Weinstein Homotopies

TL;DR

This paper addresses a specific question in symplectic topology related to Weinstein homotopies, providing partial insights into the structure and properties of Weinstein manifolds.

Contribution

It offers a partial answer to a question posed by Eliashberg concerning Weinstein homotopies, advancing understanding in symplectic topology.

Findings

Partial characterization of Weinstein homotopies

Insights into the structure of Weinstein manifolds

Progress towards answering Eliashberg's question

Abstract

We give a partial answer to a question asked by Eliashberg in one of his recent papers.

Equations

$$\hat{X}=X\cup (\partial X\times [0,\infty ))$$

$$Core(\mathcal{L})={\cap }_{t>0}{Z}^{-t}(X)$$

$$X.\varphi \ge \delta (|X{|}^2+|d\varphi {|}^2)$$

$$Z_t=Z_0+Z_{th}=Z_0+t{Z}_h,t\in [0,1]$$

$${\Gamma }_i(t)=({\gamma }_i(t),t)$$

$$\tilde{Z}(x,t)={\Psi }_i(x,t){\partial }_t+(1-{\Psi }_i(x,t))Z(x,t)$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$j^r:DiffM\to Dif{f}_M{X}^{(r)},h\mapsto h_{\ast }$$

$$\{p\in M:dim(ker(df{)}_p)=i_1\}$$

$$r_F:J^k(M,\mathbb{R})\to J^k(F,\mathbb{R}):j^{k}f(x)\mapsto j^k(f_{\mid F})(x)$$

$${\Sigma }_{\mathcal{F}}^{(i_1,i_2,...,i_k)}:={\cup }_{\{Fleafof\mathcal{F}\}}{r}_F^{-1}{\Sigma }^{(i_1,i_2,...,i_k)}$$

$${\varphi }_t=(1-t){\varphi }_0+t{\varphi }_1$$

$$Z_0.d{\varphi }_t=(1-t)Z_0.d{\varphi }_0+t{Z}_0.d{\varphi }_1$$

$$\begin{array}[]{rcl}Z_{0}.d\phi_{t}+Z_{th}.d\phi_{1}&=&(1-t)Z_{0}.d\phi_{0}+t[Z_{0}.d\phi_{1}+Z_{h}.d\phi_{1}]\\ &\geq&(1-t)\delta(|Z_{0}|^{2}+|d\phi_{0}|^{2})+t\delta^{\prime}(|Z_{0}+Z_{h}|^{2}+|d\phi_{1}|^{2})\\ &\geq&min(\delta,\delta^{\prime})[(1-t)|Z_{0}|^{2}+t|Z_{0}+Z_{h}|^{2}+(1-t)|d\phi_{0}|^{2}+t|d\phi_{1}|^{2}]\end{array}$$

$$\begin{array}[]{rcl}\frac{(Z_{0}.d\phi_{t}+Z_{th}.d\phi_{1})}{|Z_{0}+tZ_{h}|^{2}+|d\phi_{t}+d\phi_{1}|^{2}}&\geq&min(\delta,\delta^{\prime})[\frac{(1-t)|Z_{0}|^{2}+t|Z_{0}+Z_{h}|^{2}}{(1-t)^{2}|Z_{0}|^{2}+t^{2}|Z_{0}+Z_{h}|^{2}+2t(1-t)|Z_{0}||Z_{0}+Z_{h}|+|d\phi_{t}+d\phi_{1}|^{2}}\\ &&+\frac{(1-t)|d\phi_{0}|^{2}+t|d\phi_{1}|^{2}}{(1-t)^{2}|Z_{0}|^{2}+t^{2}|Z_{0}+Z_{h}|^{2}+2t(1-t)|Z_{0}||Z_{0}+Z_{h}|+|d\phi_{t}+d\phi_{1}|^{2}}]\end{array}$$

$$(Z_0.d{\varphi }_t+Z_{th}.d{\varphi }_1)\ge \tilde{\delta }[|Z_0+t{Z}_h{|}^2+|d{\varphi }_t+d{\varphi }_1{|}^2]$$

$$({\alpha }_{t^{\prime }}+{\eta }_{t^{\prime }})(Z_0+t^{\prime }{Z}_h)=({\alpha }_{t^{\prime }}(Z_0)+{\eta }_{t^{\prime }}(Z_{t^{\prime }h}))\ge \tilde{\delta }[|Z_0+t^{\prime }{Z}_h{|}^2+|{\alpha }_{t^{\prime }}+{\eta }_{t^{\prime }}{|}^2]$$

$$(\tilde{\alpha }+\tilde{\eta })(\tilde{Z})>\tilde{\delta }[|\tilde{Z}{|}^2+|\tilde{\alpha }+\tilde{\eta }{|}^2]$$

$$(M\times {\mathbb{R}}^2{)}^{(1)}=(\tilde{X}\times \mathbb{R}{)}^{(1)}\times (\tilde{X}\times \mathbb{R}{)}^{(1)}$$

$$\mathcal{R}=\{(j^1{f}_0,j^1{f}_1)\in (M\times {\mathbb{R}}^2{)}^{(1)}:L(j^1{f}_i)(\tilde{Z})>\tilde{\delta }[|\tilde{Z}{|}^2+|L(j^1{f}_i){|}^2]fori=0,1,forsome\tilde{\delta }>0\}$$

$$f_t(x)=f_0(x,t)$$

$$Z_t({\beta }_t{g}_t+(1-{\beta }_t)f_t)=[{\beta }_t{Z}_t(g_t)+(1-{\beta }_t)Z_t(f_t)]+g_t{Z}_t({\beta }_t)-f_t{Z}_t({\beta }_t)$$

$$a+x_1^2+...+x_k^2-x_{k+1}^2+...+x_{2n}^2$$

$${\psi }_s^{\ast }(df)(\tilde{Z})\ge cdf(\tilde{Z})>c\tilde{\delta }[|\tilde{Z}{|}^2+|df{|}^2]$$

Full text

1. introduction

In this paper we discuss a problem mentioned by Eliashberg in his paper [3]. He has asked if two completed Weinstein structures $(\hat{X},\lambda_{0},\phi_{0})$ and $(\hat{X},\lambda_{1},\phi_{1})$ on the same symplectic manifold $(\hat{X},\omega)$ can be homotoped through Weinstein structures. We discuss this problem and prove a weak partial result by assuming some additional conditions.

We begin with the basic definitions. Let $(X,\omega)$ be a $2n$-dimensional symplectic domain with boundary with an exact symplectic form $\omega$ and primitive form $\lambda$ i.e, $d\lambda=\omega$.

A Liouville form is a choice of a primitive form $\lambda$ such that $\lambda_{\mid\partial X}$ is a contact form on $\partial X$ and the orientation on $\partial X$ by the form $\lambda\wedge d\lambda^{n-1}_{\mid\partial X}$ coincides with its orientation as the boundary of $(X,\omega)$. The $\omega$-dual vector field $Z$ of $\lambda$ is called the Liouville vector field. $Z$ satisfies $L_{Z}\omega=\omega$ and hence its flow is conformally symplectically expanding.

Every Liouville domain $X$ can be completed in the following way.Set

[TABLE]

and extend $\lambda$ on $\hat{X}$ as $e^{s}(\lambda_{\mid\partial X})$ on the attached end. Given a Liouville domain $\mathcal{L}=(X,\omega,\lambda)$ consider the compact set

[TABLE]

It is called Core or the Skeleton of the Liouville domain.

Let $\lambda_{0}$ and $\lambda_{1}$ be two Liouville forms on a fixed symplectic manifold $(X,\omega)$, moreover let $Z_{0}$ and $Z_{1}$ be the respective Liouville vector fields. Then obviously $\lambda_{1}=\lambda_{0}+dh$ for some $h:X\to\mathbb{R}$ and $Z_{1}=Z_{0}+Z_{h}$ where $Z_{h}$ is the hamiltonian vector field for $h$.

A Liouville cobordism $(W,\omega,Z)$ is a cobordism $W$ with an exact symplectic form $\omega$ such that the Liouville vector field $Z$ points inward along $\partial_{-}W$ and outward along $\partial_{+}W$.

Remark 1.1**.**

On the infinite end of $\hat{X}$, the Liouville vector field is given by $\partial_{s}$ irrespective of the choice of the Liouville form $\lambda$ on $X$.

Now we shall define the Weinstein structures. For this we need to recall few notions. A complete vector field is a vector field whose flow exists for all forward and backward time.

Let $\phi$ be a Morse function. A vector field $X$ is called gradient-like for $\phi$ if it satisfies

[TABLE]

for some $\delta>0$ and $|X|$ is with respect to some Riemannian metric and $|d\phi|$ is with respect to its dual metric.

Definition 1.2**.**

*([7])

A Weinstein manifold $(X,\omega,Z,\phi)$ is a symplectic manifold $(X,\omega)$ with a complete Liouville vector field $Z$ which is gradient like with respect to the exhausting Morse function $\phi$. A Weinstein cobordism $(W,\omega,Z,\phi)$ is a Liouville cobordism $(W,\omega,Z)$ whose Liouville vector field $Z$ is gradient-like with respect to a Morse function $\phi$ which is constant on the boundary. A Weinstein cobordism with $\partial_{-}W=\Phi\ (empty)$ is called a Weinstein domain.*

In [3] Eliashberg has asked the following question.

Problem: Let $(\hat{X},\lambda_{0},\phi_{0})$ and $(\hat{X},\lambda_{1},\phi_{1})$ be two completed Weinstein structures on the same symplectic manifold $(\hat{X},\omega)$. Are they homotopic as Weinstein structures ?

Obviously if $Z_{0}$ and $Z_{1}$ are the respective Liouville vector fields then $Z_{1}=Z_{0}+Z_{h}$ for $h$ satisfying $\lambda_{1}=\lambda_{0}+dh$ and hence

[TABLE]

gives a homotopy of Liouville vector fields. However $(X,\omega,Z_{t})$ may not be a Liouville homotopy. We refer the reader [7] for a precise definition of Liouville homotopy.

On Weinstein cobordisms a similar result has been proved in [7] although the Weinstein structures need to flexible. We refer the reader to [7] for a precise definition of flexible Weinstein structures.

Theorem 1.3**.**

*([7])

Let $(W,\omega_{0},\lambda_{0},\phi_{0})$ and $(W,\omega_{1},\lambda_{1},\phi_{1})$ be two flexible Weinstein structures on the same cobordism $W$ with dimension $2n>4$ which coincide on $Op(\partial_{-}W)$. Let $\eta_{t}$ be a homotopy rel $Op(\partial_{-}W)$ of non-degenerate two forms on $W$ connecting $\omega_{0}$ and $\omega_{1}$. Then there exists a homotopy of flexible Weinstein structures connecting the given ones.*

Let us now return to the question asked by Eliashberg. We assume that all the zeros of $Z_{t}$ are non-degenerate for all $t\in[0,1]$. So the zeros of $Z_{t}$ executes curves $\gamma_{i}(t),\ i=1,...,k$ (say). We consider $\tilde{X}=\hat{X}\times[0,1]$ and define vector field $Z(x,t)=Z_{t}(x)$ on $\tilde{X}$. The curves $\gamma_{i}$’s define curves $\Gamma_{i}$’s on $\tilde{X}$ as follows

[TABLE]

Consider two tubular neighborhoods of $\Gamma_{i}$ as $\Gamma_{i}\subset N^{\prime}_{i}\subset N^{\prime\prime}_{i}$. Let $\Psi_{i}:\tilde{X}\to\mathbb{R}$ be cutoff functions such that $\Psi_{i}=1\ on\ N^{\prime}_{i}$ and $\Psi_{i}=0\ outside\ N^{\prime\prime}_{i}$. Define $\tilde{Z}$ on $\tilde{X}$ by canonically removing the zeros of $Z$ as follows. Define $\tilde{Z}$ close to $\Gamma_{i}$ as

[TABLE]

Let $\tilde{\mathcal{F}}$ be the foliation defined by $\tilde{Z}$. Then $\tilde{\mathcal{F}}$ is a regular foliation.

Definition 1.4**.**

We call the homotopy of the Liouville vector field $Z_{t}$ uniformly open if it satisfies

1. (1)

*All the zeros of $Z_{t}$ are non-degenerate for all *$t\in[0,1]$ 2. (2)

The foliation $\tilde{\mathcal{F}}\times\tilde{\mathcal{F}}$ on $\tilde{X}\times\tilde{X}$ is uniformly open

Please see 3.1 bellow for the definition of Uniformly open foliation. Now we state the main theorem of this paper.

Theorem 1.5**.**

Let $(\hat{X},\lambda_{0},\phi_{0})$ and $(\hat{X},\lambda_{1},\phi_{1})$ be two completed Weinstein structures on the same symplectic manifold $(\hat{X},\omega)$ and let the homotopy of the Liouville vector field $Z_{t}$ is uniformly open (1.4), moreover $Z_{0}$ and $Z_{1}$ do not have a common zero. Then $(\hat{X},\lambda_{0},\phi_{0})$ and $(\hat{X},\lambda_{1},\phi_{1})$ can joined by a homotopy of Weinstein structures for which the underlying symplectic structure $\omega$ remains fixed.

Remark 1.6**.**

In 1.3 the underlying symplectic structure is not fixed.

2. $h$-Principle

This section does not have any new result, we just recall some facts from the theory of $h$-principle which we shall need in our proof.

Let $X\to M$ be any fiber bundle and let $X^{(r)}$ be the space of $r$-jets of jerms of sections of $X\to M$ and $j^{r}f:M\to X^{(r)}$ be the $r$-jet extension map of the section $f:M\to X$. If $X=M\times N$ then $X^{(r)}$ is denoted as $J^{r}(M,N)$. A section $F:M\to X^{(r)}$ is called holonomic if there exists a section $f:M\to X$ such that $F=j^{r}f$. In the following we use the notation $Op(A)$ to denote a small open neighborhood of $A\subset M$ which is unspecified.

Let $\mathcal{R}$ be a subset of $X^{(r)}$. Then $\mathcal{R}$ is called a differential relation of order $r$. $\mathcal{R}$ is said to satisfy $h$-principle if any section $F:M\to\mathcal{R}\subset X^{(r)}$ can be homotopped to a holonomic section $\tilde{F}:M\to\mathcal{R}\subset X^{(r)}$ through sections whose images are contained in $\mathcal{R}$. Put differently, if the space of sections of $X^{(r)}$ landing into $\mathcal{R}$ is denoted by $Sec\mathcal{R}$ and the space of holonomic sections of $X^{(r)}$ landing into $\mathcal{R}$ is denoted by $Hol\mathcal{R}$ then $\mathcal{R}$ satisfies $h$-principle if the inclusion map $Hol\mathcal{R}\hookrightarrow Sec\mathcal{R}$ induces a epimorphism at [math]-th homotopy group $\pi_{0}$. $\mathcal{R}$ satisfies parametric $h$-principle if $\pi_{k}(Sec\mathcal{R},Hol\mathcal{R})=0$ for all $k\geq 0$.

Let $p:X\to M$ be a fiber bundle and by $Diff_{M}X$ we denote the fiber preserving diffeomorphisms $h_{X}:X\to X$, i.e, $h_{X}\in Diff_{M}X$ if and only if there exists diffeomorphism $h_{M}:M\to M$ such that the following diagram commutes

[TABLE]

Let $\pi:Diff_{M}X\to DiffM$ be the projection $h_{X}\mapsto h_{M}$. We call a fiber bundle $p:X\to M$ natural if there exists a homomorphism $j:DiffM\to Diff_{M}X$ such that $\pi\circ j=id$. For a natural fiber bundle $p:X\to M$ the associated jet bundle $X^{(r)}\to M$ is also natural. The lift is given by

[TABLE]

where $h_{*}(s)=J^{r}_{j(h)\circ\bar{s}}(h(m))$, $s\in X^{(r)}$, $m=p^{r}(s)\in M$ and $\bar{s}$ is a local section near $m$ which represents the $r$-jet $s$. Observe $(h^{-1})_{*}=(h_{*})^{-1}$ and hence define $h^{*}=h_{*}^{-1}$.

For a natural fiber bundle $X\to M$, a differential relation $\mathcal{R}\subset X^{(r)}$ is called $DiffM$-invariant if the action $s\mapsto h_{*}s,\ h\in DiffM$, leaves $\mathcal{R}$ invariant.

Theorem 2.1**.**

*([4])

If a relation $\mathcal{R}$ is open and $DiffM$-invariant on an open manifold $M$ then it satisfies parametric $h$-principle.*

3. Bertelson’s Uniformly Open Foliations

In this section we recall some result from [1] and [2].

Definition 3.1**.**

*([1])

A foliated manifold $(M,\mathcal{F})$ is called uniformly open if there exists a function $f:M\to[0,\infty)$ such that*

1. (1)

$f$* is proper,* 2. (2)

$f$* has no leafwise local maxima,* 3. (3)

$f$* is $\mathcal{F}$-generic.*

Remark 3.2**.**

Observe that if $dim\mathcal{F}=1$ then $(M,\mathcal{F})$ can not be uniformly open as on a one dimensional manifold, a critical point will be either a local maximum or minimum.

So let us explain the notion $\mathcal{F}$-generic. In order to do so we need to define the singularity set $\Sigma^{(i_{1},i_{2},...,i_{k})}(f)$ for a map $f:M\to W$. $\Sigma^{i_{1}}(f)$ is the set

[TABLE]

It was proved by Thom [6] that for most maps $\Sigma^{i_{1}}(f)$ is a submanifold of $M$. So we can restrict $f$ to $\Sigma^{i_{1}}(f)$ and construct $\Sigma^{(i_{1},i_{2})}(f)$ and so on. In [6] it has been proved that there exists $\Sigma^{(i_{1},...,i_{k})}\subset J^{k}(M,W)$ such that $(j^{k}f)^{-1}\Sigma^{(i_{1},...,i_{k})}=\Sigma^{(i_{1},...,i_{k})}(f)$.

Let us set $W=\mathbb{R}$ as this is the only situation we need. Let $(M,\mathcal{F})$ be a foliated manifold with a leaf $F$. Define the restriction map

[TABLE]

Define foliated analogue of the singularity set as

[TABLE]

Definition 3.3**.**

([1]) A smooth real valued function $f:M\to\mathbb{R}$ is called $\mathcal{F}$-generic if the first jet $j^{1}f\pitchfork\Sigma^{(n)}_{\mathcal{F}}$ and the second jet $j^{2}f\pitchfork\Sigma^{(i_{1},i_{2})}_{\mathcal{F}}$ for all $(i_{1},i_{2})$.

Definition 3.4**.**

*([1])

An isotopy of the manifold $M$ is a family $\psi_{t},\ t\in[0,1]$ of diffeomorphisms of $M$ such that the map $\psi:M\times[0,1]\to M\ :(x,t)\mapsto\psi_{t}(x)$ is smooth and $\psi_{0}=id_{M}$. Consider a foliation $\mathcal{F}$ on $M$. A foliated isotopy of $(M,\mathcal{F})$ is an isotopy $\psi_{t}$ of $M$ that preserves the foliation $\mathcal{F}$, that is, $(\psi_{t})_{*}(T\mathcal{F})=T\mathcal{F}$ for all $t\in[0,1]$. A relation $\mathcal{R}$ is called foliated invariant on $(M,\mathcal{F})$ if the action by foliated isotopies leaves $\mathcal{R}$ invariant.*

Theorem 3.5**.**

*([1])

On an uniformly open foliated manifold, any open, foliated invariant differential relation satisfies the parametric $h$-principle.*

In [2] Bertelson has contructed counter examples that without the uniformly open condition 3.5 fails.

4. Main Theorem

In this section we prove 1.5. Let us first set some notations. First of all we have the Liouville vector fields $Z_{0}$ and $Z_{1}=Z_{0}+Z_{h}$ and let $Z_{t}=Z_{0}+tZ_{1}$ be the homotopy of uniformly open Liouville vector field. So we have

1. (1)

$Z_{0}.\phi_{0}\geq\delta(|Z_{0}|^{2}+|d\phi_{0}|^{2})$ 2. (2)

$Z_{0}.\phi_{1}+Z_{h}.\phi_{1}\geq\delta^{\prime}(|Z_{0}+Z_{h}|^{2}+|d\phi_{1}|^{2})$

With equality occurs in the above inequalities at the zeros of $Z_{0}$ and $Z_{0}+Z_{h}$. Define

[TABLE]

Then observe that

[TABLE]

Now consider

[TABLE]

So we get

[TABLE]

Recall that (according to 1.1) on the infinite end of $\hat{X}$ the Liouville vector fields $Z_{0}$ and $Z_{0}+Z_{h}$ are equal to $\partial_{s}$. Moreover since $Z_{0}$ and $Z_{1}$ do not have a common zero $|d\phi_{t}+d\phi_{1}|>0$ and since $\Gamma_{i}$’s are compact $|d\phi_{t}+d\phi_{1}|$ is bounded bellow.

So the right hand side of the above inequality is bounded and hence the right hand side is equal to $\tilde{\delta}$ (say). So we get

[TABLE]

Without loss of generality we assume that $Z_{0}\pitchfork Z_{h}$ otherwise we can use a relative version of $h$-principle.

We replace $t$ by a new parameter $t^{\prime}=f(t)$ where $f:[0,1]\to[0,1]$ is such that $f=0$ on $[0,\epsilon]$ and $f=1$ on $[1-\epsilon,1]$. We can replace the parameter in the above inequality.

Define one forms $\alpha_{t^{\prime}}$ and $\eta_{t^{\prime}}$ as follows. First $\alpha_{t^{\prime}}$, $\alpha_{t^{\prime}}(Z_{0})=d\phi_{t^{\prime}}(Z_{0})$, $Z_{h}\in ker(\alpha_{t^{\prime}}),\ for\ t\in[\epsilon,1-\epsilon],\ t^{\prime}=f(t)$ and $\alpha_{0}=d\phi_{0},\ \alpha_{1}=d\phi_{1}$. Similarly $\eta_{t^{\prime}}(Z_{t^{\prime}h})=\eta_{t^{\prime}}(t^{\prime}Z_{h})=d\phi_{1}(t^{\prime}Z_{h})$, $Z_{0}\in ker\eta_{t^{\prime}}\ for\ t\in[\epsilon,1-\epsilon],\ t^{\prime}=f(t)$ and $\eta_{0}=d\phi_{1}=\eta_{1}$. So we have

[TABLE]

Now extending on $\tilde{X}$ and regularizing $Z_{t^{\prime}}=Z_{0}+t^{\prime}Z_{h}$ we get $\tilde{Z}$ as in 1. We extend $\alpha_{t^{\prime}}$ and $\eta_{t^{\prime}}$ to $\tilde{X}$ as $\alpha^{\prime}(x,t^{\prime})=\alpha_{t^{\prime}}(x)\ and\ \eta^{\prime}(x,t^{\prime})=\eta_{t^{\prime}}(x)$. Adjusting $\alpha^{\prime}$ and $\eta^{\prime}$ near $\Gamma_{i}$’s (1) to $\tilde{\alpha}$ and $\tilde{\eta}$ so that

[TABLE]

Now we come to the $h$-principle part. Consider $M=\tilde{X}\times\tilde{X}$ and the trivial bundle $P\times P:M\times\mathbb{R}^{2}=\tilde{X}\times\mathbb{R}\times\tilde{X}\times\mathbb{R}\to M$ where $P:\tilde{X}\times\mathbb{R}\to\tilde{X}$ is the projection on the first factor. Observe that

[TABLE]

Note that this does not happen in case of higher order jet extensions as there will be mixed derivatives.

Observe that the section space $\Gamma(\tilde{X}\times\mathbb{R})=C^{\infty}(\tilde{X},\mathbb{R})$. There is a natural affine fibration $L:(\tilde{X}\times\mathbb{R})^{(1)}\to T^{*}(\tilde{X}\times\mathbb{R})$ given by $L(j^{1}f(x))=df_{x}$ where $f\in C^{\infty}(\tilde{X},\mathbb{R})$. Define the relation $\mathcal{R}\subset(M\times\mathbb{R}^{2})^{(1)}$ as

[TABLE]

Obviously $(\tilde{\alpha}+\tilde{\eta},\tilde{\alpha}+\tilde{\eta})\in Sec\mathcal{R}$. Next we shall show that $\mathcal{R}$ is open and invariant under $\tilde{\mathcal{F}}\times\tilde{\mathcal{F}}$-foliated isotopy. This will conclude the proof of 1.5 in view of 3.5. Only thing one needs to do is the following. Let $(f_{0},f_{1})$ is a resulting solution. Choose either $f_{0}$ or $f_{1}$ say $f_{0}$. Then define $f_{t}$ as

[TABLE]

Now we have to re-introduce the singularities. Let $g_{t}$ be a family of Morse functions defined near $\Gamma_{i}$ with index same as the index of $Z$ along $\Gamma_{i}$. Let $\beta$ be a cutoff function such that $\beta=1$ on a tubular neighborhood $\mathcal{N}_{i}\supset N^{\prime\prime}_{i}$ and $\beta=0$ outside $\mathcal{N}^{\prime}_{i}\supset\mathcal{N}_{i}$.Let $\beta_{t}(x)=\beta(x,t)$. Observe

[TABLE]

Observe that $Z_{t}(\beta_{t})$ has compact support and $g_{t}$ is of the form

[TABLE]

So if we take $a$ large enough then $(g_{t}Z_{t}(\beta_{t})-f_{t}Z_{t}(\beta_{t}))>0$ and obviously compactly supported. So we get the desired result.

Lemma 4.1**.**

The relation $\mathcal{R}$ is open and invariant under the action of $\tilde{\mathcal{F}}\times\tilde{\mathcal{F}}$-foliated isotopies.

Proof.

Openness of $\mathcal{R}$ follows directly from the definition of $\mathcal{R}$.

For second part we see $\psi_{s}^{*}(df)(\tilde{Z})=df(d\psi_{s}(\tilde{Z}))\geq cdf(\tilde{Z})$, where $c$ is a positive real number. Positive as $\psi_{0}=id$ and $M\times\mathbb{R}^{2}$ is connected. So

[TABLE]

∎