)$,
where $A$ is identified with $A\times(u=1,t=0)\subset A\times
U_{1,\varepsilon}$.

###### Proof.

Let $W\subset X$ be a defining domain such that $A\subset\partial W$ and let
$A\times(-\varepsilon,\varepsilon)\subset\partial W$ be its contact
surrounding, as in [E18], so that the restriction of $\lambda$ to this
neighborhood is of the form $\pi^{*}\lambda|_{A}+dt$, where
$\pi:A\times(-\varepsilon,\varepsilon)\to A$ is the projection and
$t\in(-\varepsilon,\varepsilon)$. For every $x\in
X\setminus\mathrm{Skel}(X,\lambda)$, let $x^{s}$ denote the backwards flow of
$Z_{X}$ starting at $x$, where $s\in(-\infty,0]$. For every $y\in
A\times(-\varepsilon,\varepsilon)$, let $y^{s}\in U_{1,\varepsilon}$ denote
the backwards flow of $Z_{A}+u\frac{\partial}{\partial u}$ starting at $y$,
where $s\in(-\infty,0]$. We extend the contact embedding
$A\times(-\varepsilon,\varepsilon)\to\partial W$ which sends $y\mapsto x$ to a
Liouville embedding $A\times U_{1,\varepsilon}\to W$ by sending $y^{s}\mapsto
x^{s}$, so that by construction $Z_{A}+u\frac{\partial}{\partial u}$ is
identified with $Z_{X}$. It is clear that the embedding is determined by the
contact surrounding and its parametrization, which are unique up to
contractible choice. ∎

We now explain the hypersurface-to-nucleus conversion. Let $(X,\lambda)$ be a
Wc-manifold and let $\mathscr{A}\subset X\setminus\mathrm{Skel}(X,\lambda)$ be
a Wc-hypersurface. Fix defining Wc-domains $A$ and $W$ for $\mathscr{A}$ and
$X$ respectively so that we have an embedding $A\to\partial W$ which extends
to an embedding $A\times U_{1,\varepsilon}\to W$ as in Lemma 2.40. Consider
the exact symplectic manifold which is the result of gluing $W$ and $A\times
U_{3,\varepsilon}$ along $A\times U_{1,\varepsilon}$. Consider the Weinstein
structure on $U_{3,\varepsilon}$, thought of as the $\varepsilon$-neighborhood
of the zero section in $T^{*}(0,3)$, which is the restriction of the Weinstein
normal form for Morse-Bott boundary of index $k=1$ translated two units in the
$u$ direction, so that the critical set is $\\{t=0,u\geq 2\\}$. The
restriction of this structure to $U_{1,\varepsilon}$ is $udt$, hence we obtain
a patched up structure on the union of $W$ and $A\times U_{3,\varepsilon}$.
Moreover, by inspection the new Liouville field points outwards at the
boundary. So we may replace $U_{3,\varepsilon}$ with a smoothing
$U_{3,\mu}=\\{-\varepsilon<t<\varepsilon,\,\,0<u<\text{min}(3,\mu(|t|))\\}$
for $\mu:(0,\varepsilon)\to[1,\infty)$ a function satifying

  * (i)

$\mu(x)\to\infty$ as $x\to 0$

  * (ii)

$\mu=1$ on $(\varepsilon/2,\varepsilon)$

  * (iii)

$\mu^{\prime}<0$ on $(0,\varepsilon/2)$

and the Liouville field is still outwards pointing for the smoothed boundary.
Thus we obtain a defining domain for a Wc-manifold $(X^{A},\lambda^{A})$.
Indeed the only think that needs to be checked is the existence of a Morse-
Bott Lyapunov function, which is clear from the local models since without
loss of generality we can start with a defining domain $W$ for $X$ such that
$\partial W$ is a regular level set of the Morse-Bott Lypanuov function $\phi$
for $X$. Note that $A\times\\{t=0,u=3\\}$ is the nucleus of the new boundary
face of $X^{A}$.

By construction we have
$\mathrm{Skel}(X^{A},\lambda^{A})=\mathrm{Skel}(X,A,\lambda)$, or more
precisely

$\mathrm{Skel}(X^{A},\lambda^{A})=\mathrm{Skel}(X,A,\lambda)\cup(\mathrm{Skel}(A,\lambda|_{A})\times[1,2]).$

In other words, the backwards $\lambda$-Liouville cone on $A$ has a trivial
collar attached to it, which does not change the topology of
$\mathrm{Skel}(X,A,\lambda)$. Note that the space of possible smoothings $\mu$
is convex, hence contractible. The choice of Morse-Bott Weinstein normal form
is also contractible. Similarly the choice of collars used in the construction
is contractible. Hence we get a Wc-germ $(\mathscr{X}^{A},\lambda^{A})$ which
is well defined up to Wc-homotopy. Finally, note that different
representatives $A$ for $\mathscr{A}$ will also result in homotopic Wc-
manifolds $(X^{A},\lambda^{A})$. Hence the conversion operation only depends
on the Wc-hypersurface germ $\mathscr{A}$ up to Wc-homotopy.

######  Definition 2.41.

The Wc-germ $(\mathscr{X}^{\mathscr{A}},\lambda^{\mathscr{A}})$ is called the
hypersurface-to-nucleus conversion of the Wc-germ $X$ and the Wc-hypersurface
$\mathscr{A}$.

Figure 2.8. Converting a boundary nucleus to a Wc-hypersurface.

####  2.5.4. Cancellation of conversions

The hypersurface-to-nucleus conversion is inverse to the nucleus-to-
hypersurface conversion at the level of Weinstein homotopy classes, as follows
from a parametric Smale cancellation of Morse-Bott index $0$ and $1$ critical
points, the existence of which is immediate from the local models. This
implies the following obvious but important lemma.

######  Lemma 2.42.

Let $(X^{\mathscr{A}},\lambda^{\mathscr{A}})$ be the result of hypersurface-
to-nucleus conversion of a Wc-hypersurface $\mathscr{A}\subset
X\setminus\mathrm{Skel}(X,\lambda)$. Then the underlying Weinstein manifolds
of $(X,\lambda)$ and $(X^{\mathscr{A}},\lambda^{\mathscr{A}})$ are Wc-
homotopic.

###  2.6. Gluing and splitting

####  2.6.1. Horizontal gluing

We will now describe the basic gluing operation for Wc-manifolds. Given two
Wc-manifolds $(X_{0},\lambda_{0})$ and $(X_{1},\lambda_{1})$, we define their
horizontal gluing as follows. Pick some non-adjacent boundary faces
$P_{1},\dots,P_{l}$ of $X_{1}$ and $Q_{1},\dots,Q_{l}$ of $X_{0}$, and for
each $j$ suppose we are given a Liouville isomorphism $\phi_{j}:N_{j}\to
M_{j}$, where $N_{j}$ and $M_{j}$ are the nuclei or $P_{j}$ and $Q_{j}$
respectively. We have collar neighborhoods $N_{j}\times
T^{*}\mathcal{I}\subset X_{1}$ and $M_{j}\times T^{*}\mathcal{I}\subset X_{2}$
and we extend $\phi_{j}$ to $\Phi_{j}:P_{j}\to Q_{j}$ as
$\Phi_{j}=\phi_{j}\times\text{id}_{\mathbb{R}}:N_{j}\times{\mathbb{R}}\to
M_{j}\times{\mathbb{R}}$.

######  Definition 2.43.

The horizontal gluing of $(\mathscr{X}_{1},\lambda_{1})$ and
$(\mathscr{X}_{2},\lambda_{2})$ along $\\{\phi_{j}\\}_{j}$ is defined to be
the Wc-germ of the Wc-manifold $X_{1}\cup X_{2}$ where we identify $P_{j}\sim
Q_{j}$ via $\Phi_{j}$.

######  Example 2.44.

Suppose $M,N$ are two manifolds with boundary and we have a diffeomorphism
$\psi:\partial N\to\partial M$. Then $\psi$ lifts to a Liouville isomorphism
$\phi$ between $T^{*}(\partial N)$ and $T^{*}(\partial M)$ and we can perform
the horizontal gluing $T^{*}N\cup_{\phi}T^{*}M$ which is the same as
$T^{*}(N\cup_{\psi}M)$ for $N\cup_{\psi}M$ the smooth gluing of $N$ and $M$
given by $\psi$.

Figure 2.9. Horizontal gluing of two Wc-manifolds.

Note that the skeleton of the horizontal gluing of $X_{1}$ and $X_{2}$ is
equal to the union of the skeleta of $X_{1}$ and $X_{2}$.

####  2.6.2. Vertical gluing

We can also glue two Wc-manifolds $X_{1}$ and $X_{2}$ in a different way.

######  Definition 2.45.

Suppose we have a Wc-hypersurface $\mathscr{A}$ in $\mathscr{X}_{1}$ and a
Liouville isomorphism $\phi:\mathscr{A}\to\mathscr{N}$ for $\mathscr{N}$ the
germ of the nucleus $N$ of a boundary face of $X_{2}$. The vertical gluing of
$\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ is defined to be the composition of
two steps: a conversion a Wc-hypersurface into a boundary hypersurface, see
Section 2.5 above, followed by a horizontal gluing.

Figure 2.10. Vertical gluing is the composition of two steps: a Wc-
hypersurface to boundary nucleus conversion and a horizontal gluing.

Indeed, if we first convert $\mathscr{A}$ into a boundary nucleus, we can then
apply the horizontal gluing determined by $\phi$ as descibed above. The
resulting Wc-germ is well defined up to Wc-isomorphism. Note that the skeleton
of the vertical gluing is equal to the union of the skeleton of the pair
$(\mathscr{X}_{1},\mathscr{A})$ and the skeleton of $\mathscr{X}_{2}$. We
record for future reference the following slight strengthening of this
observation, where we denote by $\lambda^{P}$ the Liouville form
$\lambda-d(ut)$ for $t$ the defining coordinate of a boundary face $P$.

######  Lemma 2.46.

Let $(\mathscr{X}_{i},\lambda_{i})$, $i=1,2$ be two Wc-germs,
$\mathscr{A}\subset\mathscr{X}_{2}\setminus\mathrm{Skel}(\mathscr{X}_{2},\lambda_{2})$
a Wc-hypersurface, $N$ the nucleus of a boundary face $P$ of $X_{1}$ and
$\mathscr{N}$ its germ, $\phi:\mathscr{A}\to\mathscr{N}$ a Liouville
isomorphism and $(\mathscr{Y},\lambda)$ the result of vertically gluing
$\mathscr{X}_{1}$ to $\mathscr{X}_{2}$ along $\phi$. Then there exists a
symplectic embedding of germs $\Psi:\text{int}\mathscr{X}_{1}\to\mathscr{Y}$
such that the following properties hold, where we denote by
$\iota:\text{int}\mathscr{X}_{2}\to\mathscr{Y}$ the restriction of the
inclusion to the interior of $\mathscr{X}_{2}$.

  * (i)

$\iota(\mathrm{Skel}(\mathscr{X}_{2},\lambda_{2}))\cup\Psi(\mathrm{Skel}(\mathscr{X}_{1},\lambda_{1})\cap\text{int}\mathscr{X}_{1})=\mathrm{Skel}(\mathscr{Y},\lambda)$

  * (ii)

$\iota(\mathrm{Skel}(\mathscr{X}_{2},\lambda_{2}))\cap\Psi(\text{int}\mathscr{X}_{1})=\varnothing$.

  * (iii)

$\Psi^{-1}(\iota(\text{int}\mathscr{X}_{2}))={\mathcal{O}}{\it
p}\,\mathscr{N}\setminus\partial\mathscr{X}_{1}\subset\mathscr{X}_{1}$ and
$(\Psi)^{*}(\iota)_{*}\lambda_{2}=\lambda_{1}^{P}$.

###### Proof.

We recall from the gluing construction that we attached
$\mathscr{N}\times(-\varepsilon,\varepsilon)$ to the top of a cylinder
$\mathscr{A}\times U_{3,\varepsilon}\subset Y$. Since a neighborhood
${\mathcal{O}}{\it p}\,N$ of $N$ in $X_{1}$ is symplectomorphic to $N\times
T_{\varepsilon}^{*}[0,\delta)$, we can map $\mathscr{N}\times
T^{*}_{\varepsilon}(0,\delta)$ symplectomorphically to $\mathscr{A}\times
U_{3,\varepsilon}\cup\mathscr{N}\times T^{*}_{\varepsilon}[0,\delta)$,
relative to $\mathscr{N}\times T^{*}_{\varepsilon}(\delta/2,\delta)$ say, so
that the symplectomorphism extends to the rest of $\text{int}\mathscr{X}_{1}$
as the identity. Moreover, the symplectomorphism can be obtained from a
reparametrization of the $t$ coordinate, hence the Liouville forms pull back
as desired. ∎

####  2.6.3. Splitting

The converse operation to gluing is splitting.

######  Definition 2.47.

Given a Wc-manifold $(X,\lambda)$, a regularly embedded hypersurface
$(H,\partial H)\subset(X,\partial X)$ with corners called splitting for $X$ if
it satisfies the following conditions:

  * (i)

$H$ divides $X$ into two parts, $X=X_{+}\cup X_{-}$ and $X_{-}\cap X_{+}=H$,
which are manifolds with corners.

  * (ii)

the Liouville vector field $Z$ of $X$ is tangent to $H$.

  * (iii)

there exists a hypersurface with corners $(S,\partial S)\subset(H,\partial H)$
tangent to $Z$.

  * (iv)

$(S,\lambda|_{S})$ is a Wc-manifold and there exists a retraction $\pi:H\to S$
such that $\lambda|_{H}=\pi^{*}(\lambda|_{S})$.

######  Remark 2.48.

That $H$ is regularly embedded implies in particular that ${\mathcal{O}}{\it
p}\,H=H\times(-\varepsilon,\varepsilon)$.

We will refer to $S$ as the soul of the splitting hypersurface $H$ and denote
it by $\mathrm{Soul}(H)$. Note that
$\mathrm{Skel}(S,\lambda|_{S})=\mathrm{Skel}(X,\lambda)\cap H$.

######  Example 2.49.

Let $M$ be a manifold with corners and $N$ a regularly embedded codimension
$1$ submanifold with corners, so that ${\mathcal{O}}{\it
p}\,N=N\times(-\varepsilon,\varepsilon)$. Assume that $N$ divides $M$ into two
parts: $M_{+}$ and $M_{-}$, so that $M_{+}\cup M_{-}=M$. Then $H=T^{*}M|_{N}$
is a splitting hypersurface. Given a decomposition of the tubular neighborhood
of $N$, ${\mathcal{O}}{\it p}\,(N)=N\times(-\varepsilon,\varepsilon)$, we have
$S:=T^{*}N\times 0\subset T^{*}M=T^{*}N\times T^{*}(-\varepsilon,\varepsilon)$
the soul of $H$.

The splitting hypersurface $S$ can be used to split $X$ into two Wc-manifolds
in two ways, horizontal and vertical. For the horizontal splitting we view
$X_{\pm}$ as Wc-manifolds with an additional component $H$ of
$\partial_{1}X_{\pm}$. The Wc-manifold $S$ serves as the nucleus of the
additional boundary component $H$. The vertical splitting consists of a
horizontal splitting followed by a nucleus-to-hypersurface conversion
operation, transforming the nucleus $S$ into a Wc-hypersurface.

####  2.6.4. Partial horizontal gluing

One more basic operation that we will need is the creation of corners and its
associated partial horizontal gluing. Let $X$ be a Wc-manifold, $P$ a boundary
face of $X$ and $N$ the nucleus of $P$. Let $\Sigma\subset N$ be a splitting
hypersurface with soul $T$. Then there is a deformation of the Wc-structure on
$X$ which creates an additional corner along $\Sigma$ with its nucleus $T$.

More precisely, take a collar neighborhood ${\mathcal{O}}{\it p}\,P=N\times
T^{*}\mathcal{I}$, so that $\lambda=\lambda|_{N}+udt$ and extend it to
$N\times T^{*}(-\varepsilon,\varepsilon)$ with the same Liouville form. Let
$\Sigma\times(-\delta,\delta)\subset N$ be a tubular neighborhood of $\Sigma$
in $N$ and consider a smooth function $\mu:(0,\delta)\to[0,\infty)$ satisfying
the following properties:

  * (i)

$\mu(x)\to\infty$ as $x\to 0$.

  * (ii)

$\mu(x)=0$ on $[\delta/2,\delta)$

  * (iii)

$\mu^{\prime}(x)<0$ on $(0,\delta/2)$.

Set $\eta(x)={\rm min}(\mu(x),\epsilon/2)$ and consider the subset
$\Omega\subset(\Sigma\times(-\delta,\delta))\times
T^{*}(-\varepsilon,\varepsilon)$ consisting of points $(\sigma,s,t,u)$ such
that $\eta(s)+t\geq 0$. Then the union of $\Omega$ and $X$ along $P$ is a new
Wc-manifold which has a new corner along a parallel copy of $\Sigma$.

Figure 2.11. A creation of additional corners on $\mathscr{T}^{*}M$ for $M$ a
closed manifold with corners is induced from a creation of additional corners
on $M$.

Next, consider two Wc-manifolds $X_{0},X_{1}$ with boundary faces
$P_{0},P_{1}$ and respective nuclei $N_{0},N_{1}$. Let $\Sigma$ be a dividing
hypersurface for $N_{0}$ with its soul $T$, which splits $N_{0}$ into Wc-
manifolds $N_{0}^{\prime}$ and $N_{0}^{\prime\prime}$, which share a common
face $\Sigma$ with the nucleus $T$. Suppose there is a Liouville isomorphism
$\phi:N_{1}\to N_{0}^{\prime}$.

######  Definition 2.50.

The partial horizontal gluing of $\mathscr{X}_{1}$ to $\mathscr{X}_{0}$ using
$\phi$ is the Wc-germ obtained from the following two operations:

  * (a)

Creating a corner along $\Sigma$, i.e. adding $\Sigma$ to $\partial_{2}X_{0}$,
and replacing the face $N_{0}$ with two faces
$N_{0}^{\prime},N_{0}^{\prime\prime}$;

  * (b)

The horizontal gluing $X_{0}$ and $X_{1}$ along $\phi$.

As before, the partial horizontal gluing is well defined up to Wc-homotopy.

###  2.7. Wc-buildings

####  2.7.1. Iterated gluing

We now introduce the iterated vertical gluing of Wc-manifolds, which will be a
convenient notion for the constructions in this paper. Consider Wc-germs
$(\mathscr{X}_{1},\lambda_{1}),\dots,(\mathscr{X}_{k},\lambda_{k})$. We
inductively define a $k$-level Wc-building, which we will denote by
$(\mathscr{X}_{k}\to\mathscr{X}_{k-1}\to\dots\to\mathscr{X}_{1})$, as the
following presentation of a Wc-germ.

######  Definition 2.51.

A $1$-level Wc-building is $(\mathscr{X}_{1},\lambda_{1})$ itself. Suppose
that the Wc-building

$(\mathscr{Y},\mu):=(\mathscr{X}_{k}\to\mathscr{X}_{k-1}\dots\to\mathscr{X}_{2})$

is already defined. Consider a nucleus
$(\mathscr{N},\mu^{\prime}:=\mu|_{\mathscr{N}})$ of one of the boundary faces
$\mathscr{P}$ of $(\mathscr{Y},\mu)$ and let $\phi$ be an embedding
$\mathscr{N}$ as a Wc-hypersurface in
$\mathscr{X}_{1}\setminus\mathrm{Skel}(\mathscr{X}_{1},\lambda_{1})$. The
building

$(\mathscr{X}_{k}\to\mathscr{X}_{k-1}\to\cdots\to\mathscr{X}_{2}\to\mathscr{X}_{1}):=(\mathscr{X}_{k}\to\mathscr{X}_{k-1}\to\cdots\to\mathscr{X}_{2})\to\mathscr{X}_{1}$

is by definition the result of a vertical gluing of $(\mathscr{Y},\mu)$ and
$(\mathscr{X}_{1},\lambda_{1})$ along $\phi$.

Figure 2.12. A Wc-building.

So a Wc-building is a Wc-germ $\mathscr{Y}$ equipped with a choice of
presentation of its Wc-deformation class as the iterated vertical gluing
$\mathscr{Y}=(\mathscr{X}_{1}\to\cdots\to\mathscr{X}_{k})$.

####  2.7.2. Distinguished cover of a Wc-building

A Wc-building come equipped with a symplectic cover by the Wc-manifolds it is
built out of. The relevant properties of this cover are summarized in the
following lemma. Recall that given a Wc-manifold $(X,\lambda)$ the Liouville
form $\lambda$ restricted to a neighborhood of a face $P$ has the form
$\lambda:=\lambda|_{N}+udt,$ where $N$ is the nucleus of $P$ and $t$ is the
defining coordinate for the face $P$. We recall the notation $\lambda^{P}$ for
the form $\lambda|_{N}-tdu=\lambda-d(ut)$ on ${\mathcal{O}}{\it p}\,N$.

######  Lemma 2.52.

Suppose $(\mathscr{X},\lambda)$ is presented as a $k$-level building
$(\mathscr{X}_{k}\to\mathscr{X}_{k-1}\to\dots\to\mathscr{X}_{1})$. Then there
exist symplectic embeddings
$\Phi_{j}:\text{int}\mathscr{X}_{j}\to\mathscr{X}$, $j=1,\dots,k$, such that

  * (i)

$\bigcup_{j=1}^{k}(\Phi_{j}(\mathrm{Skel}(\mathscr{X}_{j},\lambda_{j})\cap\text{int}\mathscr{X}_{j})=\mathrm{Skel}(\mathscr{X},\lambda)$;

  * (ii)

$\Phi_{j}(\text{int}{\mathscr{X}_{j}})\cap\Phi_{i}(\mathrm{Skel}(\mathscr{X}_{i},\lambda_{i})\cap\text{int}\mathscr{X}_{i})=\varnothing$
if $j>i$;

  * (iii)

If
$\Phi_{j}(\text{int}{\mathscr{X}_{j}})\cap\Phi_{i}(\text{int}{\mathscr{X}_{i}})\neq\varnothing$
for $j>i$ then there exists a nucleus $\mathscr{N}$ of a boundary face of
$\mathscr{X}_{j}$ such that
$\Phi_{j}^{-1}(\Phi_{i}(\text{int}\mathscr{X}_{i}))={\mathcal{O}}{\it
p}\,\mathscr{N}\cap\text{int}{\mathscr{X}_{j}}$ and
$\Phi_{j}^{*}(\Phi_{i})_{*}\lambda_{i}=\lambda_{j}^{P}.$

Figure 2.13. Presenting a Wc-building as the union of Wc-manifolds.

###### Proof.

We inductively construct the maps
$\Phi_{i,j}:\mathscr{X}_{i}\to(\mathscr{X}_{k}\to\cdots\to\mathscr{X}_{j})$,
$i\geq j$, as follows, at the end of the inductive process we will set
$\Phi_{i}=\Phi_{i,1}$ . The map $\Phi_{k,k}$ is just the inclusion on
$\mathscr{X}_{1}$ into $\mathscr{X}$. Suppose we have defined maps
$\Phi_{i,j}$ for $i\geq j$. Recall that
$\mathscr{Y}=(\mathscr{X}_{1}\to\cdots\to\mathscr{X}_{j})$ is vertically glued
to $\mathscr{X}_{j-1}$ by a Liouville isomorphism between the nucelus
$\mathscr{N}$ of a boundary face $\mathscr{P}$ of $\mathscr{Y}$ and a Wc-
hypersurface $\mathscr{A}$ in $\mathscr{X}_{j-1}$. Let
$\Psi:\mathscr{Y}\to(\mathscr{X}_{1}\to\cdots\to\mathscr{X}_{j}\to\mathscr{X}_{j-1})$
be the symplectic embedding from Lemma 2.46, set
$\Phi_{i,j-1}=\Psi\circ\Phi_{i,j}$ and set $\Phi_{j-1,j-1}$ to be the
restriction of the inclusion
$\mathscr{X}_{j-1}\hookrightarrow(X_{1}\to\cdots\to X_{j}\to X_{j-1})$ to the
interior of $\mathscr{X}_{j-1}$. The desired properties for the resulting
$\Phi_{i}$ follow from the properties stated in Lemma 2.46. ∎

######  Remark 2.53.

Note that on the overlap
$\Phi_{i_{1}}(\mathscr{X}_{i_{1}})\cap\cdots\cap\Phi_{i_{k}}(\mathscr{X}_{i_{k}})\subset\mathscr{X}$
the pushed-forward Liouville fields $Z_{i_{1}},\ldots,Z_{i_{k}}$ coming from
$\mathscr{X}_{i_{1}},\ldots,\mathscr{X}_{i_{k}}$ all commute with each other.

####  2.7.3. Cotangent buildings

Finally, we introduce a particular class of Wc-buildings which will be
particularly relevant for our purposes.

######  Definition 2.54.

A Wc-building $(\mathscr{X}_{k}\to\cdots\to\mathscr{X}_{1})$ for which each
Wc-germ consists of a cotangent block $\mathscr{T}^{*}M_{j}$ is called a
cotangent building.

We will later show that any finite type Weinstein manifold can be realized as
the underlying Weinstein manifold of a cotangent building.

###  2.8. Legendrian submanifolds

####  2.8.1. Legendrians in cotangent bundles

Consider first a cotangent bundle $T^{*}M$, where $M$ an $n$-dimensional
compact manifold with corners. Let $\Lambda$ be an $(n-1)$-dimensional
manifold with corners.

######  Definition 2.55.

An embedding $\Lambda\to T^{*}M\setminus M$ is called Legendrian if its
projection to $S^{*}M$ is a Legendrian embedding.

With every Legendrian $\Lambda\subset T^{*}M$ we associate its Lagrangian
Liouville cone $L:=\text{Cone}(\Lambda,pdq)$ formed by forward and backwards
trajectories of the Liouville field $Z$. Note that $L$ is embedded.

The splitting $P\times\mathcal{I}^{k}$ near a $k$-face $P$ of $M$ provide
canonical splittings

(1) $T^{*}{M}|_{{\mathcal{O}}{\it p}\,{P}}=T^{*}P\times T^{*}\mathcal{I}^{k}$

We will call a Legendrian embedding $\phi:L\to\mathscr{T}^{*}M\setminus M$
adapted to the block $T^{*}M$ if $f({\mathcal{O}}{\it p}\,\partial
L)\subset{\mathcal{O}}{\it p}\,\partial M$ and for each stratum
$P\subset\partial_{k}M$ we have $L\cap{\mathcal{O}}{\it
p}\,P=L_{k}\times\mathcal{I}^{k}$ for a Legendrian
$L_{k}\subset\mathscr{T}^{*}P$.

By conical Lagrangians $L\subset T^{*}M\setminus M$ we will always mean
Lagrangian Liouville cones over Legendrians.

Denote by $\pi:T^{*}M\to M$ the cotangent bundle projection.

######  Definition 2.56.

An adapted Legendrian $\Lambda$ is called regular if
$\pi|_{\Lambda}:\Lambda\to M$ is an immersion with transverse self-
intersections.

Figure 2.14. The Legendrian on the left is regular, while the one on the right
is not.

####  2.8.2. Legendrians in Wc-manifolds

Let $(X,\lambda)$ be a $2n$-dimensional Wc-manifold.

######  Definition 2.57.

An $(n-1)$-dimensional submanifold $\Lambda\subset
X\setminus\mathrm{Skel}(X,\lambda)$ is called Legendrian if it projects to
$\partial_{\infty}X$ as an embedded Legendrian submanifold of
$\partial_{\infty}X$. Its Liouville cone $L=\mathrm{Cone}(\Lambda,\lambda)$ is
a called a conical Lagrangian.

We say that a Legendrian $\Lambda$ is adapted to the Wc-structure of
$(X,\lambda)$ if $\partial_{k}\Lambda\subset\partial_{k}X$ and moreover in a
neighborhood of any corner $P\subset\partial_{k}X$ we have
$\Lambda=\Lambda_{N}\times\mathcal{I}^{k}\subset P\times
T^{*}\mathcal{I}^{k}$, where $\Lambda_{N}$ is a Legendrian in the nucleus $N$
of the corner $P$.

Figure 2.15. An adapted Legendrian $\Lambda$ in a Wc-manifold $W$.

######  Example 2.58.

Let $M$ be a manifold and $N\subset M$ a co-oriented hypersurface. Then any
submanifold $\Lambda\subset T^{*}M\setminus M$ which 1-1 projects to the
conormal lift of $N$ in $S^{*}M$ (and is never tangent to the Liouville flow)
is Legendrian. If $N$ is adapted for the corner structure of $M$, then one can
construct a Legendrian lift $\Lambda$ which is adapted for the Wc-structure of
$T^{*}M$.

We can also think of Legendrians in $T^{*}M$ as equivalence classes of
Legendrian embeddings into defining domains for $T^{*}M$, where the
equivalence is given by translation along the Liouville flow. This gives us
the notion of a Legendrian in a cotangent block $\mathscr{T}^{*}M$. Similarly,
we have the notion of a Legendrian in a Wc-germ $\mathscr{X}$. For a
Legendrian $\Lambda$ in a defining domain $(W,\lambda|_{W})$ for
$(\mathscr{X},\lambda)$, we only take its positive Liouville cone
$\text{Cone}^{+}(\Lambda,\lambda)$, which is contained in
$W\setminus\mathrm{Skel}(W,\lambda)$. One can think of the equivalence class
of all positive Liouville cones over all Legendrians equivalent to $\Lambda$
as the whole Liouville cone $\text{Cone}(\Lambda,\lambda)\subset
X\setminus\mathrm{Skel}(X,\lambda)$.

###  2.9. Gluing and splitting of Legendrians

####  2.9.1. Gluing Legendrians

######  Lemma 2.59.

Let $(\mathscr{X},\lambda)$ be a Wc-germ,
$\Lambda\subset\mathscr{X}\setminus(\mathrm{Skel}(\mathscr{X},\lambda)$ an
adapted Legendrian and $L$ the $\lambda$-Liouville cone over $\Lambda$. Let
$(\mathscr{X}^{N},\lambda^{N})$ be the result of a nucleus-to-hypersurface
conversion of the nucleus $N$ of one of the boundary faces of $X$. Then
$\Lambda\subset\mathscr{X}^{N}$ remains Legendrian and its
$\lambda^{N}$-Liouville cone coincides with $L$.

###### Proof.

In a neighborhood of the face $P$ of $X$ whose nucleus $N$ we convert to a
boundary face we have ${\mathcal{O}}{\it p}\,P=T^{*}N\times T^{*}\mathcal{I}$
and $\Lambda=\Lambda_{N}\times\mathcal{I}$. The conversion changes
$\lambda=\lambda_{N}+udt$ only by the differential of a function of $(u,t)$,
moreover the resulting Liouville field on $T^{*}\mathcal{I}$ is tangent to
$\mathcal{I}$ (and outwards pointing at the boundary). Hence $\Lambda$ remains
Legendrian and its Liouville cone is unchanged. ∎

######  Lemma 2.60.

Let $\mathscr{X}_{0},\mathscr{X}_{1}$ be two Wc-germs, $P_{0},P_{1}$ two of
their boundary faces and $N_{0},N_{1}$ the nuclei of $P_{0},P_{1}$. Let
$\Lambda_{0}\subset\mathscr{X}_{0}\setminus\mathrm{Skel}(\mathscr{X}_{0},\lambda_{0})$
and
$\Lambda_{1}\subset\mathscr{X}_{1}\setminus\mathrm{Skel}(\mathscr{X}_{1},\lambda_{1})$
be two adapted Legendrians. Denote $\Lambda^{\prime}_{i}:=\Lambda_{i}\cap
N_{i}$, $i=0,1$. Let $\phi:N_{0}\to N_{1}$ be a Liouville isomorphism such
that $\phi(\Lambda^{\prime}_{0})=\Lambda^{\prime}_{1}$. Then the horizontally
glued Wc-manifold
$\mathscr{X}_{01}=\mathscr{X}_{0}\mathop{\cup}_{\phi}\mathscr{X}_{1}$ contains
a glued Legendrian

$\Lambda_{01}=\Lambda_{0}\mathop{\cup}\limits_{\Lambda_{1}^{\prime}=\phi|_{\Lambda_{0}^{\prime}}}\Lambda_{1}.$

###### Proof.

This is obvious by definition of adapted. ∎

####  2.9.2. Splitting Legendrians

Let $X$ be a Wc-manifold and $P\subset X$ a splitting hypersurface with the
soul $S$. A Legendrian $\Lambda\subset X\subset\mathrm{Skel}(X,\lambda)$ is
called in in splitting position if $\Lambda$ is transverse to $P$ and
$\Lambda\cap P\subset S$.

######  Lemma 2.61.

Let $X$ be a Wc-manifold, $P\subset X$ a splitting hypersurface and consider a
Legendrian $\Lambda\subset X\setminus\mathrm{Skel}(X,\lambda)$ which is in
splitting position. Then under a horizontal splitting of $X$ we can arrange it
so that $\Lambda$ splits into Legendrians with corners $\Lambda_{\pm}\subset
X_{\pm}\setminus\mathrm{Skel}(X_{\pm},\lambda_{\pm})$.

###### Proof.

It suffices to take a neighborhood of $P$ in $\mathscr{X}$ so that $\Lambda$
is of product form with respect to the neighborhood, which is clearly
possible. ∎

Let us recall that given a contact manifold $(M,\xi)$ a hypersurface $N\subset
M$ is called convex if it admits a tranverse to it contact vector field $v$.
Consider a split a tubular neighborhood
$U_{\varepsilon}:=N\times(-\varepsilon,\varepsilon)$ of a contact hypersurface
$N$ such that the contact vector field $v$ coincides with
$\frac{\partial}{\partial t}$, where $t$ is the coordinate corresponding to
the second factor. The contact structure $\xi$ on $U_{\varepsilon}$ can be
then defined by a contact form $\alpha=fdt+\lambda$, where
$f:N\to{\mathbb{R}}$ is a smooth function and $\lambda$ is a 1-form on $N$.
The set $Q=\\{f=0\\}=\\{x\in N;\;v(x)\in\xi\\}$ is called the dividing set. It
is always a regular level set of the function $f$. Denote
$N_{+}:=\\{f>0\\},N_{-}=\\{f<0\\}$. We also denote

$\widehat{Q}:=Q\times(-\varepsilon,\varepsilon)=\bigcup\limits_{t\in(-\varepsilon,\varepsilon)}v^{t}(Q)$

and call it the dividing wall. The contact condition implies that
$\beta:=\lambda|_{\Sigma}$ is a contact form on $Q$ and that the form
$\lambda_{f}:=\frac{1}{f}\lambda$ is Liouville on $N\setminus Q=N_{+}\cup
N_{-}$. The corresponding Liouville fields $Z_{\pm}$ are complete of both
halves $N_{+}$ and $N_{-}$ and for any point $x\in
N_{\pm}\setminus\mathrm{Skel}(N_{\pm})$ there exists a limit
$\lim\limits_{s\to\infty}Z_{\pm}^{s}(x)\in Q$. We say that the pair $(N,v)$,
where $N$ is a convex hypersurface and $v$ a transverse vector field, is of
Weinstein type, if $(N_{\pm},\lambda_{f})$ is Weinstein.

######  Lemma 2.62.

Let $(M,\xi)$ be a contact manifold, $N\subset(M,\xi)$ a Weinstein type convex
hypersurface and $\Lambda\subset(M,\xi)$ a Legendrian transverse to $N$. Then
there exists a contact isotopy $\phi_{s}:M\to M$, $t\in[0,1]$ such that

  * (i)

$\phi_{s}$ fixes $Q$;

  * (ii)

$\phi_{s}$ leaves $N$ invariant;

  * (iii)

there exists $\varepsilon>0$ such that $\phi_{1}(\Lambda\cap
U_{\varepsilon})\subset Q_{\varepsilon}$;

  * (iv)

$\phi_{s}$ is supported in a neighborhood $U^{\prime}\Supset U_{\varepsilon}$.

###### Proof.

Note that a contact vector field $Y:=t\frac{\partial}{\partial t}+Z_{\pm}$ on
$N_{\pm}\times(-\varepsilon,\varepsilon)$ smoothly extends to
$N\times(-\varepsilon,\varepsilon)$ as equal to $t\frac{\partial}{\partial t}$
on $Q\times(-\varepsilon,\varepsilon)$. Let us choose $\varepsilon$
sufficiently small so that the coordinate $t|_{\Lambda\cap U_{\varepsilon}}$
has no critical points. Denote $\Gamma:=\Lambda\cap N$. A general position
argument allows us to $C^{\infty}$-perturb $\Lambda$ so that
$\Gamma\cap(\mathrm{Skel}(N_{+})\cup\mathrm{Skel}(N_{-}))=\varnothing$ and the
projection of $\pi:\Gamma\to Q$ along Liouville trajectories of $N_{\pm}$ is
an embedding. Note that $\lambda|_{\Gamma}=0$, and hence $\pi(\Gamma)\subset
Q$ is Legendrian for the contact structure $\ker\beta$ on $Q$. Let us
parameterize $\Lambda\cap U_{\varepsilon}$ by an embedding
$h_{0}:\Gamma\times(-\varepsilon,\varepsilon)\to\Lambda\cap U_{\varepsilon}$
such that $h_{0}(\Gamma\times t)\subset N\times t$. Define a Legendrian
isotopy $h_{s}:\Gamma\times(-\varepsilon,\varepsilon)\to\Lambda\cap
U_{\varepsilon}$, $s\in[0,\infty)$ by the formula

$h_{s}(x,t)=Y^{s}(h_{0}(x,te^{-s})).$

Let us observe that $h_{s}(x,t)=(g_{s,t}(x),t)$, with $g_{s,0}(x)$ flowing a
$\lambda$-isotropic submanifold $\Gamma$ to $\pi(\Gamma)$ along trajectories
of $Z_{\pm}$. The flow is fixed on $Q$. Let us analyze the isotopy in a
neighborhood of $Q$. The form $\lambda_{f}$ on ${\mathcal{O}}{\it p}\,Q$ can
be written as $\frac{1}{u}\beta+dt$ for the coordinate $u=f$ on
${\mathcal{O}}{\it p}\,Q$, and hence, $Y=t\frac{\partial}{\partial
t}-u\frac{\partial}{\partial u}$. The flow $Y^{t}$ is given by
$Y^{s}(x,u,t)=(x,e^{-s}u,e^{s}t)$. Assuming that $T$ is large enough so that
$h_{T}(\Gamma\times(-\varepsilon,\varepsilon))\subset{\mathcal{O}}{\it
p}\,Q\times(-\varepsilon,\varepsilon)$ we get
$h_{T+s}(x,u,t)=(x,e^{-s}x,t)\mathop{\to}(x,0,t)$ as $s\to\infty$. Hence by
reparameterizing the isotopy to the interval $[0,1)$ and extending it to the
end-point $1$ as $(x,0,1)$. we get a smooth Legendrian isotopy
$\phi_{s}:\Gamma\times(-\varepsilon,\varepsilon)$ such that
$\phi_{1}(\Gamma\times(-\varepsilon,\varepsilon))\subset
Q\times(-\varepsilon,\varepsilon)$. The isotopy $\phi_{s}$ can be realized by
an ambient contact isotopy which keeps $N$ invariant. By cutting off this
isotopy outside a smaller neighborhood of $N$ we get the required Legendrian
isotopy of $\Lambda$. ∎

######  Lemma 2.63.

Consider a Wc-manifold $(X,\lambda)$ with a dividing hypersurface $P\subset X$
and its soul $S$. Let $\Lambda_{0}\subset X\setminus\mathrm{Skel}(X,\lambda)$
be any Legendrian. Then there exists a Legendrian isotopy $\Lambda_{t}\subset
X\setminus\mathrm{Skel}(X,\lambda)$, $t\in[0,1]$, such that $\Lambda_{1}$ is
in a splitting position.

###### Proof.

We can realize $\partial_{\infty}X$ as a smooth boundary of a neighborhood of
$\mathrm{Skel}(X,\lambda)$ and view $\Lambda,\partial_{\infty}S$ and
$\partial_{\infty}P$ as submanifolds of $\partial_{\infty}X$. Then the
boundary $\partial_{\infty}P\subset\partial_{\infty}X$ is a convex
hypersurface in the contact manifold $(\partial_{\infty}X,\lambda)$, and its
dividing set $\partial_{\infty}S$ divides $\partial_{\infty}P$ into two copies
$S_{\pm}$ of $S$. Therefore the conclusion follows from the previous lemma. ∎

####  2.9.3. Legendrians in Wc-buildings

Let $(\mathscr{X},\lambda)$ be a Wc-germ with a structure of a $k$-block
building $\mathscr{X}_{k}\to\dots\to\mathscr{X}_{1}$. We say that a Legendrian
$\Lambda\subset\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$ is
compatible with the building structure if its Liouville cone $L$ is invariant
with respect to all Liouville fields $Z_{i}$, $i=1,\dots,k$, which we can
think of as living in $\mathscr{X}$ via Lemma 2.52.

######  Lemma 2.64.

Let $(X,\lambda)$ be presented as a Weinstein building
$\mathscr{X}_{k}\to\dots\to\mathscr{X}_{1}$. Let
$\Lambda\subset\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$ be a
Legendrian. Then there exists a Legendrian isotopy $\Lambda_{t}$ starting from
$\Lambda_{0}=\Lambda$ such that $\Lambda_{1}$ is compatible with the building,
i.e. its $Z$-Liouville cone is invariant with respect to all Liouville fields
$Z_{i}$ where they are defined.

###### Proof.

By definition we have filtration by Wc-manifolds
$\mathscr{X}\supset\mathscr{X}_{\geq 2},\dots,\supset\mathscr{X}_{\geq
k-1}\supset\mathscr{X}_{k}$, where $\mathscr{X}_{\geq
j}=\mathscr{X}_{k}\to\dots\to X_{j}$. Note that we equivalently view
$\mathscr{X}$ as a 2-block building $\mathscr{X}_{\geq 2}\to\mathscr{X}_{1}$.
By definition the building $\mathscr{X}_{\geq 2}\to\mathscr{X}_{1}$ can be
split into the blocks $\mathscr{X}_{1}$ and $\mathscr{X}_{\geq 2}$ along a
splitting hypersurface $H$ which can be defined as follows. Recall that
$\mathscr{X}_{\geq 2}\to\mathscr{X}_{1}$ is obtained from $\mathscr{X}_{1}$
and $\mathscr{X}_{\geq 2}$ by taking the nucleus $N$ of one of the boundary
faces of $\mathscr{X}_{\geq 2}$, realizing it as Wc-hypersurface in
$\mathscr{X}_{1}$, converting it to a nucleus of a boundary face and then
performing the horizontal gluing along the face $H$ of $\mathscr{X}_{1}$ and
the converted face of $\mathscr{X}_{\geq 2}$. The collar neighborhood
${\mathcal{O}}{\it p}\,H$ of the face $H$ in $\mathscr{X}_{1}$ is split as
${\mathcal{O}}{\it p}\,H=H\times\mathcal{I}$. The inverse splitting can be
done by using the $H\times\varepsilon$ as a splitting surface. Note that
making the Legendrian $\Lambda$ compatible with the building
$\mathscr{X}_{\geq 2}\to\mathscr{X}_{1}$ is equivalent to moving it to a
splitting position with respect to the splitting hypersurface
$H\times\varepsilon$. Hence, using Lemma 2.64 we can move $\Lambda$ by a
Legendrian isotopy to make it compatible with the building $\mathscr{X}_{\geq
2}\to\mathscr{X}_{1}$. The splitting defines a Legendrian $\Lambda_{\geq
2}\subset\mathscr{X}_{\geq 2}\setminus\mathrm{Skel}(\mathscr{X}_{\geq
2},\lambda)$. We continue by applying again 2.64 we move $\Lambda_{\geq 2}$ by
a Legendrian isotopy to make it compatible with the building
$\mathscr{X}_{\geq 2}\to\mathscr{X}_{2}$. Continuing the process we prove the
lemma. ∎

###  2.10. Weinstein handlebodies revisited

####  2.10.1. Rounded handles

Consider an $n$-ball $D^{n}\subset S^{n-k}\times D^{k}$ and denote

$K^{n}_{k}:=(S^{n-k}\times D^{k})\setminus{\rm Int\,}D^{n}.$

Thus, $\partial K_{k}^{n}$ is manifold with two boundary components
$\partial_{-}K_{k}^{n}=S^{n-k}\times\partial D^{k-1}$ and
$\partial_{+}K_{k}^{n}:=S^{n-1}$.

Consider a Weinstein manifold with boundary $G^{n}_{k}:=T^{*}K^{n}_{k}$. It
has two boundary faces $M_{-}$ and $M_{+}$ with nuclei
$P_{-}:=T^{*}(S^{n-k}\times\partial D^{k})$ and $P_{+}:=T^{*}S^{n-1}$.

######  Definition 2.65.

We call $G^{n}_{k}$ a rounded subcritical Weinstein handle.

######  Remark 2.66.

Our rounded handles $G^{n}_{k}$ have nothing to do with round handles of index
$k$.

######  Lemma 2.67.

Let $W^{2n}$ be Weinstein domain and $\phi:\partial D^{k}\times
D^{n-k}\to\partial W$ a Legendrian embedding, $k\leq n$. Let $W^{\prime}$ be
the Weinstein domain obtained by a subcritical index $k$ Weinstein handle
attachment along the isotropic sphere $\phi(\partial D^{k}\times 0)$ with the
framing given by the Legendrian extension. Suppose that the embedding $\phi$
is extended to a Legendrian embedding $\Phi:\partial D^{k}\times
S^{n-k}\to\partial W$. Let $\Sigma=T^{*}(\partial D^{k}\times S^{n-k})$ be the
Weinstein ribbon of this embedding. Then the Weinstein structure obtained by
the vertical gluing of the rounded handle $G_{k}^{n}$ along $\Sigma$ is
homotopic to the Weinstein structure on $W^{\prime}$.

###### Proof.

The vertical gluing consists of the conversion of $\Sigma$ to a boundary
nucleus and the horizontal gluing to $G_{k}^{n}$ along the new nucleus. The
conversion is achieved by taking the union of $W$ with a collar
$T^{*}(\partial D^{k}\times S^{n-k}\times[0,1])$ and deforming the Liouville
field of $W$ near $\Sigma$ so that the result is a Weinstein manifold with
boundary. Hence the vertical gluing is the union of $W$ with the cotangent
bundle of $\widehat{K}_{k}^{n}=(\partial D^{k}\times
S^{n-k}\times[0,1])\cup(K_{n}^{k}\times 1)$. Let $L:=D^{k}\times D^{n-k}$ be
attached to $\partial D^{k}\times S^{n-k}$ along $\partial D^{k}\times
D^{n-k}$, i.e. the image of $\phi$. Then $\widehat{K}_{k}^{n}$ is
diffeomorphic to $T_{k}^{n}:=(\partial D^{k}\times
S^{n-k}\times[0,1])\cup(L\times 1)$ after smoothing corners. The underlying
Weinstein structure of the vertically glued Weinstein manifold is given along
the boundary of $T^{*}\widehat{K}_{k}^{n}$ by the Weinstein normal form for
Morse-Bott boundary of index $0$. Before smoothing, on $T^{*}T_{k}^{n}$ this
corresponds to the Weinstein normal form for Morse-Bott boundary, also of
index 0. By a Weinstein homotopy we can deform the structure to cancel the
critical points on $T^{*}(\partial D^{k}\times S^{n-k}\times[0,1])$ along the
$[0,1]$ direction, similar to a parametric Smale cancellation, where we think
of $\partial D^{k}\times S^{n-k}$ as the parameter space. We are left with the
Weinstein normal form for Morse-Bott boundary and corners on $T^{*}L\simeq
T^{*}(D^{k}\times D^{n-k})$ which has index $1$ on the boundary face $\partial
D^{k}\times D^{n-k}$ and index $0$ on the boundary face $D^{k}\times\partial
D^{n-k}$. This Weinstein structure is evidently homotopic to the standard
Morse-Weinstein normal form of index $k$ on $T^{*}(D^{k}\times D^{n-k})$,
which is by definition the Weinstein surgery of index $k$. ∎

Figure 2.16. Attaching a subcritical isotropic handle is the same as attaching
a rounded handle.

####  2.10.2. Weinstein manifolds as cotangent buildings

Recall that a cotangent building is a Wc-building $W=(B_{k}\to\dots\to B_{1})$
consisting of cotangent blocks $B_{i}:=\mathscr{T}^{*}M_{i}$.

######  Proposition 2.68.

Any Weinstein manifold germ $(\mathscr{X},\lambda)$ admits the structure of a
cotangent building with blocks $B_{j}=\mathscr{T}^{*}M_{j}$, where each
$M_{j}$ is diffeomorphic to either a disc $D^{n}$ or one of the manifolds
$K^{n}_{j}$, $j=1,\dots,n-1$ with corners added to its boundary.

###### Proof.

Consider a handle decomposition $H_{k}\to H_{k-1}\to\dots\to H_{1}$ of a
Weinstein germ $(\mathscr{X},\lambda)$. If $k=0$ then we just replace the
first index $0$ handle by $T^{*}D^{n}$. Suppose that the Weinstein structure
on the handlebody $W_{\leq k}:=H_{k-1}\to\dots\to H_{1}$ is already compatible
with a structure of a cotangent building. If the index of the handle $H_{k}$
is equal $m$ for $0<m<n$ we use Lemma 2.67 to replace the handle $H_{k}$ by
the rounded handle $G^{n}_{m}=T^{*}K^{n}_{m}$. If $m=n$ we keep it as is equal
to $T^{*}D^{n}$. In both cases we denote by $T^{*}M_{k}$ the block we need to
attach with the nucleus of the attaching face $N$ equal to $T^{*}\partial
D^{n}$ in the latter case and $\partial_{-}G^{n}_{m}=T^{*}(S^{m}\times
S^{n-m})$ in the former one. For the attaching, $N$ is realized as a Weinstein
hypersurface in $W^{<k}\setminus\mathrm{Skel}(W^{<k})$. We denote by $\Lambda$
the core Legendrian of that hypersurface. Let $L$ be the $(Z_{\leq k})$-cone
of $\Lambda$. Using Lemma 2.64 we deform $\Lambda$ via a Legendrian isotopy to
make the cone $L$ invariant with respect to the contracting fields on the
blocks $B_{1},\dots,B_{k}$. Note that this automatically gives
$\widehat{M}_{k}:=L\cup M^{k}$ a new structure of a Wc-manifold with
additional corners. The neighborhood of $\widehat{M}_{k}$ then gets the
structure of a cotangent block $B_{k}=\mathscr{T}^{*}(\widehat{M}_{k})$ and
$W$ can be obtained from the vertical gluing of $B_{k}$ to $W_{\leq k}$. ∎

Figure 2.17. If $B_{2}\to B_{1}$ already has the structure of a cotangent
building and we wish to extend this structure to a third block $B_{3}$, we
need to make sure the Legendrian $\Lambda$ along whose ribbon the block
$B_{3}$ will be attached is adapted, i.e. its Liouville cone is invariant
under the Liouville flows of the blocks $B_{1}$ and $B_{2}$.

More generally, we have the following result:

######  Proposition 2.69.

Any Wc-manifold admits the structure of a cotangent building.

To prove this we will inductively apply the following lemma.

######  Lemma 2.70.

Given any Wc-pair $(\mathscr{X},\mathscr{A},\lambda)$, where $\mathscr{X}$ and
$\mathscr{A}$ are Wc-germs which admit the structure of a cotangent building,
the hypersurface-to-nucleus conversion
$(\mathscr{X}^{\mathscr{A}},\lambda^{\mathscr{A}})$ also admits the structure
of a cotangent building.

###### Proof.

Suppose that $(\mathscr{X},\lambda)$ is a Wc-manifold which admits the
structure of a cotangent building $B_{k}\to\cdots\to B_{1}$ and
$\mathscr{A}\subset\mathscr{X}\setminus\mathrm{Skel}(\mathscr{X},\lambda)$ is
a Wc-hypersurface which also admits the structure of a cotangent building
$B_{r}^{\prime}\to\cdots\to B_{1}^{\prime}$. Then it is straightforward to
verify that the hypersurface-to-nucleus conversion
$(\mathscr{X}^{\mathscr{A}},\lambda^{\mathscr{A}})$ inherits an induced
structure of a cotangent building
$(B_{r}^{\prime}\times\mathscr{T}^{*}[0,1])\to\cdots\to(B_{1}^{\prime}\times\mathscr{T}^{*}[0,1])\to
B_{k}\to\cdots\to B_{1}$. ∎

###### Proof of Proposition 2.69 .

Given any Wc-manifold $(\mathscr{X},\lambda)$, we may convert all of its
boundary nuclei to Wc-hypersurfaces to obtain its underlying Weinstein
manifold $\widehat{X}$, and $\mathscr{X}$ can be recovered from $\widehat{X}$
by converting back the Wc-hypersurfaces to face nuclei. By Proposition 2.68 we
know that $\widehat{X}$ admits the structure of a cotangent building, and we
may assume by induction on dimension that the Wc-hypersurfaces also admit the
structure of a cotangent building. Hence it follows from Lemma 2.70 that
$\mathscr{X}$ admits the structure of a cotangent building. ∎

##  3\. Symplectic neighborhoods of arboreal Lagrangians

In this section we introduce the notion of an arboreal space and prove that
any arboreal space has a unique symplectic thickening.

###  3.1. Arboreal spaces

####  3.1.1. Pre-Ac spaces

In [AGEN20a] we associated to each signed rooted tree
$\mathcal{T}=(T,\rho,\varepsilon)$, an arboreal Lagrangian
$L_{\mathcal{T}}\subset T^{*}{\mathbb{R}}^{n(\mathcal{T})}$ which is a union
of smooth pieces, $L_{\mathcal{T}}=\bigcup_{\alpha\in
v(\mathcal{T})}L_{\alpha}$.

For fixed $c\geq 0$, and $m\geq n(\mathcal{T})+c$, set $d=m-n(\mathcal{T})+c$,
and consider the product

$\textstyle{L(\mathcal{T},m,c):=L_{\mathcal{T}}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{c}_{\geq
0}}$

We also have the smooth pieces
$L(\mathcal{T},m,c)_{\alpha}:=L_{\alpha}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{c}_{\geq
0}$.

We equip $L(\mathcal{T},m,c)$ with the structure sheaf
$\mathcal{O}(\mathcal{T},m,c)$ of functions restricted from smooth functions
on
$T^{*}{\mathbb{R}}^{n(\mathcal{T})}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{c}_{\geq
0}$ by the inclusion $L(\mathcal{T},m,c)\hookrightarrow
T^{*}{\mathbb{R}}^{n(\mathcal{T})}\times{\mathbb{R}}^{d}\times{\mathbb{R}}^{c}_{\geq
0}$.

######  Definition 3.1.

An $m$-dimensional pre-arboreal space with corners (pre-Ac space)
$(X,\mathcal{O})$ is a locally ringed Hausdorff topological space locally
modeled on

$(L(\mathcal{T},m,c),\mathcal{O}(\mathcal{T},m,c))$

for varying signed rooted trees $\mathcal{T}$, and $c\leq m-n(\mathcal{T})$.

A smooth sheet $Y\to X$ of a pre-Ac space $(X,\mathcal{O})$ is a finite proper
map from a manifold with corners such that the map is an embedding on the
interior of $Y$, and locally the map identifies the local connected components
of $Y$ with smooth pieces

$\textstyle{L(\mathcal{T},m,c)_{\alpha}\subset L(\mathcal{T},m,c)}$

for varying $\alpha\in v(\mathcal{T})$.

In this paper we shall only need to consider the cases where $c=0$ or $c=1$.
When $c,d=0$, we write $(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})$ in place
of $(L(\mathcal{T},m,c),\mathcal{O}(\mathcal{T},m,c))$.

Figure 3.1. We can take $X$ to be the interval $[-1,1]$ with $1$ glued to $0$
as an $\mathcal{A}_{2}$-singularity. Then $[-1,1]\to X$ is a smooth sheet. We
will introduce the notion of Ac-building to avoid the unnecessary
complications of this kind of gluing.

######  Remark 3.2.

Given a pre-Ac space $(X,\mathcal{O})$, we can speak about a closed embedding
into a smooth manifold $(M,C^{\infty}_{M})$. These are embeddings with closed
image $X\to M$ with a surjective map of sheaves of local rings
$C^{\infty}_{M}|_{X}\to\mathcal{O}$.

######  Remark 3.3.

One can relax the definition of a pre-arboreal space by choosing the
alternative structure sheaf of continuous real-valued functions on
$L(\mathcal{T},m,c)$ whose restrictions to each smooth piece
$L(\mathcal{T},m,c)_{\alpha},$ for $\alpha\in v(\mathcal{T})$, are smooth. One
can show that every such “weak” pre-arboreal space is diffeomorphic to a
strong one, and any weak diffeomorphism between strong pre-arboreals is
isotopic to a strong one.

####  3.1.2. Ac spaces

In [AGEN20a] we proved that any germ of an arboreal Lagrangian is
symplectomorphic to exactly one of the canonical models
$L_{\mathcal{T}}\subset T^{*}{\mathbb{R}}^{n(\mathcal{T})}$, hence in
particular isomorphic as the germ of a locally ringed Hausdorff topological
space to the corresponding $(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})$.
Going the other way, if we are given a local model
$(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})$ it is not possible to
reconstruct the symplectic geometry of $L_{\mathcal{T}}\subset
T^{*}{\mathbb{R}}^{n(\mathcal{T})}$ unless we keep track of additional
structure. This is due to the fact that not every automorphism of
$(L_{\mathcal{T}},\mathcal{O}_{T})$ can be lifted to an automorphism of
$L_{\mathcal{T}}\subset T^{*}{\mathbb{R}}^{n(\mathcal{T})}$, i.e. a germ of a
symplectomorphism preserving $L_{\mathcal{T}}$. We now explain the additional
structure that we need to keep track of.

For a signed rooted tree $\mathcal{T}=(T,\rho,\varepsilon)$, recall if we
remove the root $\mathcal{T}\setminus\rho$ what remains is a disjoint union of
rooted trees with signs along their edges. Let us write
$\mathcal{T}^{\prime}=(T,\rho,\varepsilon^{\prime})$ for the signed rooted
tree where we negate all of the signs along the edges in any collection of
components of $\mathcal{T}\setminus\rho$. Then negating corresponding
coordinates in our local models provides an isomorphism of pre-Ac spaces

$(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})\simeq(L_{\mathcal{T}^{\prime}},\mathcal{O}_{\mathcal{T}^{\prime}})$

that does not extend to an ambient symplectomorphism unless
$\mathcal{T}=\mathcal{T}^{\prime}$. Additionally, in the case of the tree
$\mathcal{A}_{2}$ with a single unsigned edge, the nontrivial involution of
the pre-Ac space $(L_{\mathcal{A}_{2}},\mathcal{O}_{\mathcal{A}_{2}})$ that
reverses the orientation on the smooth base piece $L_{\rho}\subset
L_{\mathcal{A}_{2}}$ can only be extended to an ambient anti-
symplectomorphism. We will see in Lemma 3.10 below these are the only
ambiguities.

We introduce here additional structure to overcome the above ambiguities and
hence enable us to reconstruct $L_{\mathcal{T}}\subset
T^{*}{\mathbb{R}}^{n(\mathcal{T})}$ from
$(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})$. Let us first mention some
simple data we could add. Given $(L_{\mathcal{T}},\mathcal{O}_{\mathcal{T}})$,
consider the base smooth piece $L_{\rho}\subset L_{\mathcal{T}}$, and the
hypersurface $H=\bigcup_{\alpha\not=\rho}L_{\alpha}\cap L_{\rho}\subset
L_{\rho}$. Then a coorientation of $H$ provides an embedding
$L_{\mathcal{T}}\subset T^{*}{\mathbb{R}}^{n(\mathcal{T})}$ as the
corresponding positive conormal of $H$. In particular, there is the specific
choice of coorientation that gives the canonical embedding
$L_{\mathcal{T}}\subset T^{*}{\mathbb{R}}^{n(\mathcal{T})}$ of the local
model.

To go forward, we will introduce a version of the above coorientation data
that can be easily glued in a global setting. For a signed rooted tree
$\mathcal{T}=(T,\rho,\varepsilon)$, and vertex $\alpha\in v(\mathcal{T})$,
consider the orientation bundle

$\textstyle{\kappa(\mathcal{T},m,c)_{\alpha}:=\wedge^{m}TL(\mathcal{T},m,c)_{\alpha}}$

Let $K(\mathcal{T},m,c)$, denote the space of real line bundles $\kappa\to
L(\mathcal{T},m,c)$ equipped with identifications

$\textstyle{\kappa|_{L(\mathcal{T},m,c)_{\alpha}}\simeq\kappa(\mathcal{T},m,c)_{\alpha}}$$\textstyle{\alpha\in
v(\mathcal{T})}$