# Cosimplicial cohomology of restricted meromorphic functions on foliated
manifolds

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Prague, Czech Republic<EMAIL_ADDRESS>
###### Abstract.

Starting from the axiomatic description of meromorphic functions with
prescribed analytic properties, we introduce the cosimplicial cohomology of
restricted meromorphic functions defined on foliations of smooth complex
manifolds. Spaces for double chain-cochain complexes and coboundary operators
are constructed. Multiplications of several restricted meromorphic functions
with non-commutative parameters, as well as for elements of double complex
spaces are introduced and their properties are discussed. In particular, we
prove that the construction of invariants of cosimplicial cohomology of
restricted meromorphic functions is non-vanishing, independent of the choice
of the transversal basis for a foliation, and invariant with respect to
changes of coordinates on a smooth manifold and on transversal sections. As an
application, we provide an example of general cohomological invariants, in
particular, generalizing the Godbillon–Vay invariant for codimension one
foliations.

AMS Classification: 53C12, 57R20, 17B69

###### Key words and phrases:

Meromorphic functions with prescribed analytic properties, foliations,
cohomology

##  1\. Conflict of Interest

The author states that:

1.) The paper does not contain any potential conflicts of interests.

##  2\. Data availability statement

The author confirms that:

1.) All data generated or analyzed during this study are included in this
published article.

2.) Data sharing not applicable to this article as no datasets were generated
or analyzed during the current study.

##  3\. Introduction

It is natural to consider the cohomology of various structures associated to
foliations of a smooth manifold [6, 7, 9, 11, 15, 20, 30, 32, 33]. In this
paper we construct explicitly a cohomology theory of meromorphic functions
with specified analytical properties. Restricted meromorphic functions depend
on non-commutative parameters provided by elements of an infinite-dimensional
Lie algebra as well as sets of commutative formal variables which can be
associated to local coordinates of certain complex domains. Restricted
meromorphic functions are defined subject to several axioms and restrictions
on their convergence. A particular example of such functions can be given by
bilinear pairing on the algebraic completion of spaces associated to an
infinite-dimensional Lie algebras [9, 10, 12, 13, 14, 18, 44]. For a smooth
complex manifold $M$, one needs to identify formal variables of restricted
meromorphic functions with local coordinates of domains on $M$. In the case of
a foliated manifold, additional formal parameters can be identified with local
coordinates on sections of a transversal basis. Our motivation was to
understand further the continuous cohomology [1, 5, 6, 7, 9, 11, 12, 13, 14,
15, 16, 18, 20, 22, 23, 25, 26, 29, 30, 31, 33, 37, 44] of structures defined
on foliations. One hopes to use properties of restricted meromorphic functions
to be able to describe new invariants of foliations. In particular [5], one
hopes to relate cohomology of functions with specified behavior originating
from infinite-dimensional Lie algebras-valued series on manifolds. In order to
construct a cohomology theory on foliations, we define double complex spaces
of restricted meromorphic functions on cosimplicial domains introduced in the
standard way [44]. Properties of such spaces are then studied. We prove
independence of their elements with respect to changes of the transversal
basis and coordinates on $M$ and transversal sections. An appropriate
coboundary operator is defined. To be able to study examples of cohomology
invariants, we introduce a multiplication of several elements of complex
spaces and determine its properties. The main result of this paper is the
computation of the general form of cohomology invariants. An example
generalizing the Godbillon–Vay invariant for codimension one foliations [2, 3,
4, 16, 17, 20, 32, 34, 35, 36] is derived and examined. For further
developments, we plan to study applications of restricted meromorphic
functions and the cosimplicial construction introduced in this paper for
$K$-theory, description of cohomological invariants of foliations [1, 22, 23,
37, 43, 38], deformation theory [19], Losik’s approach [16, 33],
Krichever–Novikov algebras [39, 40, 41], current algebras on manifolds [42],
factorization algebras [25], and classification of types of leaves for
foliations [2, 3, 34, 35, 36].

##  4\. Axiomatics of restricted meromorphic functions

Developing ideas of [28], we introduce in this Section the notion of
restricted meromorphic functions, i.e., meromorphic functions with prescribed
analytic behavior on open complex domains. In Sections 5–5.5 we will see that
the space of such functions underlies a cohomology theory. Restricted
meromorphic functions depend on an number of non-commutative parameters
(provided by an infinite-dimensional Lie algebra elements) as well as
commutative formal variables. First, let us set the notations we use. For a
tuple of $n$ complex formal variables ${\bf z}=(z_{1},\ldots,z_{n})$, we
dedicate the notation ${\bf z}_{l}$ for $l$ sets of $n$ values of ${\bf z}$,
namely, ${\bf
z}_{l}=\left((z_{1,1},\ldots,z_{1,n}),\ldots,(z_{l,1},\ldots,z_{l,n})\right)$.
For local coordinates on transversal sections of codimension $p$ foliation
$\mathcal{F}$ (see Subsection 5.1), the notation ${\bf z}^{\prime}$ for
$k$-sets of $p$ formal parameters is reserved. In general, for a set of $m$
elements $(y_{1},\ldots,y_{m})$ we use the notation ${\bf y}_{m}$, while for
$k_{1}\leq k_{2}\in\mathbb{Z}$, we denote ${\bf
y}_{(k_{1},k_{2})}=\left(y_{k_{1}},y_{k_{1}+1},\ldots,y_{k_{2}}\right)$. When
$m$ is arbitrary we write ${\bf y}$. In particular, for $l$ sets of $n$
algebra/group elements we denote ${\bf g}_{l}=(g_{1},\ldots,g_{l})$. When
combined with $l$ sets of $n$ formal parameters ${\bf z}_{l}$, we use the
specific ${\bf x}$-notation: ${\bf
x}_{l}=(g_{1},z_{1}),\ldots,(g_{l},z_{l}))$. For a function $A(x)$ we denote
the sequence $(A(x_{1}),...,A(x_{n}))$ by $\overline{A}({\bf x})$, while
$\overline{A}({\bf x}_{l})$ stands for
$(A(x_{1,1}),\ldots,A(x_{1,n})),\ldots,(A(x_{l,1}),\ldots,A(x_{l,n}))$. The
notations for products of functions are $A({\bf x})=(A(x_{1})$ $\ldots$
$A(x_{n}))$ and $A({\bf x}_{l})=A(x_{1,1})\ldots A(x_{1,n})$ $\ldots$
$A(x_{l,1})\ldots A(x_{l,n})$.

###  4.1. Meromorphic functions with non-commutative parameters

In this paper we consider functions $f({\bf z}_{l})$ of several complex
variables with non-commutative parameters defined on sets of open domains
extendable meromorphic functions $\mathcal{M}(f({\bf z}_{l}))$ converging on
larger domains with respect to corresponding norms. Denote by
$F_{ln}\mathbb{C}$ the configuration space of $l\geq 0$ ordered sets of $n$
complex coordinates in $\mathbb{C}^{ln}$, $F_{ln}\mathbb{C}=\\{{\bf
z}_{l}\in\mathbb{C}^{ln}\;|\;z_{i,l}\neq z_{j,l^{\prime}},i\neq j\\}$. Let
$\mathfrak{g}$ be an infinite-dimensional Lie algebra generated by
$\left\\{\xi_{i},i\in\mathbb{Z}\right\\}$, and $\mathcal{G}({\bf x}_{l})$ be
the space of complex-valued functions depending on $l$-pairs of elements
${\mathfrak{g}}$ and ${\bf z}_{l}$. Abusing notations, for
$F\in\mathcal{G}({\bf x}_{l})$ we call a linear map with the only possible
poles at $z_{i,l}=z_{j,l^{\prime}}$, $i\neq j$, $1\leq l$, $l^{\prime}\leq n$,
$F:{\bf x}_{l}\mapsto\mathcal{M}(F({\bf x}_{l}))$, the meromorphic function in
${\bf z}_{l}$.

Next, let us state further properties of meromorphic functions we require. We
define the left action of the permutation group $S_{ln}$ on $F({\bf x}_{l})$
by $\sigma(F)({\bf x}_{l})=F({\bf g}_{l},{\bf z}_{\sigma(i)})$, $1\leq i\leq
l$. Denote by $(T_{G})_{i}$ the translation operator $f(z_{i})\mapsto
f(z_{i}+z_{0})$ acting on the $i$-th entry. We then define the action of
partial derivatives on an element $F({\bf x}_{l})$

$\partial_{z_{i}}F({\bf x}_{l})=F((T_{G})_{i}{\bf g}_{l},{\bf
z}_{l}),\qquad\sum\limits_{i\geq 1}\partial_{z_{i}}F({\bf x}_{l})=T_{G}F({\bf
x}_{l}),$ (4.1)

and call it $T_{G}$-derivative property. For $z\in\mathbb{C}$, let

$e^{zT_{G}}F({\bf x}_{l})=F({\bf g}_{l},{\bf z}_{l}+z).$ (4.2)

Let ${\rm Ins}_{i}(A)$ denotes the operator of multiplication by
$A\in\mathbb{C}$ at the $i$-th position. Then we define

$F({\bf g}_{l},{\rm Ins}_{i}(z)\;{\bf z}_{l})=F({\rm
Ins}_{i}(e^{zT_{G}})\;{\bf g}_{l},{\bf z}_{l}),$ (4.3)

equal as equal power series expansions in $z$, in particular, absolutely
convergent on the open disk $|z|<\min_{i\neq
j}\\{|z_{i,l}-z_{j,l^{\prime}}|\\}$. A meromorphic function has
$K_{G}$-property if for $z\in\mathbb{C}^{\times}$ satisfies $z{\bf z}_{l}\in
F_{ln}\mathbb{C}$,

$z^{K_{G}}F({\bf x}_{l})=F\left(z^{K_{G}}{\bf g}_{l},z\;{\bf z}_{l}\right).$
(4.4)

Let us recall the notion of a shuffle. For $m\in\mathbb{N}$ and $1\leq p\leq
m-1$, let $J_{m;p}$ be the set of elements of $S_{m}$ which preserves the
order of the first $p$ numbers and the order of the last $(m-p)$ numbers, that
is, $J_{m,p}=\\{\sigma\in
S_{m}\;|\;\sigma(1)<\ldots<\sigma(p),\;\sigma(p+1)<\ldots<\sigma(m)\\}$. Let
$J_{m;p}^{-1}=\\{\sigma\;|\;\sigma\in J_{m;p}\\}$. For some meromorphic
functions we require the property:

$\sum_{\sigma\in J_{ln;p}^{-1}}(-1)^{|\sigma|}\sigma(F({\bf
g}_{\sigma(i)},{\bf z}_{l}))=0.$ (4.5)

Let ${\mathcal{W}}(\mathfrak{g})$ be the space of universal enveloping algebra
$U(\mathfrak{g})$-valued formal series in ${\bf z}_{l}$. We will consider
meromorphic functions $F({\bf x}_{l})$ depending on elements ${\bf
g}_{l}\in{\mathcal{W}}(\mathfrak{g})$ and satisfying the above conditions of
this subsection. In what follows we concentrate on the case
$\mathcal{G}(\mathfrak{g})=G$ where $G$ is the algebraic completion of
$\mathcal{W}(\mathfrak{g})$. In order to define a specific space of restricted
meromorphic functions associated to $\mathcal{W}(\mathfrak{g})$ and satisfying
certain properties we have to work with the algebraic completion of
$\mathcal{W}(\mathfrak{g})$. In particular, for that purpose, we have to
consider elements of a $\mathcal{W}(\mathfrak{g})$ with inserted exponentials
of the grading operator $K_{G}$, i.e., of the form
$\sum\limits_{m\in\mathbb{Z}}z^{-m-1}\;a^{K_{G}}g_{m}\;b^{K_{G}}g$. For
general $a$, $b$, $z\in\mathbb{C}^{\times}$, such elements do not satisfy the
properties needed to construct a appropriate cohomology theory of restricted
meromorphic functions. Thus we have to extend $\mathcal{W}(\mathfrak{g})$
algebraically and analytically, i.e., to consider
$G=\overline{\mathcal{W}}(\mathfrak{g})=\prod_{m\in\mathbb{Z}}\mathcal{W}_{(m)}=(\mathcal{W}^{\prime})^{*}(\mathfrak{g})$,
as well as include extra elements to make the structure of
$\mathcal{W}(\mathfrak{g})$ compatible with the descending filtration with
respect to the grading subspaces, and analytic properties with respect to
formal parameters ${\bf z}_{l}$. Then $G$ has the structure that is complete
in topology determined by the filtration. When a particular set ${\bf z}_{m}$
of formal parameters is specified for $G$ we denote it as $G_{{\bf z}_{m}}$.
We assume that $G$ is endowed with non-degenerate bilinear pairing $(.\;;.)$
(not to be confused with the notation $(g,z)$ for parameters), and denote by
$\widetilde{G}$ the dual space to $G$ with respect to this pairing. The norm
determining convergence of a non-commutative parameter meromorphic functions
can be in particular taken as a bilinear pairing mentioned above. For ${\bf
g}_{l}\in G$, denote ${\bf x}_{l}=({\bf g}_{l},{\bf z}_{l})$. In this paper we
will deal with meromorphic functions given by $F:{\bf
x}_{l}\mapsto\mathcal{M}\left(\left(\vartheta,F({\bf x}_{l})\right)\right)$,
and converging in ${\bf z}_{l}$ on certain complex domains ${\bf V}_{l}$.

###  4.2. Restricted meromorphic functions

In this subsection, following [28], we give the definition of meromorphic
functions with prescribed analytical behavior on a complex domain. Let us
assume that the space $G$ is endowed with a grading
$G=\bigcup_{m\in\mathbb{Z},m>m_{0}}G_{(m)}$ bounded from below with respect to
the grading operator $K_{G}$. We denote by $P_{m}:G\to G_{(m)}$, the
projection of $G$ on $G_{(m)}$. For each element $g\in G$, and $x=(g,z)$,
$z\in\mathbb{C}$ let us associate the differential form
$\nu_{G}(x)=\sum\limits_{m\in\mathbb{C}}g_{m}\;z^{-m}\;dz^{\mbox{\rm wt}\
(g)}$, where $\mbox{\rm wt}\ (g)$ is the weight with respect to the grading
operator $K_{G}g=\mbox{\rm wt}\ (g)g$. Finally, we formulate the definition of
restricted meromorphic functions with extra sets of parameters. Due to the
nature of axiomatics of meromorphic functions described in Subsection 5.2, we
have to restrict their analytic behavior to be able to introduce corresponding
cohomology theory. In particular, we would like to make $F({\bf x}_{l})$ to
play the role of cochains. Corresponding coboundary operators are supposed to
include insertions into $F({\bf x}_{l})$ of extra ${\bf x}$-dependence by
means of extra $\overline{\nu}_{G}({\bf x})$-forms. The result of such
insertions should remain in $G$. As a formal sum, such map has to be
absolutely convergent in ${\bf z}_{l}$. For this purpose we formulate the
following definition representing extra restricting conditions on meromorphic
functions introduced in Subsection 5.2.

######  Definition 1.

We assume that there exist positive integers
$\beta(g_{l^{\prime},i},g_{l",j})$ depending only on $g_{l^{\prime},i}$,
$g_{l^{\prime\prime},j}\in G$ for $i$, $j=1,\dots,(l+k)n$, $k\geq 0$, $i\neq
j$, $1\leq l^{\prime},l^{\prime\prime}\leq n$. Let ${\bf l}_{n}$ be a
partition of $(l+k)n=\sum_{i\geq 1}l_{i}$, and $k_{i}=l_{1}+\cdots+l_{i-1}$.
For $\zeta_{i}\in\mathbb{C}$, define for $\widetilde{k}_{i}=k_{i}+l_{i}$,
$f_{i}=F\left(\overline{\nu}_{G}({\bf g}_{\widetilde{k}_{i}},{\bf
z}_{\widetilde{k}_{i}}-\zeta_{i})\right)$, for $i=1,\dots,ln$. We then call a
meromorphic function $F({\bf x}_{l})$ satisfying properties (4.1)–(4.4), a
restricted meromorphic function if under the following conditions on domains,
$|z_{k_{i}+p}-\zeta_{i}|+|z_{k_{j}+q}-\zeta_{j}|<|\zeta_{i}-\zeta_{j}|$, for
$i$, $j=1,\dots,k$, $i\neq j$, and for $p=1,\dots$, $l_{i}$, $q=1$, $\dots$,
$l_{j}$, the function $\sum\limits_{{\bf
m}_{n}\in\mathbb{Z}^{n}}F\left(\overline{P_{m_{i}}f_{i}},{\bm{\zeta}}_{l}\right)$,
is absolutely convergent to an analytic extension in ${\bf z}_{l+k}$,
independently of complex parameters ${\bm{\zeta}}_{l}$, with the only possible
poles on the diagonal of ${\bf z}_{l+k}$ of order less than or equal to
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$. In addition to that, for
${\bf g}_{l+k}\in G$, the series
$\sum_{q\in\mathbb{C}}F\left(\overline{\nu}_{G}(x_{k})\;{\bf
P}_{q}\left(F\left({\bf x}_{(k,k+l)}\right)\right)\right)$, is absolutely
convergent when $z_{i}\neq z_{j}$, $i\neq j$ $|z_{i}|>|z_{s}|>0$, for
$i=1,\dots,k$ and $s=k+1,\dots,l+k$, and the sum can be analytically extended
to a meromorphic function in ${\bf z}_{l+k}$ with the only possible poles at
$z_{i}=z_{j}$ of orders less than or equal to
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$.

For an arbitrary $\theta\in\widetilde{G}$ and $l\geq 0$ complex variables
${\bf z}_{l}$ defined in domains ${\bf V}_{l}$, let us introduce the following
vector $\overline{F}({\bf x}_{l})=\left[F\left({\bf g}_{l},{\bf z}_{l}\;{\bf
dz}_{{\it i}(ln)}\right)\right]$, containing meromorphic functions where ${\it
i}(j)$, $j=1,\ldots,ln$, are cycling permutations of $(1,\dots,ln)$ starting
with $j$. For $k$ elements ${\bf g}^{\prime}_{k}$, $k\geq 0$, we call the
space of all vectors $\overline{F}({\bf x}_{l})$ combined with $k$-sets of
forms $\overline{\nu}_{G}({\bf x}^{\prime}_{k})$ depending on $p$ complex
variables ${\bf z}^{\prime}_{k}$ defined in domains ${\bf U}_{k}$ satisfying
$T_{G}$\- and $K_{G}$-properties (4.1), (4.4), and (4.5), the space
$\Theta(G,{\bf V}_{l},{\bf U}_{k})$ of restricted meromorphic functions.

###  4.3. Coordinate change invariance

In this subsection we show that vectors $\overline{F}({\bf
x}_{l})\in\Theta(G,{\bf V}_{l},{\bf U}_{k})$ containing restricted meromorphic
functions are canonical with respect to changes of sets of formal parameters.
Let ${\rm Aut}\;{\mathcal{O}}^{(ln)}={\rm Aut}_{\mathbb{C}}[[{\bf z}_{l}]]$ be
the group of formal automorphisms of $ln$-dimensional formal power series
algebra ${\mathbb{C}}[[{\bf z}_{l}]]$. In what follows, we assume that $G$ is
equipped with the action of operators $\left(z^{m+1}\partial_{z}\right)$,
$m\in\mathbb{N}$, and its Lie subalgebra ${\rm Der}_{0}\;{\mathcal{O}}^{(ln)}$
of $\mathfrak{g}$ is given by the Lie algebra of ${\rm
Aut}\;{\mathcal{O}}^{(ln)}$. Since the vector fields
$\left(z^{m+1}\partial_{z}\right)$ act on $G$ as operators of degree $(-m)$,
the action of the Lie subalgebra ${\rm Der}_{+}\;{\mathcal{O}}^{(ln)}$ is
locally nilpotent. The operator $\left(z\partial_{z}\right)$ acts as the
grading operator $K_{G}$, which is diagonalizable with integral eigenvalues.
Thus, the action of ${\rm Der}\;{\mathcal{O}}^{(ln)}$ on $G$ can be
exponentiated to an action of ${\rm Aut}\;{\mathcal{O}}^{(ln)}$. We write an
element of ${\rm Aut}\;{\mathcal{O}}^{(ln)}$ as ${\bf
z}\to{\bm{\rho}}_{n}({\bf z})$, where elements of ${\bm{\rho}}_{n}$ are
$\rho_{i}({\bf z})=\sum_{{i_{1}\geq 0,\ldots,i_{n}\geq
0,}\quad{\sum\limits_{j=1}^{n}i_{j}\geq 1}}a_{{\bf i}_{k}}\;{\bf z}_{k}^{{\bf
i}_{k}}$, where $a_{{\bf i}_{k}}\in{\mathbb{C}}$, and are the images of
$\rho_{i}$, $i=1,\ldots,l$ in the finite dimensional ${\mathbb{C}}$-vector
space. In order to represent the action of the group ${\rm
Aut}\;{\mathcal{O}}^{(ln)}$ on the variables ${\bf z}$ of $\overline{F}$ in
terms of an action on elements ${\bf g}_{l}$, we have to transfer (as in $n=1$
case of [8]) to an exponential form of the transformations $\rho_{i}({\bf z})$
with corresponding coefficients $\beta^{(j)}_{{\bf r}_{l}}\in\mathbb{C}$
recursively found [21] in terms of coefficients $a^{(i)}_{{\bf r}_{l}}$.

Next, we recall the general definition of a torsor [8]. Let $\mathfrak{G}$ be
a group, and $S$ a non-empty set. Then $S$ is called a $\mathfrak{G}$-torsor
if it is equipped with a simply transitive right action of $\mathfrak{G}$,
i.e., given $s_{1}$, $s_{2}\in S$, there exists a unique $\mu\in\mathfrak{G}$
such that $s_{1}\cdot\mu=s_{2}$, where the right action is given by
$s_{1}\cdot(\mu\mu^{\prime})=(s_{1}\cdot\mu)\cdot\mu^{\prime}$. The choice of
any $s_{1}\in S$ allows us to identify $S$ with $\mathfrak{G}$ by sending
$s_{1}\cdot\mu$ to $\mu$. Finally, we state

######  Lemma 1.

The vector $\overline{F}({\bf x}_{l})\in\Theta(G,{\bf V}_{l},{\bf U}_{k})$ is
invariant with respect to changes of formal variables.

###### Proof.

Consider the vector $\overline{F}(\widetilde{\bf x}_{l})=\left[F\left({\bf
g}_{l},\widetilde{\bf z}_{l}\;{\bf d}\widetilde{\bf z}_{{\it
i}(ln)}\right)\right]$. Note that
$d\widetilde{z}_{j}=\sum_{i=1}^{ln}dz_{i}\;{\partial_{z_{i}}\rho_{j}}$,
$\partial_{z_{i}}\rho_{j}={\partial\rho_{j}}/{\partial z_{i}}$. By definition
of the action of ${\rm Aut}_{n}\;{\mathcal{O}}^{(1)}_{ln}$, for
$d\widetilde{\bf z}_{i}$, we have $\overline{F}(\widetilde{\bf x}_{l})={\rm
R}({\bm{\rho}}_{ln})\;\left[F\left({\bf g}_{l},{\bf z}_{l}\;{\bf
d}\widetilde{\bf z}_{i(ln)}\right)\right]={\rm
R}({\bm{\rho}}_{ln})\;\left[F\left({\bf g}_{l},{\bf
z}_{l}\;\sum{}^{ln}_{j=1}\;\partial_{j}\rho_{i(ln)}\;dz_{j}\right)\right]$,
with ${\rm R}({\bm{\rho}}_{ln})=$
$\left[\widehat{\partial}_{J}\rho_{i(I)}\right]=\left[\begin{array}[]{c}\widehat{\partial}_{J}\rho_{i_{1}(I)},\widehat{\partial}_{J}\rho_{i_{2}(I)},\ldots,\widehat{\partial}_{J}\rho_{i_{ln}(I)}\end{array}\right]^{T}$.
The index operator $J$ takes the value of index $z_{j}$ of arguments in the
vector $\overline{F}(\widetilde{\bf x}_{l})$ while the index operator $I$
takes values of index of differentials $dz_{i}$ in each entry of the vector
$\overline{F}$. The index operator $i(I)=(i_{1}(I),\ldots,i_{ln}(I))$ is given
by consequent cycling permutations of $I$. Taking into account the property
(4.2), we define the operator

$\widehat{\partial}_{J}\rho_{a}=\exp\left(-\sum{}_{{\bf
q}_{ln},\;\sum\limits_{i=1}^{ln}q_{i}\geq 1,\;1\leq J\leq
ln}r_{J}\;\beta^{(a)}_{{\bf
r}_{n}}\;({\bm{\zeta}})_{ln}\;\partial_{z_{J}}\right),$ (4.6)

which contains index operators $J$ as index of a dummy variable $\zeta_{J}$
turning into $z_{j}$, $j=1,\ldots,ln$. (4.6) acts on each argument of maps $F$
in the vector $\overline{F}$. Due to properties of $G$ required above, the
action of operators $R\left({\bm{\rho}}_{ln}\right)$ on ${\bf g}_{l}\in G$
results in a sum of finitely many terms. By using (4.2) and linearity of the
mapping $\overline{F}$, we obtain $\overline{F}(\widetilde{\bf
x}_{l})=\overline{F}({\bf g}_{l},\widetilde{\bf z}_{l}\;{\bf
d}\widetilde{z}_{l})=\left[F\left({\bf g}_{l},{\bf z}_{l}\;{\bf
dz}_{i(ln)}\right)\right]$. We then conclude that the vector $\overline{F}$ is
invariant, i.e., $\overline{F}(\widetilde{\bf x}_{l})=\overline{F}({\bf
x}_{l})$. Definition 1 of restricted meromorphic functions $\overline{F}({\bf
x}_{l})\in\Theta\left(G,{\bf V}_{l},{\bf U}_{k}\right)$ consists of two
conditions on $\overline{F}({\bf x}_{l})$ and $\overline{\nu}_{G}({\bf
z}^{\prime}_{k})$. The first requires existence of positive integers
$\beta^{n}_{m}(g_{i},g_{j})$ depending on $g_{i}$, $g_{j}$ only, and the
second restricts orders of poles of corresponding sums. The insertions of a
sequence of $k$ forms $\overline{\nu}_{G}({\bf x}^{\prime})$ which are present
in Definition 1 of prescribed rational functions keep elements $\overline{F}$
invariant with respect to coordinate changes. Thus, elements of $\Theta(G,{\bf
V}_{l},{\bf U}_{k})$-spaces are invariant under the action of the group ${\rm
Aut}_{ln+kp}\;{\mathcal{O}}^{(1)}_{ln+kp}$. ∎

##  5\. Cosimplicial double complex defined for a foliation

In this Section we introduce spaces for double complexes used to define the
restricted meromorphic functions cohomology of a foliation $\mathcal{F}$ of
codimension $p$ on a smooth complex manifold $M$ of dimension $n$.

###  5.1. Foliation holonomy embeddings and transversal basis

Let us first recall [11] some definitions concerning transversal basis and
holonomy embeddings for a foliation $\mathcal{F}$. A transversal section is an
embedded $p$-dimensional submanifold $U\subset M$ everywhere transverse to all
leaves of $\mathcal{F}$. Suppose $\alpha$ is a path between two points $p$ and
$\widetilde{p}$ on the same leaf of $\mathcal{F}$. Let $U$ and $\widetilde{U}$
be transversal sections comming through points $p$ and $\widetilde{p}$. Then
$\alpha$ determines a transport along the leaves from a neighborhood of $p$ in
$U$ to a neighborhood of $\widetilde{p}$ in $\widetilde{U}$. Thus it is
assumed that there exists a germ of a diffeomorphism ${\rm
hol}(\alpha):(U,p)\longrightarrow(\widetilde{U},\widetilde{p})$ which is
called the holonomy of the path $\alpha$. In the case that the transport above
is defined in all of $U$ and embeds $U$ into $\widetilde{U}$, this embedding
$h:U\hookrightarrow\widetilde{U}$ is called the holonomy embeddings. Now
recall the definition of the transversal basis for a folitaion $\mathcal{F}$.
A transversal basis for $\mathcal{F}$ is a family $\mathcal{U}$ of transversal
sections $U\subset M$ with the property that, if $\widetilde{U}$ is any
transversal section through a given point $p\in M$, there exists a holonomy
embedding $h:U\hookrightarrow\widetilde{U}$ with $U\in\mathcal{U}$ and $p\in
h(U)$.

###  5.2. Restricted meromorphic functions associated to a foliation

Suppose $M$ is endowed with a coordinate chart $\mathcal{V}=\left\\{{\bf
V}_{m},m\in\mathbb{Z}\right\\}$. Let ${\bf p}_{l}$ be a set of $l$ points in
${\bf V}_{l}$. Let us identify $l$-sets of $n$-tuples of formal variables
${\bf z}_{l}$ with $l$ sets of $n$ local coordinates on ${\bf V}_{l}$. Note
that according to our construction, $M$ can be infinite-dimensional. Thus, in
that case, we consider $l$ infinite sets of complex coordinates. For another
$k$ sets of $p$-tuples of points ${\bf p}^{\prime}_{k}$ on transversal
sections ${\bf U}_{k}$ of the transversal basis $\mathcal{U}$ of foliation
$\mathcal{F}$, we take $k$ sets of $p$-tuples of ${\bf x}^{\prime}_{k}=({\bf
g}^{\prime}_{k},{\bf z}^{\prime}_{k})$, identify formal parameters ${\bf
z}^{\prime}_{k}$ with local coordinates of $k$ points ${\bf p}^{\prime}_{k}$,
and consider corresponding series $\overline{\nu}_{G}\left({\bf
x}^{\prime}_{k}\right)$. While the choice of the initial point
$p^{\prime}_{0}$ is arbitrary, we assume that other points are related by the
holonomy embeddings ${\bf h}_{k}$, $p^{\prime}_{0}\stackrel{{\scriptstyle
h_{1}}}{{\longrightarrow}}p^{\prime}_{1}\stackrel{{\scriptstyle
h_{1}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle
h_{k}}}{{\longrightarrow}}p^{\prime}_{k}$. It may happened that $z_{1,j}$
coincides with some $z_{p,j}$. In order to work with objects having coordinate
invariant formulation on $M$ we consider restricted meromorphic functions
$\overline{F}({\bf x}_{l})\in\Theta\left(G,\mathcal{V},\mathcal{U}\right)$. In
[8], they proved that $\nu_{G}(x)$ is an invariant object with respect to
changes of coordinate in one-dimensional complex case. In Section 6.2 we
proved that elements of the space $\Theta(G,{\bf V}_{l},{\bf U}_{k})$ and sets
of forms $\overline{\nu}_{G}({\bf x}^{\prime}_{k})$ are invariant with respect
to change of coordinates, i.e., to the group of coordinate transformations
${\rm Aut}\;{\mathcal{O}}^{(ln)}$, $\widetilde{\bf z}_{l}\mapsto{\bf z}_{l}$,
and corresponding differentials.

###  5.3. Spaces for cosimplicial double complexes

In [18] the original approach to cohomology of vector fields of manifolds was
initiated. We find another approach to cohomology of Lie algebra of vector
fields on a manifold in the cosimplicial setup in [13, 44]. Taking into
account the standard methods of defining canonical (i.e., independent of the
choice of covering $\mathcal{U}$) cosimplicial object [13, 44] as well as the
${\rm\check{C}}$ech-de Rham cohomology construction [11] for a foliation, we
consider restricted meromorphic functions $F({\bf x}_{l})$ with $k$ sets of
forms $\overline{\nu}_{G}({\bf x}^{\prime}_{k})$ and give the following
definition of a cosimplicial double complex for $G$.

For ${\bf x}_{l}$ with local coordinates ${\bf z}_{l}\in\mathcal{V}$, and a
transversal basis $\mathcal{U}=\left\\{U_{i},i\in I\right\\}$ for
$\mathcal{F}$ on $M$, consider vectors $\overline{F}({\bf
x}_{l})\in\Theta(G,{\bf V}_{l},{\bf U}_{k})$ defined in Subsection 4.2. Then
let us associate to any subset $\left\\{i_{1}<\cdots\right.$
$\left.<i_{k}\right\\}$ of $I$, the space of restricted meromorphic functions
with $k$ sets of $\overline{\nu}_{G}({\bf x}^{\prime})$-forms with local
coordinates ${\bf z}^{\prime}\in{\bf U}_{l}$, satisfying Definition 1, and
defined on the intersection of transversal sections

$C^{l}_{k}(G,\mathcal{V},\mathcal{U},\mathcal{F})=\Theta\left(G,\mathcal{V},\bigcap_{U_{i_{0}}\stackrel{{\scriptstyle
h_{i_{0}}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle
h_{i_{k-1}}}}{{\longrightarrow}}U_{i_{k}},\;i_{1}\leq\ldots\leq i_{k},\;k\geq
0}U_{i_{k}},\right),$ (5.1)

where the intersection ranges over all $k$-tuples of holonomy embeddings ${\bf
h}_{k}$, among transversal sections of foliation in $\mathcal{U}$. Since we
assume that the points ${\bf p}^{\prime}_{k}$, are related by holonomies, each
consequent arrow $h_{j}$ in the holonomy sequence in the intersection (5.1)
introduces $p$ $\nu_{G}$-forms in addition to the initial $(j-1)$ sets of
forms with formal parameters.

###  5.4. Properties of the double complex spaces
$C_{k}^{l}(G,\mathcal{V},\mathcal{U},\mathcal{F})$

In this subsection we fix $G$ and $\mathcal{F}$ and omit them and
$\mathcal{V}$ from notations where it is possible, and study properties of
$C_{k}^{l}(\mathcal{U})$ spaces. Let us set $C_{k}^{0}(\mathcal{U})=G$. Then
we have

######  Lemma 2.

The spaces (5.1) are non-zero, and $C_{k}^{l}(\mathcal{U})\subset
C_{k-1}^{l}(\mathcal{U})$.

###### Proof.

Recall the conditions on $k$ $\overline{\nu}_{G}({\bf x})$-forms given in
Definition (1). Since exists the lower limit on domain of absolute convergence
given in Definition 1, the extension of the tuple of $k$-homology embeddings
in (5.1) by another embedding preserves the conditions applied to the mappings
$\overline{F}({\bf x})$ which belong to the spaces (5.1). Thus, (5.1) is non-
zero. ∎

######  Lemma 3.

For any $l$, $p\geq 0$, the construction (5.1) of double complex spaces
$C_{k}^{l}(\mathcal{U})$ does not dependent of the choice of transversal basis
$\mathcal{U}$.

###### Proof.

Suppose we consider another transversal basis $\mathcal{U}^{\prime}$ for
$\mathcal{F}$. The of the double complex spaces then assumes that vectors
$\overline{F}$ on $U^{\prime}_{j}$ are defined on all transversal sections of
$\mathcal{U}^{\prime}$. According to the definition of the transversal basis
given above, for each transversal section $U_{i}$ which belongs to the
original basis $\mathcal{U}$ in (5.1) there exists a holonomy embedding
$\widetilde{h}_{i}:U_{i}\hookrightarrow U_{j}^{\prime}$, i.e., it embeds
$U_{i}$ into a section $U_{j}^{\prime}$ of our new transversal basis
$\mathcal{U}^{\prime}$. Then then statement of the proposition follows. ∎

In what follows, we omit $\mathcal{U}$ from notations of (5.1). Next, we prove
that the construction of spaces (5.1) for the chain-cochain double complex is
independent of the choice of coordinates on $\mathcal{V}$ and $\mathcal{U}$.

######  Proposition 1.

Elements $\overline{F}\left({\bf x}_{l}\right)\in C^{l}_{k}$ with ${\bf
x}_{l}\in\mathcal{V}$ and the forms $\overline{\nu}_{G}({\bf
x}^{\prime}_{k})$, with ${\bf x}^{\prime}_{k}\in{\bf U}_{k}$ of $\mathcal{F}$
are canonical, i.e., independent on changes $({\bf z}_{l},{\bf
z}^{\prime}_{k})\mapsto\left(\widetilde{\bf z}_{l},\widetilde{\bf
z}^{\prime}_{k}\right)={\bm{\rho}}_{l+k}\left({\bf z}_{l},{\bf
z}^{\prime}_{k}\right)$ of local coordinates on $\mathcal{V}$ and
$\mathcal{U}$.

###### Proof.

In Lemma 1 we proved that elements of $\overline{F}({\bf
x}_{l})\in\Theta(G,\mathcal{V},\mathcal{U})$ are coordinate change-invariant.
The construction of the double complex spaces (5.1) assumes that
$\overline{F}({\bf x}_{l})\in C^{l}_{k}$ satisfies conditions of definition of
a $\Theta$-space and with extra conditions of Definition 1 on $k$ sets of $p$
$\overline{\nu}_{G}({\bf x}_{k})$-forms. In [8] they proved in one-dimensional
complex case, that the form $\nu_{g}(x)$ containing the $\mbox{\rm wt}\
(g_{i})$-power of the differential $dz$ is invariant with respect to the
action of the group ${\rm Aut}\;{\mathcal{O}}^{(1)}$. Here we prove that
$\overline{\nu}_{G}({\bf x}^{\prime}_{k})$ are invariant with respect to the
change of $k$ sets of $p$ local coordinates ${\bf
z}^{\prime}_{p}\mapsto\widetilde{\bf z}^{\prime}_{p}({\bf z}^{\prime}_{p})$ on
a transversal section of $\mathcal{F}$. Let ${\bf z}^{\prime}$ be coordinates
on a coordinate chart around a point $p^{\prime}$ on a transversal section
$U\in\mathcal{U}$. Define a $\mbox{\rm wt}\ (g^{\prime}_{i})$-differential on
coordinate chart around $p^{\prime}$ with values in $End\;(G)_{{\bf
z}^{\prime}}$ as follows: identify $End\;G_{{\bf z}^{\prime}}$ with
$End\;\;G_{{\bf z}^{\prime}}$ using the coordinates ${\bf z}^{\prime}$. Let
$\widetilde{\bf z}^{\prime}=({\bm{\rho}})_{p}({\bf z}^{\prime})$, be another
$k$-set of coordinates on an $p$-dimensional coordinates on transversal
sections. Let us express the set of $\mbox{\rm wt}\
(g^{\prime}_{i})$-differentials on $D^{(p^{\prime}),\times}_{p^{\prime}}$
$\nu_{G}(g^{\prime}_{i},\widetilde{z}^{\prime}_{i})$, $i=1,\ldots,p$, in terms
of of the coordinates ${\bf z}^{\prime}$. We would like to show that it
coincides with the set of $\mbox{\rm wt}\ (g_{i}^{\prime})$-differentials
$\overline{\nu}_{G}({\bf z}^{\prime})$. We will use the notion of torsors in
order to prove the independence of formal series operators multiplied by some
power of differentials for for elements $g\in G$ of $\mbox{\rm wt}\
(g)\in\mathbb{Z}_{+}$ such that ${{\bf z}^{\prime}}^{m+1}\partial_{{\bf
z}^{\prime}}g=0$, for $m>0$. The general case then follows. Consider a vector
$(g^{\prime}_{i},{\bf z}^{\prime})\in G_{{\bf z}^{\prime}}$ with
$g^{\prime}_{i}\in G$. Then the same vector equals
$\left(R_{i}^{-1}\left({\bm{\rho}}_{p}\right)g^{\prime}_{i},\widetilde{\bf
z}^{\prime}\right)$, i.e., it is identified with
$R_{i}^{-1}\left({\bm{\rho}}_{p}\right)g^{\prime}_{i}\in G$, using the
coordinates $\widetilde{\bf z}^{\prime}$. Here
$R_{i}\left({\bm{\rho}}_{p}\right)$ is an operator representing transformation
of ${\bf z}^{\prime}\to\widetilde{\bf z}^{\prime}$, as an action on $G$.
Therefore if we have an operator on $G_{{\bf z}^{\prime}}$ which is equal to a
${\rm Aut}\;{\mathcal{O}}^{(p)}$-torsor $S$ under the identification
$End\;\;G_{{\bf z}^{\prime}}\in End\;\;G$ using the coordinates
$\widetilde{\bf z}^{\prime}$, then this operator equals
$R_{i}\left({\bm{\rho}}_{p}\right)\;S\;R_{i}^{-1}\left({\bm{\rho}}_{p}\right)$,
under the identification $End\;\;G_{{\bf z}^{\prime}}\in End\;\;G^{(i)}$ using
the coordinates $\left(g^{\prime}_{i},{\bf z}^{\prime}\right)$. Thus, in terms
of the coordinates $(g_{i}^{\prime},{\bf z}^{\prime})$, the differential
$\nu_{G}(g^{\prime}_{i},\widetilde{z}^{\prime}_{i})$ becomes
$\nu_{G}(g^{\prime}_{i},z^{\prime}_{i})=R_{i}(\rho)\;\nu_{G}\left(g^{\prime}_{i},\rho({\bf
z}^{\prime})\right)\;R_{i}^{-1}(\rho)$. According to Definition (5.1),
elements $\mathcal{F}({\bf x}_{l})$ satisfy conditions of Definition 1 with
the number $kp$ of $\overline{\nu}_{g}({\bf z}^{\prime}_{k})$ forms. Thus we
see that $\mathcal{F}$ is a canonical object suitable for $C^{l}_{k}$. ∎

###  5.5. Chain-cochain operators

Let us fix $G$ and $\mathcal{F}$ and skip them from notations. Recall
notations provided in Section 4. Denote by ${\bf
x}_{l,\widehat{i_{1}},\ldots,\widehat{i_{m}}}$, $1\leq i_{1}\leq\ldots\leq
i_{m}\leq l$, the set ${\bf x}$ with $({\bf x}_{i_{1}},\ldots,{\bf
x}_{i_{m}})$ tuples of ${\bf x}$ being omitted. For $\overline{F}\in
C_{k}^{l}$, let us define the operator $D^{l}_{k}$ by

$\displaystyle D^{l}_{k}\overline{F}({\bf x}_{l})$ $\displaystyle=$
$\displaystyle T_{1}(\overline{\nu}_{G}({\bf x}_{1})).\overline{F}\left({\bf
x}_{l+1,\widehat{1}}\right)$ (5.2) $\displaystyle+$
$\displaystyle\sum_{i=1}^{l}(-1)^{i}\;T_{i}(\overline{\nu}_{G}({\bf
x}_{i}))\;T_{i+1}(\overline{\nu}_{G}({\bf x}_{i+1})).\overline{F}\left({\bf
x}_{l+1,\widehat{i},\widehat{i+1}}\right)$ $\displaystyle+$
$\displaystyle(-1)^{l+1}T_{1}(\overline{\nu}_{G}({\bf
x}_{l+1})).\overline{F}\left({\bf x}_{\widehat{l+1}}\right),$

where $\overline{\nu}_{G}({\bf
x}_{m})=\left[\nu_{G}(x_{m,1}),\ldots,\nu_{G}(x_{m,n})\right]^{T}$, and
$T_{i}(\gamma).\overline{F}({\bf x}_{l+1})$ denotes insertion of $\gamma$ at
$i$-th position of $\overline{F}({\bf x}_{l+1})$. Next, we have

######  Proposition 2.

The operator (5.2) forms the double complex $D^{l}_{k}:C_{k}^{l}\to
C_{k-1}^{l+1}$ on the spaces (5.1) (when lower index is zero the sequence
terminates) and $D_{k-1}^{l+1}\circ D^{l}_{k}=0$.

###### Proof.

Note that $(l+1)n$ formal variables ${\bf z}_{l+1}$ in the Definition (5.1)
are identified with coordinates of $l+1$ arbitrary points on
$\mathcal{V}\subset M$ not related to coordinates on transversal sections. By
Proposition 2.8 of [28], $D^{l}_{k}\overline{F}({\bf x}_{l})$ satisfies
Definition (1) for $(k-1)$ $\overline{\nu}_{G}({\bf x}^{\prime}_{k-1})$-forms
and has the $T_{G}$-derivative (4.1) property and the $K_{G}$-conjugation
(4.2) properties according of Subsection 5.2. So $D^{l}_{k}\overline{F}({\bf
x}_{l})\in C_{k-1}^{l+1}$ and $D^{l}_{k}$ is indeed a map whose image is in
$C_{k-1}^{l+1}$. In [28] we find the construction of double chain-cochain
complex for $n=1$ case and various $l\geq 0$. In particular, (c.f. Proposition
4.1), the chain condition for such double complex was proven. Consider now
$D^{l}_{k}\overline{F}({\bf x}_{l+1})$. By construction of the coboundary
operator in each component of the $\overline{F}$ a $n=1$ case of the action of
$D^{l}_{k}$ is realized. Thus, according to Proposition 4.1 of [28], each
component of $D^{l+1}_{k-1}\circ D^{l}_{k}$ vanishes. ∎

According to Proposition 2 one defines the $(l,k)$-th restricted meromorphic
function cosimplicial cohomology $H^{l}_{k,\;cos}(G,\mathcal{F})$ of a
foliation $\mathcal{F}$ of $M$ to be $H_{k,\;cos}^{l}(G,\mathcal{F})={\rm
Ker}\;D^{l}_{k}/\mbox{\rm Im}\;D^{l-1}_{k+1}$.

##  6\. The multiplication of elements of several double complex spaces
$C^{l}_{k}$

In this Section we fix $G$, $\mathcal{F}$, $\mathcal{V}$ and $\mathcal{U}$,
and skip them from further notations. In order to introduce and study
cohomology invariants associated to a foliation of codimension $p$, we first
have to define a multiplication among elements of several double complex
spaces $C^{l}_{k}$ for various $l$ and $k$. The simplest way to define such a
multiplication is to associate it to a sum of products of restricted
meromorphic functions over a basis in $G$, and powers of a complex parameter.
The formal parameters ${\bf z}_{l_{i}}$, $i\geq 1$ are identified with local
coordinates of $l_{i}$ points ${\bf p}_{l_{i}}$ on $M$. Some $r$ coordinates
of points among ${\bf p}_{l_{i}}$ may coincide with coordinates of points
among ${\bf p}_{l_{j}}$, $i\geq 1$, $j\geq 1$, $ik\neq j$. Similarly, on
transversal section, some $t$ coordinates of points may coincide ${\bf
p}^{\prime}_{l_{i}}$ with coordinates of points ${\bf p}^{\prime}_{l_{j}}$ In
that case we keep only one from each pair of coinciding parameters. As it
follows from definition of the configuration space $F_{ln}\mathbb{C}$ in
Subsection 5.2, in the case of coincidence of two formal parameters they are
excluded from $F_{ln}\mathbb{C}$. Thus, we require that the set of formal
parameters $(\widetilde{\widetilde{\bf z}}_{l_{1}+\ldots+l_{q}-r})$ would
belong to $F_{l_{1}+\ldots+l_{q}-r}\mathbb{C}$. This leads to the fall off of
the total number of formal parameters for
$\Theta(l_{1}+\ldots+l_{q}-r,k_{1}+\ldots+k_{s}-t)$, $q$, $s\geq 1$. Let
$\\{g_{m}\\}$ be a $G_{(m)}$-basis, and $\overline{u}_{m}$ be the dual of
$g_{m}$ with respect to a non-degenerate bilinear pairing $\left(.\ ,.\right)$
on $G$.

Let us introduce the multiplication of elements of $q$ double complex spaces
$C^{l_{i}}_{k_{i}}$, $1\leq i\leq q$, associated with the same foliation of
codimention $p$, with the image in another double complex space
$C^{l_{1}+\ldots+l_{1}-r}_{k_{1}+\ldots+k_{s}-t}$. We assume the same
$\mathcal{V}$ and $\mathcal{U}$ for these spaces. This multiplication is
coherent with respect to the original coboundary operator (5.2), and the
symmetry property (4.5). Recall that a $C_{l}^{k}$-space is defined by means
of $\overline{F}({\bf x}_{l_{i}})$ satisfying $L_{V}(0)$-conjugation,
$L_{V}(-1)$-derivative conditions, (4.5), and composable with $k$ vertex
operators. For $\overline{F}({\bf x}_{l_{i}})\in\Theta(l_{i},k_{i})$,
introduce the multiplication

$\displaystyle*_{q}:\times_{i=1}^{q}\Theta(l_{i},k_{i})\to\Theta(l_{1}+\ldots+l_{q}-r,k_{1}+\ldots+k_{s}-t),$
(6.1) $\displaystyle\overline{F}({\zeta_{a,i}};{\bf
x}_{l_{i}};q)=\sum\limits_{m\in\mathbb{Z}}\lambda^{m}\sum_{g_{m}\in
G_{(m)}}\prod_{i=1}^{q}\nu_{G}\left(\overline{F}({\bf
y}_{l_{i}}),\zeta_{2,i}\right),$ (6.2)

for $1\leq i\leq q$, $\lambda\in\mathbb{C}$, ${\bf y}_{l_{i}}=({\bf
x}_{l_{i}},(\overline{g}_{m},\zeta_{1,i}))$, where
$\overline{g}_{m}\in\widetilde{G}$. The dependence of (6.2) on $\lambda$ is
assumed via the relations $\zeta_{1,i}\zeta_{2,i}=\lambda$, $1\leq i\leq q$.
We require also the absolute convergence of the restricted meromorphic
functions given by

${\mathcal{M}}^{(i)}=\sum_{g_{m}\in G_{(m)}}\nu_{G}\left(\overline{F}({\bf
y}_{l_{i}}),\zeta_{2,i}\right),$ (6.3)

in powers of $\lambda$ with respect to a norm with some radiuses of
convergence $R_{a,i}$ for $|\zeta_{a,i}|\leq R_{a,i}$, $1\leq i\leq q$. By the
standard reasoning $\overline{F}({\zeta_{a,i}};{\bf x}_{l_{i}};q)$ does not
depend on the choice of a basis of $\\{g_{m}\\}$ of $G_{(m)}$,
$m\in\mathbb{Z}$. In the next subsection we prove that the multiplication
(6.2) converges as a series in $\lambda$.

###  6.1. The proof of the multiplication convergence

In this subsection we prove that the multiplication of several restricted
meromorphic functions defined above converges subject to converging of
indivicual elements related to $\overline{F}({\bf x}_{l_{i}})$. Consider the
product (6.2),

$\displaystyle||\overline{F}(\zeta_{a,i};{\bf
x}_{i};q)||=\left|\left|\sum\limits_{m\in\mathbb{Z}}\lambda^{m}\sum_{g_{m}\in
G_{(m)}}\prod_{i=1}^{q}\nu_{G}\left(\overline{F}({\bf
y}_{l_{i}}),\zeta_{2,i}\right)\right|\right|$
$\displaystyle\qquad\qquad\qquad\qquad=\left|\left|\sum\limits_{m\in
Z}\lambda^{m}\left(\overline{F}(\zeta_{a,i};{\bf
x}_{i};q)\right)_{m}\right|\right|$
$\displaystyle=\left|\left|\sum_{m\in\mathbb{Z}}\sum_{g_{m}\in
V_{m}}\sum\limits_{n\in\mathbb{C}}\lambda^{m-n-1}\;\prod_{i=1}^{q}\left({\mathcal{M}}^{(i)}\right)_{n}\right|\right|,$
(6.4)

as a formal series in $\lambda$ for $|\zeta_{a,i}|\leq R_{a,i}$,
$\left|\lambda\right|\leq R_{1,i}R_{2,i}$. For
$R_{i}=max\left\\{R_{1,i},R_{2,i}\right\\}$, we apply Cauchy’s inequality to
the coefficient forms (6.3) to find

$\left|\left|\left({\mathcal{M}}^{(i)}\right)_{n}\right|\right|\leq{M_{i}}{R_{i}^{-n}},$
(6.5)

with $M_{i}=\sup_{\left|\zeta_{a,i}\right|\leq R_{i},\left|\lambda\right|\leq
r}\left|\left|\mathcal{M}^{(i)}\right|\right|$. Using (6.5) we obtain for
(6.4) for $M=\min\left\\{M_{1},\ldots,M_{q}\right\\}$, and
$R=\max\left\\{R_{1},\ldots,R_{q}\right\\}$,

$\displaystyle\left|\left|\left(\overline{F}(\zeta_{a,i};{\bf
x}_{i};q)\right)_{m}\right|\right|\leq\prod_{i=1}^{q}\left|\left|\left(\mathcal{M}^{(i)}\right)_{n}\right|\right|\leq\prod_{i=1}^{q}M_{i}R_{i}^{-m+n+1}\leq
M{R^{-m+n+1}}.$

We see that (6.2) is absolute convergent as a formal series in $\lambda$. The
extra poles can only appear at $z_{i,j}=z_{i^{\prime},j^{\prime}}$, $1\leq
i\neq i^{\prime}\leq q$, $1\leq j\leq l_{i}-r_{i}$, $1\leq j^{\prime}\leq
l_{i^{\prime}}-r_{{}^{\prime}i}$, where $r_{i}$ denotes the number of excluded
parameters in for $C^{l_{i}-r_{i}}_{m_{i}-t_{i}}$.

###  6.2. Properties of $*_{q}$-multiplication of $C^{l_{i}}_{k_{i}}$-spaces

We study properties of the multiplication of several elements of the double
complex spaces $C^{l_{i}}_{k_{i}}$, $1\leq i\leq q$.

According to the previous subsection, the result of multiplication it as an
absolutely converging function in $\lambda$ on the configuration space
$F\mathbb{C}_{(l_{1}+\ldots+l_{q}-r)n}$ of ${\bf z}_{l_{i}}$, $1\leq i\leq q$,
with only possible poles at $z_{i}=z_{i^{\prime}}$,
$\widetilde{z}_{j}=\widetilde{z}_{j^{\prime}}$, and $z_{i}=\widetilde{z}_{j}$,
$1\leq i,i^{\prime}\leq l_{1}$, $1\leq j,j^{\prime}\leq l_{2}-r$, with
excluded $\widehat{z}_{l_{j}}$, and parametrized by
$\zeta_{a,i}\in\mathbb{C}$, with all monomials $(z_{i_{l}}-z_{j_{l}})$, $1\leq
l\leq r$, excluded from (6.2). We will omit $\zeta_{a,i}$ from further
notations for $\overline{F}(\zeta_{a,i};{\bf x}_{l_{i}};q)$. We define the
action of differentiation
$\partial_{i,j}=\partial_{z_{i,j}}={\partial}/{\partial_{z_{l_{i},j}}}$,
$1\leq i\leq q$, where $r_{i}$ is the number of skipped parameters for in the
$i$-th space, on $\overline{F}({\bf x}_{l_{i}};q)$ with respect to the $j$-th
entry, $1\leq j\leq l_{i}$, of the $i$-th set $({\bf x}_{l_{i}})$ of
parameters, $1\leq i\leq q$, as follows

$\partial_{i,j}\overline{F}({\bf
x}_{l_{i}};q)=\sum_{s=1}^{q}\sum_{m=1}^{l_{s}}\partial^{\delta_{i,s}\delta_{m,i}}_{z_{m,s}}\nu_{G}\left(\overline{F}({\bf
x}_{l_{i}}),\zeta_{i}\right).$

We then define the action of an element $\sigma\in S_{q}$ on the
multiplication of $\overline{F}({\bf
x}_{l_{i}};q)\in\Theta(l_{1}+\ldots+l_{q}-r,k_{1}+\ldots+k_{s}-t)$, as

$\displaystyle\sigma(\overline{F})\left({\bf
x}_{l_{i}};q\right)=\overline{F}\left({\bf
x}_{(\sigma(l_{i}))},q\right)=\nu_{G}\left(\overline{F}\left({\bf
x}_{(\sigma(l_{i}))}\right),\zeta_{i}\right).$ (6.6)

It is elementary to check the following

######  Lemma 4.

The multiplication (6.2) satisfies the $T_{G}$-derivative (4.2), $K_{G}$\-
conjugation (4.4), (4.5) properties, and Definition 1. ∎

Therefore, the definition of multiplication of several restricted meromorphic
functions multiplication results in a restricted meromorphic function and we
obtain

######  Proposition 3.

For $\overline{F}({\bf x}_{l_{i}})\in C^{l_{i}}_{k_{i}}$ the multiplication
$\overline{F}\left({\bf x}_{l_{i}};q\right)$ (6.6) belongs to the space
$C^{l_{1}+\ldots+l_{q}-r}_{k_{1}+\ldots+k_{s}-t}$, i.e.,
$*_{q}:\times_{i=1}^{q}C^{l_{i}}_{k_{i}}\to
C_{k_{1}+\ldots+k_{q}-t}^{l_{1}+\ldots+l_{s}-r}$. ∎

###  6.3. Coboundary operator acting on the multiplication space

Since the result of multiplication (6.6) of elements of the spaces
$C_{k_{i}}^{l_{i}}$, $1\leq i\leq q$, belongs to
$C^{l_{1}+\ldots+l_{q}-r}_{k_{1}+\ldots+k_{s}-t}$, thus the multiplication
admits the action of $D^{l_{1}+\ldots+l_{q}-r}_{k_{1}+\ldots+k_{s}-t}$ defined
in (5.2). The coboundary operator (5.2) possesses a version of Leibniz law
with respect to the multiplication (6.6). Indeed, by elementary computation we
get

######  Lemma 5.

For $\overline{F}({\bf x}_{l_{i}})\in C_{k_{i}}^{l_{i}}$, $1\leq i\leq q$, the
action of $D_{k_{1}+\ldots+k_{s}-t}^{l_{1}+\ldots+l_{q}-r}$ on the
multiplication is given by
$D_{k_{1}+\ldots+k_{s}-t}^{l_{1}+\ldots+l_{q}-r}\left(*_{q}\overline{F}({\bf
x}_{l_{i}})\right)=*_{q}(-1)^{l_{i}-r_{i}}\left(D^{l_{i}-r_{i}}_{k_{i}-s_{i}}\overline{F}({\bf
x}_{l_{i}-r_{i}})\right)$. ∎

######  Remark 1.

Checking (5.2) we see that an extra arbitrary element $g_{l_{i}+1}\in G$, as
well as corresponding extra arbitrary formal parameter $z_{l_{i}+1}$ appear as
a result of the action of $D^{l_{i}-r_{i}}_{k_{i}-s_{i}}$, $1\leq i\leq q$, on
$\overline{F}({\bf x}_{l_{i}-r_{i}})\in C^{l_{i}-r_{i}}_{k_{i}-s_{i}}$ mapping
it to $C^{l_{i}-r_{i}+1}_{k_{i}-s_{i}-1}$.

###  6.4. Relations to ${\bf\check{C}}$ech-de Rham cohomology in
Crainic–Moerdijk construction

Recall the construction of the ${\rm\check{C}}$ech-de Rham cohomology by
Crainic and Moerdijk [11] for a foliation $\mathcal{F}$ of codimension $p$ on
a smooth manifold $M$. Let $\mathcal{U}$ be a transversal basis for
$\mathcal{F}$. Consider the double complex
$C^{k,l}=\prod_{U_{0}\stackrel{{\scriptstyle
h_{1}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
h_{p}}}{{\longrightarrow}}U_{k}}\omega^{l}(U_{0})$, where the multiplication
ranges over all $k$-tuples of holonomy embeddings between transversal sections
from a fixed transversal basis $\mathcal{U}$. The vertical differential is
defined as $(-1)^{k}d:C^{k,l}\to C^{k,l+1}$, where $d$ is the ordinary de Rham
differential. The horizontal differential $\delta:C^{k,l}\to C^{k+1,l}$, is
given by $\delta=\sum\limits_{i=1}^{k}(-1)^{i}\delta_{i}$, where
$\delta_{i}\omega({\bf h}_{k+1})=\delta_{i,0}\;h_{1}^{*}\;\omega({\bf
h}_{(2,k+1)})+(1-\delta_{i,0}-\delta_{i,k+1})\;\omega({\bf
h}_{(1,i-1)},h_{i+1}h_{i},{\bf h}_{(i+2,k+1)})+\delta_{i,k+1}\omega({\bf
h}_{k})$, expressed in terms of differential forms. This double complex
constitutes a bigraded differential algebra endowed with a natural
multiplication $(\omega\;\eta)({\bf
h}_{k+k\,^{\prime}})=(-1)^{kk\,^{\prime}}\;\omega({\bf h}_{k})\;({\bf
h})_{k}^{*}\;.\eta\left({\bf h}_{(k+1,k+k^{\prime})}\right)$, for $\omega\in
C^{k,l}$ and $\eta\in C^{k^{\prime},l^{\prime}}$, thus $(\omega\cdot\eta)({\bf
h}_{k+k^{\prime}})\in C^{k+k^{\prime},l+l^{\prime}}$. The cohomology of this
complex is called the ${\rm\check{C}}$ech-de Rham cohomology of the leaf space
$M/\mathcal{F}$ with respect to the transversal basis $\mathcal{U}$. The
${\rm\check{C}}$ern-de Rham cohomology of a foliated smooth manifold
introduced in [11] results from restricted meromorphic function cohomology
introduced in this paper. Indeed, it can be seen by making the following
associations: $h_{i}\sim g_{i},\;i=1,\ldots,{ln}$, $\omega({\bf h}_{l.n})\sim
F({\bf x}_{l})$, $h^{*}(h_{1})\ldots h^{*}(h_{l.n})({\bf
z}_{ln})\sim{\bf\omega}_{G}({\bf z}_{l,n})$.

##  7\. Examples: invariants of foliations

In this Section, using the double complex construction of Section 5, we find
cohomology invariants of foliations, in particular, a generalization of the
Godbillon–Vey invariant [20] for codimension one foliations. We call a map
$\overline{F}^{l}_{k}\in C_{k}^{l}$ closed if
$D^{l}_{k}\overline{F}^{l}_{k}=0$. For $k\geq 1$. It is exact if there exists
$\overline{F}_{k-1}^{l+1}\in C_{k-1}^{l+1}$ such that
$\overline{F}_{k-1}^{l+1}=D^{l}_{k}\overline{F}^{l}_{k}$. Taking into account
the correspondence with the ${\rm\check{C}}$ech-de Rham complex due to [11],
we reformulate the derivation of a generalization of Godbillon–Vey invariant
in restricted meromorphic functions terms. For $\overline{F}^{l}_{k}\in
C^{l}_{k}$ we call the cohomology class of mappings
$\left[\overline{F}^{l}_{k}\right]$ the set of all closed forms that differs
from $\overline{F}^{l}_{k}$ by an exact mapping, i.e., for
$\overline{F}^{l-1}_{k+1}\in C^{l-1}_{k+1}$,
$\left[\overline{F}^{l}_{k}\right]=\overline{F}^{l}_{k}+D^{l-1}_{k+1}\overline{F}^{l-1}_{k+1}$,
assuming that both parts of the last formula belongs to the same space
$C^{l}_{k}$.

###  7.1. Example: the general case of $q=2$

For $\Phi^{k}_{m}\in C^{k}_{m}$ and $\Phi^{k^{\prime}}_{m^{\prime}}\in
C^{k^{\prime}}_{m^{\prime}}$, let us introduce the commutator
$\Phi^{k}_{m}*\Phi^{k^{\prime}}_{m^{\prime}}=[\Phi^{k}_{m},_{*_{2}}\Phi^{k^{\prime}}_{m^{\prime}}]_{-}$
with respect to the multiplication $*_{2}$. Next, we obtain the main result of
this paper

######  Theorem 1.

The orthogonality condition for elements of double complex spaces provides for
elements of the double complex spaces of (5.1) the non-vanishing cohomology
invariants of the form
$\left[\left(D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}\right)*\Phi^{n_{0}}_{m_{0}}\right]$,
$\left[\left(D^{n}_{m}\Phi^{n}_{m}\right)*\Phi^{n}_{m}\right]$,
$\left[\left(D^{n_{i}}_{m_{i}}\Phi^{n_{i}}_{m_{i}}\right)*\Phi^{n_{i}}_{m_{i}}\right]$,
for $i=1,\ldots,l$, for some $l\in\mathbb{N}$, with non-vanishing
$\left(D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}\right)*\Phi^{n_{0}}_{m_{0}}$,
$\left(D^{n}_{m}\Phi^{n}_{m}\right)*\Phi^{n}_{m}$, and
$\left(D^{n_{i}}_{m_{i}}\Phi^{n_{i}}_{m_{i}}\right)*\Phi^{n_{i}}_{m_{i}}$.
These classes are independent on the choices of $\Phi^{n_{0}}_{m_{0}}\in
C^{n_{0}}_{m_{0}}$, $\Phi^{n}_{m}\in C^{n}_{m}$, and $\Phi^{n_{i}}_{m_{i}}\in
C^{n_{i}}_{m_{i}}$.

###### Proof.

For $q=2$ let us consider the most general case. For non-negative $n_{0}$,
$n$, $n_{1}$, $m_{0}$, $m$, $m_{1}$, let $\Phi^{n_{0}}_{m_{0}}\in
C^{n_{0}}_{m_{0}}$, $\Phi^{n}_{m}\in C^{n}_{m}$, and $\Phi^{n_{1}}_{m_{1}}\in
C^{n_{1}}_{m_{1}}$. For $\Phi^{n}_{m}$ and $\Phi^{n_{1}}_{m_{1}}$, let $r_{0}$
be the number of common vertex algebra elements (and formal parameters), and
$t_{0}$ be the number of common vertex operators $\Phi^{n}_{m}$ and
$\Phi^{n_{1}}_{m_{1}}$ are composable to. Note that we assume $n$, $n_{1}\geq
r_{0}$, $m$, $m_{1}\geq t_{0}$. Taking into account the orthogonality
condition

$\Phi^{n}_{m}*D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=0,$

implies that there exist $C^{n_{1}}_{m_{1}}\in C^{n_{1}}_{m_{1}}$, such that
$D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=\Phi^{n}_{m}*\Phi^{n_{1}}_{m_{1}}$.
From the last equations we obtain $n_{0}+1=n+n_{1}-r_{0}$,
$m_{0}-1=m+m_{1}-t_{0}$. Note that we have extra conditions following from the
last identities: $n_{0}+1\geq 0$, $m_{0}-1\geq 0$. The conditions above for
indexes express the double grading condition for the double complex (5.1). As
a result, we have a system in integer variables satisfying the grading
conditions above. Consequently acting by corresponding coboundary operators we
obtain the following relations:

$\displaystyle\Phi^{n}_{m}*D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=0,$ (7.1)
$\displaystyle
D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=\Phi^{n}_{m}*\Phi^{n_{1}}_{m_{1}},$
$\displaystyle
D^{n}_{m}\Phi^{n}_{m}*\Phi^{n_{1}}_{m_{1}}+(-1)^{n}\Phi^{n}_{m}*D^{n_{1}}_{m_{1}}\Phi^{n_{1}}_{m_{1}}=0,$
$\displaystyle D^{n}_{m}\Phi^{n}_{m}*D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=0,$
$\displaystyle
D^{n_{0}}_{m_{0}}\Phi^{n_{0}}_{m_{0}}=D^{n}_{m}\Phi^{n}_{m}*\Phi^{n_{2}}_{m_{2}},$
$\displaystyle
D^{n}_{m}\Phi^{n}_{m}*D^{n_{i}}_{m_{i}}\Phi^{n_{i}}_{m_{i}}=0,\;\;$
$\displaystyle
D^{n_{i}}_{m_{i}}\Phi^{n_{i}}_{m_{i}}=D^{n}_{m}\Phi^{n}_{m}*\Phi^{n_{i+1}}_{m_{i+1}},$

where $\Phi^{n_{i}}_{m_{i}}\in C^{n_{i}}_{m_{i}}$, and $n_{i}$, $m_{i}$,
$i\geq 2$ satisfy relations $n_{i}=n+n_{i+1}-r_{i+1}$,
$m_{i}=m+m_{i+1}-t_{i+1}$. The sequence of relations (7.1) does not cancel
until the conditions on indexes given above fulfill. Thus, we see that the
orthogonality condition for the double complex together with the action of
coboundary operator $D^{n}_{m}$, and the multiplication (6.6), provides
invariants of the theorem.

Let $\Phi^{u}_{v}$, $\phi$ be one of generators $\Phi^{n_{0}}_{m_{0}}$,
$\Phi^{n}_{m}$, $\Phi^{n_{i}}_{m_{i}}$, $1\leq i\leq l$,
$(u,v)=(n_{0},m_{0})$, $(n,m)$, $(n_{i},m_{i})$. Let us show now the non-
vanishing property of
$\left(\left(D^{n}_{m}\Phi^{u}_{v}\right)*\Phi^{u}_{v}\right)$. Indeed,
suppose $\left(D^{n}_{m}\Phi^{u}_{v}\right)*\Phi^{u}_{v}=0$. Then there exists
$\gamma\in C^{n^{\prime}}_{m^{\prime}}$, such that
$D^{n}_{m}\Phi^{u}_{v}=\gamma*\Phi^{u}_{v}$. Both sides of the last equality
should belong to the same double complex space but one can see that it is not
possible since we obtain $m^{\prime}=t-1$, i.e., the number of common vertex
operators for the last equation is greater than for one of multipliers. Thus,
$\left(D^{n}_{m}\Phi^{u}_{v}\right)*\Phi^{u}_{v}$ is non-vanishing. By the
substitution $\Phi^{u}_{v}\mapsto\left(\Phi^{u}_{v}+\Phi\right)$,
$\Phi^{n}_{m}\in C^{n}_{m}$ for it is easy to see that
$\left[\left(D^{n}_{m}\Phi^{u}_{v}\right)*\Phi^{u}_{v}\right]$ is invariant,
i.e., it does not depend on the choice of $\Phi^{n}_{m}$. ∎

###  7.2. Example: codimention one foliation of a three-dimensional manifold

For a three-dimensional smooth complex manifold, consider a codimension one
foliation $\mathcal{F}$. Following the construction of Definition (5.1), we
take $k$-tuples of one-dimensional transversal sections. For $g_{j}\in G$,
$w_{j}\in U_{j}$, for each section we attach the form $\nu_{G}(x_{j})$,
$x_{j}=(g_{j},w_{j})$. We then work with mappings $\varphi\in C_{k}^{3l}$. As
in the setup of differential forms, a mapping $\varphi\in C_{k}^{3l}$ is
associated to a codimension one foliation. As we see from the definition of
the action of the derivative, it satisfies properties similar to differential
forms. The integrability condition for mapping $\overline{F}^{l}_{k}\in
C^{l}_{k}$ and $D^{l}_{k}\;\overline{F}_{0}\in C^{l+1}_{k_{1}}$ has the form
$\overline{F}_{0}*D^{l}_{k}\;\overline{F}_{0}=0$. It results with the
Frobenius theorem, i.e., that there exist
$\overline{F}_{2}\in{C}^{l^{\prime}}_{k^{\prime}}$, such that
$D^{l}_{k}{\overline{F}_{0}}=\overline{F}_{0}*\overline{F}_{2}$, which
uniquely determines a foliation with parameters of $\nu_{G}$-forms satisfying
Definition 1 conditions. In this case we obtain a generalization of
Godbillon–Vey invariant in terms of the cohomology classes
$\left[\left(D^{l}_{k}\Delta\right)\right.$ $*$ $\left.\Delta\right]$, for
$\Delta=\overline{F}$, $\overline{F}_{1}$, $\overline{F}^{\prime}$, and
combinations $(l,k)=(1,2)$, $(0,3)$, $(1,t)$ correspondingly. The case $t=1$
corresponds to the classical Godbillon–Vey invariant. Examples of higher $q$
will be considered elsewhere.

## Acknowledgements

The author would like to thank Ya. V. Bazaikin and A. Galaev, for related
discussions. The author’s research of the author is supported by the Academy
of Sciences of the Czech Republic (RVO 67985840).

## References

  * [1] D. W. Anderson, Chain functors and homology theories. Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971), pp. 1–12. Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971.
  * [2] Bazaikin, Ya. V., Galaev, A. S.: Losik classes for codimension one foliations, J. Inst. Math. Jussieu 21 (2022), no. 4, 1391–1419.
  * [3] Ya. V. Bazaikin, A. S. Galaev, and P. Gumenyuk. Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class, Math. Z. (2022), no. 2, 1335–1349.
  * [4] Bazaikin, Ya. V., Galaev, A. S., Zhukova, N. I.: Chaos in Cartan foliations. Chaos 30 (2020), no. 10, 103116.
  * [5] R. Bott, G.Segal, The cohomology of the vector fields on a manifold, Topology Volume 16, Issue 4, 1977, Pages 285–298.
  * [6] R. Bott, Lectures on characteristic classes and foliations. Springer LNM 279 (1972), 1–94.
  * [7] R. Bott and A. Haefliger, On characteristic classes of $\Gamma$-foliations, Bull. Amer. Math. Soc. 78 (1972), 1039–1044.
  * [8] Frenkel, E.; Ben-Zvi, D. Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88\. American Mathematical Society, Providence, RI, 2001. xii+348 pp.
  * [9] I. N. Bernstein and B. I. Rozenfeld, Homogeneous spaces of infinite-dimensional Lie algebras and characteristic classes of foliations, Rus. Math. Surv. 28 (1973), no. 4, 107–142.
  * [10] C. Chevalley and S. Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.
  * [11] M. Crainic and I. Moerdijk, Čech-De Rham theory for leaf spaces of foliations. Math. Ann. 328 (2004), no. 1–2, 59–85.
  * [12] B. L. Feigin and D. B. Fuchs, Cohomology of Lie groups and Lie algebras, Itogi NT, Current problems in mathematics, fundamental directions, 1998.
  * [13] Feigin, B. L.: Conformal field theory and Cohomologies of the Lie algebra of holomorphic vector fields on a complex curve. Proc. ICM, Kyoto, Japan, 71-85 (1990)
  * [14] D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultunt Bureau, New York, 1986.
  * [15] D. B. Fuchs, Characteristic classes of foliations. Russian Math. Surveys, 28 (1973), no. 2, 1–16.
  * [16] Galaev A. S. Comparison of approaches to characteristic classes of foliations, arXiv:1709.05888
  * [17] Galaev, A. S., Zhukova, N. I.: Attractors of Cartan foliations, arXiv:1703.07597.
  * [18] I. M. Gelfand and D. B. Fuchs, Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold. I, II, Funktional. Anal, i Prilozen. 3 (1969), no. 3, 32-52; ibid. 4 (1970), 23-32.
  * [19] Gerstenhaber, M., Schack, S. D.: Algebraic Cohomology and Deformation Theory, in: Deformation Theory of Algebras and Structures and Applications, NATO Adv. Sci. Inst. Ser. C 247, Kluwer Dodrecht 11-264 (1988)
  * [20] Ghys E. L’invariant de Godbillon-Vey. Seminaire Bourbaki, 41–eme annee, n 706, S. M. F. Asterisque 177–178 (1989)
  * [21] Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and multiplications. Translated from the Russian. Eighth edition, Elsevier Academic Press, Amsterdam, 2015. 1133 pp.
  * [22] A. Haefliger Homotopy and integrability. 1971 Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School) pp. 133–163 Lecture Notes in Mathematics, Vol. 197 Springer, Berlin
  * [23] Haefliger, A.: Sur la cohomologie de l’algèbre de Lie des champs de vecteurs. Ann. Sci. ENS, $4^{\grave{e}me}$ série, t. 9, 503-532 (1976)
  * [24] D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962), 191–204.
  * [25] B. Hennion, M. Kapranov Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras, arXiv:1811.05032, 2018
  * [26] Hinich, V., Schechtman, V.: Deformation Theory and Lie algebra Homology I. I. Algebra Colloq. 4 (1997), no. 2, 213–240; II no. 3, 291–316.
  * [27] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67.
  * [28] Huang Y.-Zh. A cohomology theory of grading-restricted vertex algebras. Comm. Math. Phys. 327 (2014), no. 1, 279-307.
  * [29] Kawazumi, N.: On the complex analytic Gel’fand-Fuks cohomology of open Riemann surfaces. Ann. Inst. Fourier, Grenoble 43, 3, 655-712 (1993)
  * [30] Khoroshkin, A. S. Characteristic classes of flags of foliations and Lie algebra cohomology. Transform. Groups 21 (2016), no. 2, 479–518.
  * [31] Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Springer Grundlehren 283 Berlin Heidelberg New York 1986
  * [32] H. B. Lawson, Foliations. Bull. Amer. Math. Soc. 80 (1974), no. 3, 369–418.
  * [33] Losik, M. V.: Orbit spaces and leaf spaces of foliations as generalized manifolds, arXiv:1501.04993.
  * [34] S. P. Novikov. The topology of foliations. (Russian) Trudy Moskov. Mat. Obsch. 14 1965 248–278.
  * [35] S. P. Novikov. Topology of generic Hamiltonian foliations on Riemann surfaces. Mosc. Math. J. 5 (2005), no. 3, 633–667, 743.
  * [36] S. P. Novikov. Topology of Foliations given by the real part of holomor- phic one-form, preprint, 2005, arXive math.GT/0501338.
  * [37] Patras, F., Thomas, J.-C. Cochain algebras of mapping spaces and finite group actions. Topology Appl. 128 (2003), no. 2-3, 189–207.
  * [38] Quillen, D.: Rational Homotopy Theory. App. B, Ann. Math. (2) 90, 205-295 (1969)
  * [39] Schlichenmaier, M. Krichever-Novikov type algebras. An introduction. Lie algebras, Lie superalgebras, vertex algebras and related topics, 181-220, Proc. Sympos. Pure Math., 92, Amer. Math. Soc., Providence, RI, 2016.
  * [40] Schlichenmaier, M., Krichever-Novikov algebras for more than two points, Lett. Math. Phys. 19(1990), 15-165.
  * [41] Krichever-Novikov algebras for more than two points: explicit generators, Lett. Math. Phys. 19(1990), 327-336.
  * [42] Sheinman, O. K. Global current algebras and localization on Riemann surfaces. Mosc. Math. J. 15 (2015), no. 4, 833–846.
  * [43] Smith, S. B. The homotopy theory of function spaces: a survey. Homotopy theory of function spaces and related topics, 3–39, Contemp. Math., 519, Amer. Math. Soc., Providence, RI, 2010.
  * [44] Wagemann, F.: Differential graded cohomology and Lie algebras of holomorphic vector fields. Comm. Math. Phys. 208 (1999), no. 2, 521–540