On smooth families of exact forms

TL;DR

This paper presents an algorithm and explicit formulas for smoothly computing primitive families of exact forms on manifolds, using a cech-de Rham complex approach.

Contribution

It introduces a novel algorithm and explicit formulas for constructing smooth primitive families of exact forms in parameterized settings.

Findings

Algorithm for smooth primitive family computation

Explicit formulas for primitives

Application of cech-de Rham complex in the method

Abstract

For a smooth family of exact forms on a smooth manifold, an algorithm for computing a primitive family smoothly dependent on parameters is given. The algorithm is presented in the context of a diagram chasing argument in the \v{C}ech-de Rham complex. In addition, explicit formulas for such primitive family are presented.

Equations

$${\eta }_x={\int }_0^1\frac{1}{t}{\lambda }_t^{\ast }{\iota }_I{\omega }_{x}dt,$$

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$$\delta :\Pi {\Omega }^q(U_{{\alpha }_0\dots {\alpha }_p})\to \Pi {\Omega }^q(U_{{\alpha }_0\dots {\alpha }_{p+1}})$$

$$(\delta \omega {)}_{{\alpha }_0\dots {\alpha }_{p+1}}=\sum _{i=0}^{p+1}(-1{)}^i{\omega }_{{\alpha }_0\dots {\hat{\alpha }}_i\dots {\alpha }_{p+1}}{\mid }_{U_{{\alpha }_0\dots {\alpha }_i\dots {\alpha }_{p+1}}}.$$

$$\{d_0{\tau }_x^{\alpha }{\}}_{\alpha \in J}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$({\delta }_1\{d_0{\tau }_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_0{\alpha }_1}=d_0{\tau }_x^{{\alpha }_1}{|}_{U_{{\alpha }_0{\alpha }_1}}-d_0{\tau }_x^{{\alpha }_0}{|}_{U_{{\alpha }_0{\alpha }_1}}=w_x{|}_{U_{{\alpha }_0{\alpha }_1}}-w_x{|}_{U_{{\alpha }_0{\alpha }_1}}=0.$$

$$({\delta }_0\{{\tau }_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_0{\alpha }_1}$$

$$=F^{{\alpha }_1}(-,x){|}_{U_{{\alpha }_0{\alpha }_1}}-F^{{\alpha }_0}(-,x){|}_{U_{{\alpha }_0{\alpha }_1}}\in {\mathcal{C}}^{\infty }(U_{{\alpha }_0{\alpha }_1}).$$

$$C_x^{{\alpha }_0{\alpha }_1}=({\delta }_0\{{\tau }_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_0{\alpha }_1}.$$

$${\tilde{C}}_x^{\alpha }={\tau }_x^{\alpha }-{\eta }_x{|}_{U_{\alpha }}.$$

$${\tau }_x^{\alpha }={\eta }_x{|}_{U_{\alpha }}+{\tilde{C}}_x^{\alpha },$$

$${\tau }_x^{{\alpha }_1}{|}_{U_{{\alpha }_0{\alpha }_1}}-{\tau }_x^{{\alpha }_0}{|}_{U_{{\alpha }_0{\alpha }_1}}$$

$$=({\delta }_0\{{\eta }_x{|}_{U_{\alpha }}+{\tilde{C}}_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_0{\alpha }_1}$$

$$=({\delta }_0\{{\tilde{C}}_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_0{\alpha }_1}$$

$$={\tilde{C}}_x^{{\alpha }_1}-{\tilde{C}}_x^{{\alpha }_0}.$$

$$C_x^{{\alpha }_0{\alpha }_1}={\tilde{C}}_x^{{\alpha }_1}-{\tilde{C}}_x^{{\alpha }_0}$$

$$C_x^{{\alpha }_0{\alpha }_1}+C_x^{{\alpha }_1{\alpha }_2}=({\tilde{C}}_x^{{\alpha }_1}-{\tilde{C}}_x^{{\alpha }_0})+({\tilde{C}}_x^{{\alpha }_2}-{\tilde{C}}_x^{{\alpha }_1})={\tilde{C}}_x^{{\alpha }_2}-{\tilde{C}}_x^{{\alpha }_0}=C_x^{{\alpha }_0{\alpha }_2}.$$

$${\delta }_0(\{{\tilde{\tau }}_x^{\alpha }{\}}_{\alpha \in J}{)}_{{\alpha }_i{\alpha }_j}$$

$$=({\tau }_x^{{\alpha }_j}{|}_{U_{{\alpha }_i{\alpha }_j}}-C_x^{{\alpha }_0{\alpha }_j})-({\tau }_x^{{\alpha }_i}{|}_{U_{{\alpha }_i{\alpha }_j}}-C_x^{{\alpha }_0{\alpha }_i})$$

$$=({\tau }_x^{{\alpha }_j}{|}_{U_{{\alpha }_i{\alpha }_j}}-{\tau }_x^{{\alpha }_i}{|}_{U_{{\alpha }_i{\alpha }_j}})-(C_x^{{\alpha }_0{\alpha }_j}-C_x^{{\alpha }_0{\alpha }_i})$$

$$=({\tilde{C}}_x^{{\alpha }_j}-{\tilde{C}}_x^{{\alpha }_i})-(C_x^{{\alpha }_0{\alpha }_j}-C_x^{{\alpha }_0{\alpha }_i})\text{mbox}by(\text{ref}B).$$

$$=({\tilde{C}}_x^{{\alpha }_j}-{\tilde{C}}_x^{{\alpha }_i})-(({\tilde{C}}_x^{{\alpha }_j}-{\tilde{C}}_x^{{\alpha }_0})-({\tilde{C}}_x^{{\alpha }_i}-{\tilde{C}}_x^{{\alpha }_0}))$$

$$=0.$$

$${\tau }_x{|}_{U_{\alpha }}={\tau }_x^{\alpha }-A_x^{\alpha }+A_x^{{\alpha }_0}$$

$$A_x^{\alpha }={\tau }_x^{\alpha }-{\eta }_x{|}_{U_{\alpha }}$$

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$${\delta }_k(\{{\tau }_x^{\alpha }-{\eta }_x{|}_{U_{\alpha }}\})={\delta }_k(\{{\tau }_x^{\alpha }\})-{\delta }_k(\{{\eta }_x{|}_{U_{\alpha }}\})={\delta }_k(\{{\tau }_x^{\alpha }\})-\{0\}={\delta }_k(\{{\tau }_x^{\alpha }\}).$$

$$\{0\}$$

$$d_{k-1}(\{a_x^{\alpha }{\}}_{\alpha \in J})=\{{\tau }_x^{\alpha }-{\eta }_x{|}_{U_{\alpha }}{\}}_{\alpha \in J}.$$

$${\delta }_k{d}_{k-1}(\{a_x^{\alpha }{\}}_{\alpha \in J})={\delta }_k\{{\tau }_x^{\alpha }{\}}_{\alpha \in J}.$$

$$d_{k-1}{\delta }_{k-1}(\{a_x^{\alpha }{\}}_{\alpha \in J})={\delta }_k\{{\tau }_x^{\alpha }{\}}_{\alpha \in J}.$$

$$g_x^{{\alpha }_0{\alpha }_1}={\delta }_{k-1}(\{a_x^{\alpha }{\}}_{\alpha \in J})$$

$$a_x^{\alpha }={\int }_0^1\frac{1}{t}{\lambda }_t^{\ast }{\iota }_I({\tau }_x^{\alpha }-{\eta }_x{|}_{U_{\alpha }})dt={\int }_0^1\frac{1}{t}{\lambda }_t^{\ast }{\iota }_I({\tau }_x^{\alpha })-{\int }_0^1\frac{1}{t}{\lambda }_t^{\ast }{\iota }_I({\eta }_x{|}_{U_{\alpha }}).$$

$$g_x^{{\alpha }_0{\alpha }_1}$$

Full text

1. Introduction

In various problems of differential and symplectic geometry, the following fact plays a fundamental role [4].

Theorem 1**.**

Let $\{\omega_{x}\}_{x\in\mathcal{P}}$ be a family of exact $(k+1)$-forms on a smooth manifold $M$, which smoothly depends on a parameter $x$ that takes values in an open set $\mathcal{P}\subseteq\mathbb{R}^{n}$. There then exists a smooth family of $k$-forms $\{\tau_{x}\}_{x\in\mathcal{P}}$ on $M$ such that $d\tau_{x}=\omega_{x}$ for each $x\in\mathcal{P}$.

Even when this fact is commonly used, it is difficult to find a complete proof of Theorem 1 in the existent bibliography. Furthermore, in cases when a complete proof is given, the algorithm for find such primitives or collating formulas remain more or less obscure because of the number of steps involved. For example, for a general case, a sketch of the proof of this theorem is given in [2]. A more detailed proof for a manifold $M$ of a finite type is given in [6]. In addition, in the case when $M$ is compact, Theorem 1 can be proven by using the Hodge theory [3], [4].

In the present article, for an arbitrary manifold $M$ (without boundary) and by using elementary tools, we give a complete constructive proof of Theorem 1 consisting of a shorter algoritm than the proofs shown in [2] and [6]. In addition, we get explicit collating formulas for the primitive family. Even when only elementary arguments are necesary for our proof, we decided to present it in the context of a diagram chasing argument in the Čech-de Rham complex [1] because in this way the proof is easier to follow.

Finally, in Section 4 we give an application of Theorem 1 proving Theorem 2, which states the following: Let $(M,\mathcal{F})$ be a foliated manifold such that the foliation $\mathcal{F}$ is given by a locally trivial submersion $p:M\to B$. If $H^{r}_{DR}(p^{-1}(b))=0$ for all $b\in B$ and an integer $r$, then $H^{r}(M,\phi^{0}(\mathcal{F}))=0$.

Now we will proceed to describe the steps of the algorithm, which is divided into two cases. In the first case, we construct a family of exact $1$-forms and in the second one a family of exact $k$-forms such that $k\geq 2$ is constructed.

Before the algorithm, we prove the assertion of Theorem 1 in the local case by applying the standard argument of the Poincaré lemma for forms that are smoothly dependent on parameters. In this way, associated with a countable good cover $\{U_{\alpha}\}_{\alpha\in J}$ of $M$, for each fixed $x$ we get a collection $\{\tau^{\alpha}_{x}\}_{\alpha\in J}$ of $k$-forms (i.e, an element in $\Pi\Omega^{k}(U_{\alpha})$) such that for each $\alpha$ the family $\{\tau_{x}^{\alpha}\}_{x\in\mathcal{P}}$ depends smoothly on the parameter $x$.

Now we will describe the case for constructing a family of exact [math]-forms given a family of $1$-forms as in Theorem 1. The algorithm starts (Step 1) with the Čech-de Rham complex associated with the good covering of $M$, and by using a commutative square in such a complex, we construct a family of constants $C^{\alpha_{0}\alpha_{1}}_{x}$ that smoothly depends on the parameter $x$, and this family is defined for each $\alpha_{0}$, $\alpha_{1}$ in $J$ such that the intersection of $U_{\alpha_{0}}$ and $U_{\alpha_{1}}$ is not empty. In Step 2, by using the hypothesis of our main theorem, we define auxiliary constants $\widetilde{C}^{\alpha}_{x}$, which do not necessarily smoothly depend on the parameter $x$, but this allows us to smoothly extend the definition of the constants $C^{\alpha_{0}\alpha_{1}}_{x}$ for all $\alpha_{0}$, $\alpha_{1}$ in $J$. In the Step 3, we define a new smooth family of [math]-forms $\{\widetilde{\tau}_{x}^{\alpha}\}_{\alpha\in J}$ on the good cover $\{U_{\alpha}\}_{\alpha\in J}$ by fixing any element $\alpha_{0}$ in $J$ and adding the constant $-C_{x}^{\alpha_{0}\alpha}$ to the [math]-form $\tau_{x}^{\alpha}$ for each $\alpha$.

In the last step (Step 4), by using a diagram chasing argument in the Čech-de Rham complex, we prove that the family $\{\widetilde{\tau}_{x}^{\alpha}\}$ defined in Step 3 determines a family of 0-forms stated in Theorem 1.

Let us describe now the steps for constructing a smooth family of $k$-forms $\{\tau_{x}\}_{x\in\mathcal{P}}$ for a given family $\{\omega_{x}\}_{x\in\mathcal{P}}$ of exact $(k+1)$-forms. In the first step (Step 1), by using a diagram chasing argument in the Čech-de Rham complex associated with the good cover of $M$ for each fixed $x$ in $\mathcal{P}$, we define an element $\{a_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$. Such an element depends on the element $\{\tau_{x}^{\alpha}\}_{\alpha\in J}$ and depends on the primitives of the collection of $(k+1)$-forms $\{w_{x}\}$ on $M$. When $\alpha$ in $J$ is fixed, the family $\{a_{x}^{\alpha}\}_{x\in\mathcal{P}}$ is not necessarily smooth on $x$. In Step 2, however, for each fixed $x$, we define an element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0}\alpha_{1}\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha_{0}\alpha_{1}})$ that depends on the element $\{a_{x}^{\alpha}\}_{\alpha\in J}$ and that for each fixed pair $\alpha_{0}$, $\alpha_{1}$ the family $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{x\in\mathcal{P}}$ smoothly depends on $x$.

In Step 3, we prove that the element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0}\alpha_{1}\in J}$ is a cocycle in $\Pi\Omega^{k-1}(U_{\alpha_{0}\alpha_{1}})$. By then using the exactness of the Čech-de Rham complex, we define an element $\{G_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$, which depends on such a cocycle and we prove that for each fixed $\alpha$ in $J$ the family $\{G_{x}^{\alpha}\}_{\alpha\in J}$ smoothly depends on $x$. In Step 4, for each $x$ we define a new element $\{\widetilde{\tau}_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k}(U_{\alpha})$ by adding $\tau_{x}^{\alpha}$ to $dG_{x}^{\alpha}$ for each $\alpha$ in $J$. Thus for each $\alpha$ the family $\{\widetilde{\tau}_{x}^{\alpha}\}_{x\in\mathcal{P}}$ smoothly depends on $x$. We then prove by a diagram chasing argument in the Čech-de Rham complex that the family $\{\widetilde{\tau}_{x}^{\alpha}\}_{\alpha\in J}$ determines the family of $k$-forms requested in Theorem 1.

2. The local case

Let $M$ be a smooth manifold of dimension $d$. We recall that an open cover $\{U_{\alpha}\}$ of $M$ is called a good cover if all non-empty finite intersections $U_{\alpha_{0}}\cap\dots\cap U_{\alpha_{p}}$ are diffeomorphic to $\mathbb{R}^{d}$. Notice that all smooth manifold $M$ has a good cover [1]. Actually, every smooth manifold $M$ that is second countable has a countable good cover. Indeed, by [7] $M$ has a countable open cover $\{V_{r}\}$ consisting of sets with compact closures. Let $\{U_{s}\}$ be an open good cover for $M$. For each $r$, $\bar{V_{r}}$ is covered by a finite number of elements in $\{U_{s}\}$. By taking the union of such finite collections when $r$ varies in all the elements of $\{V_{r}\}$, we get a countable coordinate good cover $\{U_{\alpha}\}$ for the manifold $M$.

We will denote by $U_{\alpha_{0}\cdots\alpha_{p}}$ the intersection $U_{\alpha_{0}}\cap\cdots\cap U_{\alpha_{p}}$ of the open sets $U_{\alpha_{0}},\ldots,U_{\alpha_{p}}$ in the good cover. Furthermore, we denote by $\Omega^{q}(M)$ the $\mathbb{R}$-vector space of $q$-forms on $M$ and define $\Pi\Omega(U_{\alpha_{0},\dots,\alpha_{p}})$ as the product of the collection $\{\Omega^{q}(U_{\alpha_{0},\dots,\alpha_{p}})\}_{(\alpha_{0},\dots,\alpha_{p})\in J^{p}}$.

On the other hand, let $\{\omega_{x}\}_{x\in\mathcal{P}}$ be a family of closed $(k+1)$-forms ($k\geq 0$) over the $d$-manifold $M$, which smoothly depends on the parameter $x$ in $\mathcal{P}\subseteq\mathbb{R}^{n}$. Thus, by the Poincaré lemma (as in [5]) for each $x\in\mathcal{P}$ and any chart $(U,u:U\to\mathbb{R}^{d})$ on $M$ with $U$ an open contractible subset of $M$, there exists a primitive $\eta_{x}$ of $\omega_{x}$ given by the formula

[TABLE]

where $\lambda:\mathbb{R}\times U\rightarrow U$ is defined by $\lambda(t,m)=\lambda_{t}(m)=u^{-1}(tu(m))$, $I$ is the vector field $I(m)=m$ and $\iota_{I}$ is the insertion operator (cf. [5]). It is clear that the $k$-forms $\eta_{x}$ in (1) smoothly depend on $x$. Therefore, we have a smooth family $\{\eta_{x}\}_{x\in\mathcal{P}}$ of primitives.

Let $\{U_{\alpha}\}_{\alpha\in J}$ be a fixed good cover of the manifold $M$ where the family of indexes $J$ is a countable ordered set. It follows that for each $\alpha$ in $J$ there exists a collection $\{\tau^{\alpha}_{x}\}_{x\in\mathcal{P}}$ with the following properties: If $k=0$, then $\tau^{\alpha}_{x}\in\Omega^{0}(U_{\alpha})$ for each $x$ in $\mathcal{P}$, $d\tau_{x}^{\alpha}=\omega_{x}|_{U_{\alpha}}$ for each $x$ in $\mathcal{P}$ and the family ${\{\tau^{\alpha}_{x}\}}_{x\in\mathcal{P}}$ is smoothly dependent on $x$ in $\mathcal{P}$, i.e. the function $F^{\alpha}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ defined by $F^{\alpha}(m,x)=\tau^{\alpha}_{x}(m)$ is $\mathcal{C}^{\infty}$; otherwise ($k\geq 1$), $\tau^{\alpha}_{x}\in\Omega^{k}(U_{\alpha})$ for each $x$ in $\mathcal{P}$, $d\tau_{x}^{\alpha}=\omega_{x}|_{U_{\alpha}}$ for each $x$ in $\mathcal{P}$, and the family ${\{\tau^{\alpha}_{x}\}}_{x\in\mathcal{P}}$ is smoothly dependent on the parameter $x$ in $\mathcal{P}$. This means that for each $\alpha$ and for each collection of smooth vector fields $X_{1},\dots,X_{k}$ on $U_{\alpha}$ the function $F^{\alpha}_{X_{1},\dots,X_{k}}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ defined by $F^{\alpha}_{X_{1},\dots,X_{k}}(m,x)=(\tau^{\alpha}_{x}(X_{1},\dots,X_{k}))(m)$ is $\mathcal{C}^{\infty}$.

We will use such functions $F^{\alpha}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ and $F^{\alpha}_{X_{1},\dots,X_{k}}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$, in our algorithm in the next section.

3. An algorithm to construct a smooth primitive family

In this section, we present an algorithm to construct a smooth family of $k$-forms $\{\tau_{x}\}_{x\in\mathcal{P}}$ on a manifold $M$ such that $d\tau_{x}=\omega_{x}$ for each $x\in\mathcal{P}$ for a given family $\{\omega_{x}\}_{x\in\mathcal{P}}$ of exact $(k+1)$-forms, which smoothly depends on a parameter $x$ in an open set $\mathcal{P}\subseteq\mathbb{R}^{n}$.

The algorithm is divided into two cases. The first one is when $\{\omega_{x}\}_{x\in\mathcal{P}}$ is a family of exact $1$-forms, and the second case is for exact $k$-forms such that $k\geq 2$.

3.1. Constructing a smooth family of exact 1-forms

Let $M$ be a smooth $d$-manifold, and let $\{U_{\alpha}\}_{\alpha\in J}$ be a fixed good cover of $M$, with $J$ a countable ordered set. If $\{\omega_{x}\}_{x\in P}$ is a smooth family of exact $1$-forms on $M$, then there exists a family $\{\eta_{x}\}_{x\in\mathcal{P}}$ of [math]-forms in $\Omega^{0}(M)$ with the property that $d\eta_{x}=\omega_{x}$ for each $x$ in $\mathcal{P}$.

For each $\alpha$ in $J$, let $\{\tau^{\alpha}_{x}\}_{x\in\mathcal{P}}$ be the collection of 0-forms defined in Section 2, and let $F^{\alpha}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ be the $\mathcal{C}^{\infty}$-map defined by $F^{\alpha}(m,x)=\tau^{\alpha}_{x}(m)$.

Step 1. Defining a smooth family of constants

Let us associate the Čech-de Rham complex with the good cover $\{U_{\alpha}\}$ of the manifold $M$; see [1]. We have the following commutative diagram with exact rows:

[TABLE]

In every vertical arrow, $d$ denotes the differential operator induced by the usual differential operator of the de Rham complex. Moreover, for every $q\in\mathbb{Z}$ let us introduce the coboundary operator

[TABLE]

defined as follows: If $\omega$ belongs to $\Pi\Omega^{q}(U_{\alpha_{0}\dots\alpha_{p}})$, then $\omega$ has components $\omega_{\alpha_{0}\dots\alpha_{p}}$ in $\Omega^{q}(U_{\alpha_{0}\dots\alpha_{p}})$, and the components of $\delta\omega$ are given by

[TABLE]

Now the family $\{\tau^{\alpha}_{x}\}_{\alpha\in J}$ is an element in the $\mathbb{R}$-vector space $\Pi\Omega^{0}(U_{\alpha_{0}})$ for each fixed $x\in\mathcal{P}$. Thus, by evaluating the element $\{\tau^{\alpha}_{x}\}_{\alpha\in J}$ in the diagram (2), we get

[TABLE]

where $\delta_{1}\{d_{0}\tau_{x}^{\alpha}\}_{\alpha\in J}=\{0\}$ because

[TABLE]

We remember that the function $F^{\alpha}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ given by $F^{\alpha}(m,x)=\tau_{x}^{\alpha}(m)$ is $\mathcal{C}^{\infty}$ for all $\alpha$ in $J$. For each fixed $x$ in $\mathcal{P}$ the function $F^{\alpha}(-,x):U_{\alpha}\rightarrow\mathbb{R}$ then belongs to $\mathcal{C}^{\infty}(U_{\alpha})$ for all $\alpha$ in $J$. Therefore, for each fixed $x$ in $\mathcal{P}$ the function $(\delta_{0}\{\tau^{\alpha}_{x}\}_{\alpha\in J})_{\alpha_{0}\alpha_{1}}$ is in $\mathcal{C}^{\infty}(U_{\alpha_{0}\alpha_{1}})$ for all $U_{\alpha_{0}\alpha_{1}}\neq 0$ because

[TABLE]

Since the above diagram commutes, however, we get $d_{0}((\delta_{0}\{\tau^{\alpha}_{x}\}_{\alpha\in J})_{\alpha_{0}\alpha_{1}})=0$ on the open connected $U_{\alpha_{0}\alpha_{1}}$. On the other hand, for every smooth function from a connected manifold into another manifold such that the differential is zero, the map must be constant in the domain; see [7]. Therefore, we have for each $x\in\mathcal{P}$ that the function $(\delta_{0}\{\tau^{\alpha}_{x}\}_{\alpha\in J})_{\alpha_{0}\alpha_{1}}$ is a constant function that only depends on the indexes $\alpha_{0}$, $\alpha_{1}$ and the $x$ chosen.

For each $x$, we define a special family of constants $\{C^{\alpha_{0}\alpha}_{x}\}$, which smoothly depends on $x$.

We define for each $\alpha_{0},\alpha_{1}\in J$ such that $U_{\alpha_{0}\alpha_{1}}\neq 0$ and for each fixed $x\in\mathcal{P}$ the constants in $\mathbb{R}$ given by

[TABLE]

We note that when we let the parameter $x\in\mathcal{P}$ vary we have that $C_{x}^{\alpha_{0}\alpha_{1}}\in C^{\infty}(\mathcal{P})$ because the function $C_{(-)}^{\alpha_{0}\alpha_{1}}=F^{\alpha_{1}}(*,-)-F^{\alpha_{0}}(*,-):\mathcal{P}\rightarrow\mathbb{R}$ is $\mathcal{C}^{\infty}$, and it does not depend on the point $*$ chosen in $U_{\alpha_{o}\alpha_{1}}$.

Step 2. Extending the definition of the smooth constants

Since $d_{0}\{\eta_{x}|_{U_{\alpha}}\}_{\alpha\in J}=d_{0}\{\tau_{x}^{\alpha}\}_{\alpha\in J}$ implies that $d_{0}(\tau_{x}^{\alpha}-\eta_{x}|_{U_{\alpha}})=0$ for all $\alpha$ in $J$, then $\tau_{x}^{\alpha}-\eta_{x}|_{U_{\alpha}}$ is a constant for each fixed $x\in\mathcal{P}$. We define for each $\alpha\in J$ and for each fixed $x\in P$ the constant

[TABLE]

Note that $\widetilde{C}_{x}^{\alpha}$ is not necessarily a smooth function when the parameter $x$ in $\mathcal{P}$ varies. Thus, we get

[TABLE]

and then for all $\alpha_{0}$, $\alpha_{1}$ in $J$ such that $U_{\alpha_{0}\alpha_{1}}\neq 0$, we have

[TABLE]

From (3) and (4) we get that

[TABLE]

for all $\alpha_{0}$, $\alpha_{1}$ in $J$ such that $U_{\alpha_{0}\alpha_{1}}\neq 0$.

Now we will generalize the definition of the constants $C_{x}^{\alpha_{0}\alpha_{1}}$ for any $\alpha_{0}$, $\alpha_{1}$ in $J$, even if $U_{\alpha_{0}\alpha_{1}}$ is empty. We define

[TABLE]

for all $\alpha_{0}$, $\alpha_{1}$ in $J$.

Next, we will check that $C_{x}^{\alpha_{0}\alpha}\in\mathcal{C}^{\infty}(\mathcal{P})$ for all $\alpha_{0}$, $\alpha$ in $J$. We only need check on the case when $U_{\alpha_{0}\alpha}$ is empty. We choose any point $m_{0}$ in $U_{\alpha_{0}}$. We take a path $\gamma:[0,1]\rightarrow M$ from $m_{0}$ to any fixed point $m_{1}$ in $U_{\alpha}$. Since $[0,1]$ is compact, we can choose a finite collection $U_{\alpha_{0}},U_{\alpha_{1}},\ldots,U_{\alpha_{k-1}},U_{\alpha_{k}}=U_{\alpha}$ such that $U_{\alpha_{i}}\cap U_{\alpha_{i+1}}\neq\emptyset$ for each $i$. We will prove the case $k=2$, and the general case follows by induction.

We suppose that $U_{\alpha_{0}\alpha_{2}}$ is empty. Since $U_{\alpha_{0}\alpha_{1}}\neq\emptyset$ and $U_{\alpha_{1}\alpha_{2}}\neq\emptyset$, we have that $C_{x}^{\alpha_{0}\alpha_{1}}$, $C_{x}^{\alpha_{1}\alpha_{2}}$ are in $\mathcal{C}^{\infty}(\mathcal{P})$, so is their sum. On the other hand,

[TABLE]

Therefore, $C_{x}^{\alpha_{0}\alpha_{2}}$ is in $\mathcal{C}^{\infty}(\mathcal{P})$.

Thus, we have that $C_{x}^{\alpha_{i}\alpha_{j}}$ is defined for each pair $\alpha_{i}$, $\alpha_{j}$ in $J$ as $\widetilde{C}_{x}^{\alpha_{j}}-\widetilde{C}_{x}^{\alpha_{i}}$ and $C_{x}^{\alpha_{i}\alpha_{j}}\in\mathcal{C}^{\infty}(\mathcal{P})$.

Step 3. Redefining the family of 0-forms

By adding the family of constants $\{C^{\alpha_{0}\alpha}_{x}\}$ that we got in the last step to the family of [math]-forms $\{\tau^{\alpha}_{x}\}$ which we got in Step 1, we define a new family $\{\widetilde{\tau}^{\alpha}_{x}\}$, which has the same properties as the family $\{\tau^{\alpha}_{x}\}$ except that now the $\delta$ operator applied to this new family is zero. We proceed as follows.

We choose any fixed $\alpha_{0}\in J$, and we define $\widetilde{\tau}^{\alpha}_{x}=\tau^{\alpha}_{x}-C^{\alpha_{0}\alpha}_{x}$ for each $\alpha$ in $J$ and for each $x$ in $\mathcal{P}$. Thus, we have $\widetilde{\tau}^{\alpha}_{x}\in\Omega^{0}(U_{\alpha})$ and $d_{0}\widetilde{\tau}^{\alpha}_{x}=d_{0}(\tau^{\alpha}_{x}-C^{\alpha_{0}\alpha}_{x})=d_{0}(\tau^{\alpha}_{x})-d_{0}(C^{\alpha_{0}\alpha}_{x})=d_{0}(\tau^{\alpha}_{x})-0=\omega_{x}|_{U\alpha}$ for each fixed $x$ in $\mathcal{P}$ and for all $\alpha$ in $J$. In addition, the new family $\{\widetilde{\tau}^{\alpha}_{x}\}_{x\in\mathcal{P}}$ is smoothly depending on $x$ in $\mathcal{P}$ like the family $\{\tau^{\alpha}_{x}\}_{x\in\mathcal{P}}$. Now, however, the family $\{\widetilde{\tau}^{\alpha}_{x}\}_{x\in\mathcal{P}}$ has the additional property that $\delta_{0}\{\widetilde{\tau}^{\alpha}_{x}\}_{x\in\mathcal{P}}=\{0\}$ because

[TABLE]

Step 4. Diagram chasing in the Čech-de Rham complex

Since the rows in diagram (2) are exact, then the collection $\{\widetilde{\tau}^{\alpha}_{x}\}_{x\in\mathcal{P}}$ determines a collection of global [math]-forms $\{\tau_{x}\}_{\ x\in\mathcal{P}}$ in $\Omega^{0}(M)$. Such a collection is smoothly dependent on the parameter $x$ in $\mathcal{P}$ and $d_{0}\tau_{x}=\omega_{x}$ for each $x$ in $\mathcal{P}$.

In summary for case $k=1$, we define the smooth family of primitives $\tau_{x}$ by

[TABLE]

for each $\alpha\in J$, where the constants $A_{x}^{\alpha}$ are defined by

[TABLE]

for each $\alpha\in J$ and $\alpha_{0}$ is any fixed element in $J$.

3.2. Constructing a smooth family of exact k-forms

For any $k\geq 2$, we now construct a smooth family of $k$-forms $\{\tau_{x}\}_{x\in\mathcal{P}}$ on a manifold $M$ such that $d\tau_{x}=\omega_{x}$ for each $x\in\mathcal{P}$, for a given family $\{\omega_{x}\}_{x\in\mathcal{P}}$ of exact $(k+1)$-forms, which smoothly depends on a parameter $x$ in an open set $\mathcal{P}\subseteq\mathbb{R}^{n}$.

For each $\alpha$ in $J$, let $\{\tau^{\alpha}_{x}\}_{x\in\mathcal{P}}$ be the collection of $k$-forms defined in Section 2, and for each collection of smooth vector fields $X_{1},\dots,X_{k}$ on $U_{\alpha}$, let $F^{\alpha}_{X_{1},\dots,X_{k}}:U_{\alpha}\times\mathcal{P}\rightarrow\mathbb{R}$ be the $\mathcal{C}^{\infty}$-function defined by $F^{\alpha}_{X_{1},\dots,X_{k}}(m,x)=(\tau^{\alpha}_{x}(X_{1},\dots,X_{k}))(m)$.

Step 1. Defining an element $\{a_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$ via a diagram chasing in the Čech-de Rham complex

Associated with the good cover $\{U_{\alpha}\}_{\alpha\in J}$ of the smooth manifold $M$, we have a Čech-de Rham complex [1], which implies that we have the following commutative diagram with exact horizontal and vertical arrows:

[TABLE]

By hypothesis, there exists a family $\{\eta_{x}\}_{x\in\mathcal{P}}$ of $k$-forms on $M$ with the property $d\eta_{x}=\omega_{x}$ for each $x$ in $\mathcal{P}$. For each $x\in\mathcal{P}$ the element $\{\tau_{x}^{\alpha}-\eta_{x}|_{U_{\alpha}}\}_{\alpha\in J}$ then belongs to $\Pi\Omega^{k}(U_{\alpha})$. This element is not necessarily smoothly dependent on $x$. In addition, since $d\eta_{x}=\omega_{x}$ we get that $\delta_{k}(\{\eta_{x}|_{U_{\alpha}}\}_{\alpha\in J})=\{0\}$ for each $x$ in $\mathcal{P}$. Thus,

[TABLE]

Therefore, by evaluating the element $\{\tau_{x}^{\alpha}-\eta_{x}|_{U_{\alpha}}\}_{\alpha\in J}$ in diagram (5), we get the following commutative diagram

[TABLE]

Indeed, since vertical arrows are exact in diagram (5) (by the Poincaré lemma), there exists an element $\{a_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$, which is not necessarily smoothly dependent on $x$ such that for each $x\in\mathcal{P}$ we have that

[TABLE]

Thus,

[TABLE]

In addition, since square (III) in diagram (5) commutes, we get

[TABLE]

Step 2. Defining a smooth element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0}\alpha_{1}\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha_{0}\alpha_{1}})$ depending on $\{a_{x}^{\alpha}\}_{\alpha\in J}$

We define an element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0},\alpha_{1}\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha_{0}\alpha_{1}})$ by setting

[TABLE]

for each $\alpha_{0}$, $\alpha_{1}$ in $J$. We will proceed to prove that for each fixed $\alpha_{0}$, $\alpha_{1}$ in $J$ the family $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{x\in\mathcal{P}}$ varies smoothly with respect to the parameter $x$ in $\mathcal{P}$.

Explicitly, we have for each $\alpha$ in $J$ that

[TABLE]

Therefore,

[TABLE]

where the last expression smoothly depends on $x$. Therefore, we obtain that $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{x\in\mathcal{P}}$ varies smoothly with respect to the parameter $x$ in $\mathcal{P}$ for each fixed $\alpha_{0}$, $\alpha_{1}$ in $J$.

Step 3. Defining a smooth element $\{G_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$ on terms of the element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0}\alpha_{1}\in J}$

Since

[TABLE]

then the element $\{g_{x}^{\alpha_{0}\alpha_{1}}\}_{\alpha_{0},\alpha_{1}\in J}$ is a cocycle on $\Pi\Omega^{k-1}(U_{\alpha_{0}\alpha_{1}})$. Furthermore, because of the exactness of the Čech-de Rham complex, there exists an element $\{G_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k-1}(U_{\alpha})$ such that

[TABLE]

as is shown on the following diagram:

[TABLE]

Moreover, for each $\alpha$ in $J$ the family $\{G_{x}^{\alpha}\}_{\alpha\in J}$ can be chosen such that for each $\alpha$ in $J$ the family $\{G_{x}^{\alpha}\}_{x\in\mathcal{P}}$ varies smoothly on $x$. This is a consequence of this family being defined by using a partition of the unity and by using sums of elements $g_{x}^{\beta\alpha}$ in $\Omega^{k-1}(U_{\beta\alpha})$. In fact for each $\alpha$ in $J$ we can define

[TABLE]

where $\{\rho_{\beta}:M\rightarrow\mathbb{R}\}_{\beta\in J}$ is a partition of unity subordinate to the good cover $\{U_{\alpha}\}_{\alpha\in J}$ of $M$ as in the proof of exactness of the generalized Mayer-Vietoris sequence in [1]. In order to check explicitly that the family $\{G_{x}^{\alpha}\}_{x\in\mathcal{P}}$ varies smoothly on $x$, let $X_{1}$, $\dots$, $X_{k-1}$ be smooth vector fields on $U_{\alpha}$. Thus, the function

[TABLE]

defined as

[TABLE]

is such that

[TABLE]

where the functions

[TABLE]

defined for each $\beta$ as

[TABLE]

are $\mathcal{C}^{\infty}$ because for each $\beta$ the family $\{g_{x}^{\beta\alpha}\}_{x\in\mathcal{P}}$ is smooth on $x$. Since the sum (7) is finite, we get that the function $F^{\alpha}_{X_{1},\dots,X_{k-1}}$ is $\mathcal{C}^{\infty}$, so the family $\{G_{x}^{\alpha}\}_{x\in\mathcal{P}}$ varies smoothly on $x$.

Step 4. Diagram chasing in the Čech-de Rham complex

From diagram (6) and since the square $(\mathrm{III})$ in diagram (5) commutes, we have that

[TABLE]

where $\{d_{k-1}G_{x}^{\alpha}\}_{\alpha\in J}$ is in $\Pi\Omega^{k}(U_{\alpha})$.

By using the element $\{\tau_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k}(U_{\alpha})$, we define a new element $\{\widetilde{\tau}_{x}^{\alpha}\}_{\alpha\in J}$ in $\Pi\Omega^{k}(U_{\alpha})$ by setting

[TABLE]

for each $\alpha$ in $J$. Therefore, the element $\{\widetilde{\tau}_{x}^{\alpha}\}_{\alpha\in J}$ is such that

[TABLE]

and

[TABLE]

Therefore, we obtain the following commutative diagram:

[TABLE]

Since the bottom row is exact, there then exists a family $\{\tau_{x}\}_{x\in\mathcal{P}}$ in $\Pi\Omega^{1}(M)$ that smoothly depends on $x$ and such that

[TABLE]

But $r:\Pi\Omega^{k}(M)\rightarrow\Pi\Omega^{1}(U_{\alpha})$ in the generalized Mayer-Vietoris sequence is by definition the restriction [1]. Since square (I) from diagram (5) commutes, we get

[TABLE]

Diagram (8) means that

[TABLE]

however, or equivalently

[TABLE]

for each $\alpha$ in $J$. Since $\{U_{\alpha}\}_{\alpha\in J}$ is a cover for $M$, we have that

[TABLE]

for each $x$ in $\mathcal{P}$. Therefore, $\{\tau_{x}\}_{x\in\mathcal{P}}$ is the required family stated in the theorem.

In summary, for the case $k\geq 2$ we define the family of smooth primitives $\tau_{x}$ by

[TABLE]

for each $\alpha\in J$, where $a_{x}^{\alpha}$ is the element in $\Omega^{k-1}(U_{\alpha})$ defined by

[TABLE]

and where $\{\rho_{\beta}:M\rightarrow\mathbb{R}\}_{\beta\in J}$ is a partition of unity subordinate to the good cover $\{U_{\alpha}\}_{\alpha\in J}$ of $M$.

4. Applications

We concluded with an application in proving the following theorem.

Theorem 2**.**

Let $(M,\mathcal{F})$ be a foliated manifold such that the foliation $\mathcal{F}$ is given by a locally trivial submersion $p:M\to B$. If $H^{r}_{DR}(p^{-1}(b))=0$ for all $b\in B$ and an integer $r$, then $H^{r}(M,\phi^{0}(\mathcal{F}))=0$.

Proof.

Since $M\xrightarrow{p}B$ is a locally trivial bundle, then there exists a countable good cover $\{U_{\alpha}\}_{\alpha\in J}$ of $B$ such that for each $U$ in $\{U_{\alpha}\}_{\alpha\in J}$ there exists a homeomorphism $\varphi_{U}$ such that the following diagram commutes:

[TABLE]

Moreover, for each $U$ and for each $b$ in $U$, we have the composition

[TABLE]

where $i_{b}$ is the inclusion, and we have the induced composition

[TABLE]

Let $[w]$ be an element in $\Omega^{r}(\mathcal{F})$ such that $d_{\mathcal{F}}[w]=0$. To prove the theorem, we need to prove that there exists an element $[\tau]$ in $\Omega^{r-1}(\mathcal{F})$ such that $d_{\mathcal{F}}[\tau]=[w]$; however, this claim is equivalent to the following statements:

• (i)

$d\tau-w$ belongs to $\Omega^{r}(M)$.

• (ii)

$(d\tau-w)_{m}(u_{1},\dots,u_{r})=0$ for each $b\in B$, for all $m\in p^{-1}(b)$ and for all $u_{1},\dots,u_{r}\in T_{m}p^{-1}(b)$.

• (iii)

$i_{b}^{*}((d\tau-w)|_{p^{-1}(U)})=0$ for each $b\in U$ and for each $U$ in the good cover.

• (iv)

For each $b\in U$ and for each $U$ in the good cover,

[TABLE]

We will prove (iv). By hypothesis we have that $d_{\mathcal{F}}[w]=0$, then $i_{b}^{*}(w|_{p^{-1}(U)})$ is closed for each $b\in U$ and for each $U$ in the good cover. On the other hand, also by hypothesis all closed forms in $\Omega(p^{-1}(b))$ are exact for each $b\in B$. This happens, however, if and only if all closed forms in $\Omega(b\times F)$ are exact for each $b\in U$ and all $U$ in the good cover; this is a consequence of the isomorphism $(\varphi|_{U})^{*}:\Omega(p^{-1}(b))\to\Omega(b\times F)$, commutes with $d$.

Therefore, we get that $(\varphi_{U}|_{b})^{*}i_{b}^{*}(w|_{p^{-1}(U)})\in\Omega(b\times F)$ is closed for each $b\in U$ and all $U$ in the good cover. Thus, $(\varphi_{U}|_{b})^{*}i_{b}^{*}(w|_{p^{-1}(U)})\in\Omega(b\times F)$ is exact for each $b\in U$ and all $U$ in the good cover.

Since for each fixed $U$ in the good cover the family

[TABLE]

is a smooth family of exact $(r-1)$-forms on $F$ depending on the parameter $b$, we have by Theorem 1 that there exists a smooth family

[TABLE]

of $(r-1)$-forms on $F$ depending on the parameter $b$ such that

[TABLE]

for each $b\in U$.

On the other hand, we have the following commutative diagram:

[TABLE]

where $\widetilde{i}_{b}$ is the inclusion. Such a diagram induces the following commutative diagram:

[TABLE]

Let $\{\rho_{\alpha}:M\rightarrow\mathbb{R}\}_{\alpha\in J}$ be a partition of unity subordinated to the good cover $\{U_{\alpha}\}_{\alpha\in J}$ of $B$. We define $\tau\in\Omega^{r-1}(B\times F)$ as follows: for each $(b,f)\in B\times F$ and each $u_{1},\dots,u_{r-1}\in T_{(b,f)}(B\times F)$ we set

[TABLE]

Now we will prove first that $\widetilde{i_{b}}^{*}(\tau|_{U\times F})\in\Omega^{r-1}(b\times F)$ is such that

[TABLE]

for each $U$ in the good cover and for each $b\in U$.

For each $U\in\{U_{\alpha}\}_{\alpha\in J}$, $b\in U$ and $u_{1},\dots,u_{r-1}\in T_{(b,f)}(b\times F)$, we get

[TABLE]

Therefore, for each $(b,f)\in b\times F$ we have that

[TABLE]

That means

[TABLE]

for each $U$ and for each $b\in U$ as was established.

Now since diagram (10) commutes, we have

[TABLE]

and by substituting equation (12) in equation (11) we get

[TABLE]

Moreover, since $(\varphi_{U}|_{b\times F})^{*}$ is an isomorphism we get

[TABLE]

for each $U$ and each $b\in U$.

Now we define the element $\tau_{U}\in\Omega(p^{-1}(U))$ by

[TABLE]

so we get

[TABLE]

for each $U$ and each $b\in U$.

On the other hand, for each $U$, $V$ in $\{U_{\alpha}\}$,

[TABLE]

Therefore, the collection $\{\tau_{U}\}$ with $U$ in $\{U_{\alpha}\}_{\alpha\in J}$ defines a global form $\widetilde{\tau}\in\Omega^{r-1}(M)$ such that $\widetilde{\tau}|_{p^{-1}(U)}=\tau_{U}$. Thus, by equation (13), we have that

[TABLE]

for each $U$ in the good cover and for each $b\in U$. Such element satisfies equation (9), and we conclude with the proof of the theorem. ∎

Acknowledgements

The author Rafael Ramos acknowledges Dr. Yuri Vorobiev for the useful discussions and advice. The research was partially supported by a CONACyT research grant.