# A local-to-global analysis of Gelfand-Fuks cohomology

Lukas Miaskiwskyi

###### Abstract.

We present a novel proof technique to construct the Gelfand-Fuks spectral
sequence for diagonal Chevalley-Eilenberg cohomology of vector fields on a
smooth manifold, performing a local-to-global analysis through a notion of
generalized good covers from the theory of factorization algebras and
cosheaves. This approach yields a unified way to deal with the problem of
comparing “sheaf-like” data over different Cartesian powers of the manifold,
and is easily generalized to the study of other cohomology theories associated
to geometric objects. Independently, we lay out a detailed and easily
accessible exposition on the continuous Chevalley-Eilenberg cohomology of
formal vector fields and of vector fields on Euclidean space, modernizing and
elaborating on well-established literature on the subject by Bott, Fuks, and
Gelfand.

###### Contents

  1. 1 Introduction
  2. 2 The Lie algebra of formal vector fields
    1. 2.1 Definition and first properties
    2. 2.2 Stable cohomology of formal vector fields
    3. 2.3 A spectral sequence for the cohomology of formal vector fields
  3. 3 Gelfand–Fuks cohomology on Euclidean space
    1. 3.1 Definitions and calculation
    2. 3.2 Gelfand–Fuks cohomology of Euclidean space
  4. 4 Cosheaf-theoretic aspects of Gelfand-Fuks cohomology
  5. 5 Gelfand–Fuks cohomology for smooth manifolds
    1. 5.1 Diagonal Filtration
    2. 5.2 Generalized good covers
    3. 5.3 The Čech-Bott-Segal double complex
    4. 5.4 Spectral sequences for diagonal cohomology
  6. A Cosheaves and Čech homology
  7. B The Hochschild-Serre Spectral Sequence for locally convex Lie algebras

##  1\. Introduction

Beginning around fifty years ago, a plethora of literature has been created to
understand the continuous Chevalley-Eilenberg cohomology $\mathfrak{X}(M)$ of
the vector fields on a smooth manifold $M$. This cohomology carries the name
_Gelfand–Fuks cohomology_ , in reference to the authors who opened the
investigation of this subject with a series of highly novel papers [1, 2, 3].
Initially, it was hoped that this cohomology might contain invariants for the
smooth structure of $M$, hence be a potential tool for a classification of
smooth structures on a given topological manifold.
Unfortunately, these hopes were denied by a paper by Bott and Segal, which
showed that the Gelfand–Fuks cohomology was isomorphic to the singular
cohomology of a mapping space that can be functorially constructed from $M$,
and from which no new invariants arise [4]. Regardless, these explorations
brought with them a lot of applications, for example in the theory of
foliations [5] or for the construction of the Virasoro algebra [6]. Further,
many related open problems are still being pursued, like the continuous
cohomology of the Lie algebra of symplectic, Hamiltonian or divergence-free
vector fields on symplectic/Riemannian manifolds [7, 8].
The goal of this document is two-fold: For one, we want to present a novel
proof technique, using so-called _$k$ -good covers_ from the theory of functor
calculus and factorization algebras (cf. [9]), to construct well-known
spectral sequences which calculate the Gelfand–Fuks cohomology of certain
smooth manifolds, cf. [10, Theorem 2.4.1a, 2.4.1.b]. This approach is inspired
by the treatment of Gelfand–Fuks cohomology in the framework of factorization
algebras in a preprint by Kapranov and Hennion [11]. The idea is to use a
local-to-global analysis to reconstruct the spectral sequence for the
Gelfand–Fuks cohomology of the manifold from the Gelfand–Fuks cohomology of a
local part. The use of $k$-good covers solves the problem that this
reconstruction necessarily compares data between different Cartesian powers
$M,M^{2},M^{3},\dots$ of a manifold $M$. This is a natural problem to
encounter here, since cochains for Gelfand–Fuks cohomology are maps on
multiple copies of $\mathfrak{X}(M)$. An advantage of this approach over
previous proofs is that it is easily generalizable to other situations, as we
plan to show in future work [12].
The other goal of this document is to lay out a streamlined, detailed and
relatively elementary path to the fundamental results of Gelfand–Fuks
cohomology, accessible to any researcher with a solid understanding of
homological algebra, sheaf theory and differential geometry. To this end, we
largely follow the general strategies in [10, 13], filling in nontrivial
details that have been left to the reader in the original literature,
modernizing some of the language used, and replacing some of the arguments
with ones which the author perceives as clearer. We make no a claim to
exhaustiveness: we restrict ourselves to Gelfand–Fuks cohomology with trivial
coefficients, and direct the reader to [14] for an overview of the study of
other coefficient modules.

We begin in Section 2 with a study of continuous cohomology of the Lie algebra
of formal vector fields, i.e. vector fields whose coefficient functions are
formal power series. They represent the infinitesimal counterpart of
$\mathfrak{X}(M)$ and their cohomology is calculated using a spectral sequence
over which one can get full control. This section is largely a review of [3].
In Section 3, we tie the cohomology of formal vector fields to the
Gelfand–Fuks cohomology of Euclidean space, which may itself be understood as
the local counterpart to Gelfand–Fuks cohomology. This section is a review of
[13]. The Section 4, we examine the transformation behavior of Gelfand–Fuks
cohomology on Euclidean space under diffeomorphisms. The proofs and
formulations are original, though the results are implicitly used in the
literature. This prepares the local-to-global analysis of the Gelfand–Fuks
cohomology on an arbitrary smooth manifold, which we carry out in Section 5.
We give a variation of the well-known spectral sequences that calculate
Gelfand–Fuks cohomology for a class of orientable, smooth manifolds. For the
sake of completion, we explain how it allows a full calculation of the
Gelfand–Fuks cohomology of the circle $S^{1}$ and may be used to make certain
general statements about finite-dimensionality of the Gelfand–Fuks cohomology.
Acknowledgement. The author would like to thank Bas Janssens, Tobias Diez, and
Milan Niestijl for interesting discussions and many useful suggestions. He was
supported by the NWO Vidi Grant 639.032.734 “Cohomology and representation
theory of infinite-dimensional Lie groups”.

##  2\. The Lie algebra of formal vector fields

In this section, we mainly elaborate on the methods given in [10, Chapter 2.2]
and [3] to analyze the Lie algebra of formal vector fields, an infinitesimal
version of the Lie algebra of vector fields on a smooth manifold. We also use
methods from [15, Corollary 1] to analyze stable Chevalley-Eilenberg
cohomology of this space.

###  2.1. Definition and first properties

######  Definition 2.1 (Formal vector fields).

Let $n\in\mathbb{N}$. We define the _Lie algebra of formal vector fields_
$W_{n}$ to be equal to the topological Lie algebra

(2.1) $\displaystyle W_{n}:=\mathbb{R}\llbracket
x_{1},\dots,x_{n}\rrbracket\otimes\mathbb{R}^{n}.$

Its topology is induced by the projective limit topology of
$\mathbb{R}\llbracket x_{1},\dots,x_{n}\rrbracket$ and its Lie bracket is
given by

(2.2) $\displaystyle[f\partial_{i},g\partial_{j}]:=f\frac{\partial g}{\partial
x_{i}}\cdot\partial_{j}-g\frac{\partial f}{\partial
x_{j}}\cdot\partial_{i},\quad f,g\in\mathbb{R}\llbracket
x_{1},\dots,x_{n}\rrbracket.$

######  Remark 2.2.

There is a more geometric definition of $W_{n}$ which we will use in Section
3, as the space of infinity-jets $J_{p}^{\infty}\mathfrak{X}(\mathbb{R}^{n})$
of vector fields at an arbitrary point $p\in\mathbb{R}^{n}$. Any choice of
local frame around $p$ induces a continuous Lie algebra isomorphism

(2.3) $\displaystyle J_{p}^{\infty}\mathfrak{X}(\mathbb{R}^{n})\cong W_{n}.$

We first examine the structure of $W_{n}$.

######  Definition 2.3.

The element $E:=\sum_{i=1}^{n}x_{i}\partial_{i}\in\mathfrak{g}_{0}$ is the
_Euler vector field_ in $W_{n}$. The eigenspaces

(2.4) $\displaystyle\mathfrak{g}_{k}=\\{X\in W_{n}:[E,X]=k\cdot X\\}$

give $W_{n}=\widehat{\bigoplus}_{k\geq\mathbb{Z}}\mathfrak{g}_{k}$ as the
completion of a graded Lie algebra. Elements of $\mathfrak{g}_{k}$ are called
_homogeneous (of degree $k$)_.

More explicitly, we find for all $k\in\mathbb{Z}$

(2.5)
$\displaystyle\mathfrak{g}_{k}=\left\\{\sum_{i=1}^{n}p_{i}\partial_{i}\in
W_{n}:p_{i}\text{ homogeneous polynomials of degree }k+1\right\\}.$

In particular, ${\operatorname{\mathfrak{g}}}_{k}=0$ if $k<-1$, and in low
orders, we have Lie algebra isomorphisms:

(2.6)
$\displaystyle\mathfrak{g}_{-1}=\operatorname{span}\left\\{\partial_{i}:i=1,\dots,n\right\\}\cong\mathbb{R}^{n},\quad\mathfrak{g}_{0}=\operatorname{span}\left\\{x_{i}\partial_{j}:i,j=1,\dots,n\right\\}\cong\mathfrak{gl}_{n}(\mathbb{R}).$

######  Definition 2.4.

Let ${\operatorname{\mathfrak{g}}}$ be a topological Lie algebra. Its
_continuous Chevalley-Eilenberg cohomology_ is the cohomology of the cochain
complex

(2.7) $\displaystyle
C^{\bullet}({\operatorname{\mathfrak{g}}}):=\bigoplus_{k\geq
0}C^{k}({\operatorname{\mathfrak{g}}}).$

By $C^{k}({\operatorname{\mathfrak{g}}})$ we denote the space of multilinear,
skew-symmetric, continuous maps

(2.8) $c:{\operatorname{\mathfrak{g}}}^{k}\to\mathbb{R},$

and the differential $d:C^{\bullet}({\operatorname{\mathfrak{g}}})\to
C^{\bullet+1}({\operatorname{\mathfrak{g}}})$ of the complex is

(2.9) $\begin{gathered}dc(X_{1},\dots,X_{k+1})=\sum_{1\leq i<j\leq
k+1}(-1)^{i+j-1}c([X_{i},X_{j}],X_{1},\dots,X_{k+1})\quad\forall
X_{i}\in{\operatorname{\mathfrak{g}}}.\end{gathered}$

The space $C^{\bullet}({\operatorname{\mathfrak{g}}})$ assumes the structure
of a differential graded algebra with the wedge product

(2.10) $\displaystyle(c_{1}\wedge c_{2})$
$\displaystyle(X_{1},\dots,X_{k+l}):=$
$\displaystyle\frac{1}{k!l!}\sum_{\sigma\in\Sigma_{k+l}}\operatorname{sign}(\sigma)c_{1}(X_{\sigma(1)},\dots,X_{\sigma(k)})c_{2}(X_{\sigma(k+1)},\dots,X_{\sigma(k+l)})$

for $c_{1}\in C^{k}({\operatorname{\mathfrak{g}}}),c_{2}\in
C^{l}({\operatorname{\mathfrak{g}}}),X_{1},\dots,X_{k+l}\in{\operatorname{\mathfrak{g}}}$.

######  Remark 2.5.

If ${\operatorname{\mathfrak{g}}}$ is finite-dimensional, the continuity
assumption for cochains in $C^{\bullet}({\operatorname{\mathfrak{g}}})$ is
redundant. If ${\operatorname{\mathfrak{g}}}=W_{n}$ with its projective
topology, then the continuity assumption on $c\in C^{\bullet}(W_{n})$ just
means that there is a $k\in\mathbb{Z}$ so that $c(X,\cdot,\dots,\cdot)=0$ for
all $X$ with $\deg X>k$. In particular, $c$ is only nonzero on a finite-
dimensional subspace of $\Lambda^{\bullet}W_{n}$.

Recall the following:

######  Definition 2.6.

Let $\mathfrak{g}$ be a Lie algebra, $Y\in\mathfrak{g}$ and $c\in
C^{k}(\mathfrak{g})$ for some $k\geq 0$.

  * i)

Denote the natural Lie algebra action of an element $Y$ on
$C^{k}(\mathfrak{g})$ by $Y\cdot c$; the formula is given for
$Y,X_{1},\dots,X_{k}\in\mathfrak{g}$ by

(2.11) $\displaystyle(Y\cdot c)(X_{1},\dots,X_{k})$
$\displaystyle:=-\sum_{i=1}^{k}c(X_{1},\dots,[Y,X_{i}],\dots,X_{k}),$

and $Y\cdot c=0$ if $c\in C^{0}({\operatorname{\mathfrak{g}}})$.

  * ii)

Denote by $Y\intprod c\in C^{k-1}(\mathfrak{g})$ the _interior product of $c$
with $Y$,_ which is defined via

(2.12) $\displaystyle(Y\intprod
c)(X_{1},\dots,X_{k-1})=c(Y,X_{1},\dots,X_{k-1})$

and $Y\intprod c=0$ if $c\in C^{0}({\operatorname{\mathfrak{g}}})$.

A straightforward calculation yields the following homotopy relation:

######  Lemma 2.7.

Let $\mathfrak{g}$ be a Lie algebra, $c\in
C^{\bullet}({\operatorname{\mathfrak{g}}})$ and $Y\in\mathfrak{g}$. Then we
have the following chain homotopy formula:

(2.13) $\displaystyle d(Y\intprod c)+Y\intprod dc=-Y\cdot c$

A well-known corollary of the previous statement is:

######  Corollary 2.8.

The action of a Lie algebra $\mathfrak{g}$ on its cochains
$C^{\bullet}({\operatorname{\mathfrak{g}}})$ commutes with the Chevalley-
Eilenberg differential, and the induced action on $H^{\bullet}(\mathfrak{g})$
is trivial.

Using the grading of $W_{n}$ induced by the Euler vector field $E$, we can
also define a grading of the cochains:

######  Definition 2.9.

Let $r\in\mathbb{Z}$ and $k\geq 0$ an integer. Define

(2.14) $\displaystyle C^{k}_{(r)}(W_{n}):=\\{c\in C^{k}(W_{n}):E\cdot
c=-r\cdot c\\}.$

######  Remark 2.10.

More explicitly, $c\in C^{k}_{(r)}(W_{n})$ if and only if we have for all
homogeneous formal vector fields $X_{1},\dots,X_{k}\in W_{n}$

(2.15) $\displaystyle\sum_{i=1}^{k}\deg X_{i}\neq r\implies
c(X_{1},\dots,X_{k})=0.$

######  Proposition 2.11 ([10], Section 1.5 and 2.2).

The spaces $C^{\bullet}_{(r)}(W_{n})\leavevmode\nobreak\ $ fulfil the
following properties:

  * i)

We have
$C^{\bullet}(W_{n})=\bigoplus_{r\in\mathbb{Z}}C^{\bullet}_{(r)}(W_{n}).$

  * ii)

For all $r\in\mathbb{Z}$, the spaces $C^{\bullet}_{(r)}(W_{n})$ are
subcomplexes of $C^{\bullet}(W_{n})$.

  * iii)

For all $r,s\in\mathbb{Z}$ we have

(2.16) $\displaystyle C^{\bullet}_{(r)}(W_{n})\wedge
C^{\bullet}_{(s)}(W_{n})\subset C^{\bullet}_{(r+s)}(W_{n}).$

  * iv)

If $r<-k$, then $C^{k}_{(r)}(W_{n})=0$.

  * v)

The inclusion $C^{\bullet}_{(0)}(W_{n})\subset C^{\bullet}(W_{n})$ induces an
algebra isomorphism

(2.17) $\displaystyle H^{\bullet}(C^{\bullet}_{(0)}(W_{n}))\cong
H^{\bullet}(W_{n}).$

###### Proof.

i) The direct sum decomposition follows since every $c\in C^{k}(W_{n})$ is
zero on homogeneous vector fields of sufficiently high degree, and its
evaluation on any collection $X_{1},\dots,X_{k}\in W_{n}$ can be uniquely
decomposed into summands of homogeneous vector fields.
ii) This follows since the action of $E$ commutes with the Lie algebra
differential by Lemma 2.7.
iii) This is due to

(2.18) $\displaystyle E\cdot(c_{1}\wedge c_{2})=(E\cdot c_{1})\wedge
c_{2}+c_{1}\wedge(E\cdot c_{2})\quad\forall c_{1},c_{2}\in
C^{\bullet}(W_{n}).$

iv) Due to the pidgeonhole principle, any collection of $k$ elements in
$W_{n}$ whose degrees sum up to a value smaller than $-k$ must have an element
with degree smaller $-1$. Such an element is necessarily zero, which shows the
statement.
v) Lemma 2.7 shows that for all $c\in C^{\bullet}_{(r)}(W_{n})$ we have

(2.19) $\displaystyle d(E\intprod c)+E\intprod(dc)=-r\cdot c.$

Thus, for $r\neq 0$, the map $-\frac{1}{r}(E\intprod\cdot)$ defines a chain
homotopy between the identity and zero for the cochain complex
$C^{\bullet}_{(r)}(W_{n})$, and hence
$H^{\bullet}(C^{\bullet}_{(r)}(W_{n}))=0$. We conclude that all cohomology
classes of $C^{\bullet}(W_{n})$ admit a representative fully contained in
$C^{\bullet}_{(0)}(W_{n})$. This shows that the inclusion induces an
isomorphism of vector spaces. By statement iii) $C_{(0)}^{\bullet}(W_{n})$ is
a subalgebra of $C^{\bullet}(W_{n})$ with respect to the wedge product, hence
the inclusion induces an algebra isomorphism on cohomology. ∎

In the following we will write
$H^{\bullet}_{(r)}(W_{n}):=H^{\bullet}(C^{k}_{(r)}(W_{n}))$.

###  2.2. Stable cohomology of formal vector fields

We first focus on certain low-dimensional cohomology, the so-called _stable
cohomology_ of $W_{n}$, due to Guillemin and Shnider. They prove in [15,
Corollary 1] that $H^{k}(W_{n})$ is trivial in dimension $k=1,\dots,n$. Note
that their paper makes much more general statements, in particular about
stable cohomology of formal Lie algebras corresponding to other classical
vector field Lie algebras, e.g. formal Hamiltonian and divergence-free vector
fields.

######  Definition 2.12.

Define for all $r\in\mathbb{Z}$,

(2.20) $\displaystyle\partial C^{\bullet}_{(r)}(W_{n}):=\\{\partial_{i}\cdot
c\in C^{\bullet}_{(r+1)}(W_{n}):c\in C^{\bullet}_{(r)}(W_{n})\\},$

and

(2.21) $\displaystyle\partial
C^{\bullet}(W_{n}):=\bigoplus_{r\in\mathbb{Z}}\partial
C^{\bullet}_{(r)}(W_{n}).$

Recall that $\partial_{i}\cdot c$ denotes the action of
$\partial_{i}\in\mathfrak{g}_{-1}$ on the cochain $c$ (see Definition 2.6).

By Corollary 2.8, the Lie algebra action of ${\operatorname{\mathfrak{g}}}$ on
$C^{\bullet}(W_{n})$ commutes with the Chevalley-Eilenberg differential, and
thus:

######  Lemma 2.13.

For all $r\in\mathbb{Z}$, the space $\partial C^{\bullet}_{(r)}(W_{n})$ is a
subcomplex of $C^{\bullet}_{(r+1)}(W_{n})$.

We need one more preparing definition, since the degree zero component of
cochain complexes is often troublesome.

######  Definition 2.14 (Reduced Complex).

If $C^{\bullet}$ is a cochain complex, define the _reduced complex_
$\tilde{C}^{\bullet}$ as

(2.22) $\displaystyle\tilde{C}^{0}=0,\quad\tilde{C}^{k}:=C^{k}\quad\forall
k\geq 1,$

equipped with the inherited differential from $C^{\bullet}$.
We denote by $\tilde{H}^{\bullet}$ the cohomology of the reduced complex.

The aim of this section is the construction of a Koszul complex relating the
complexes $C_{(r)}^{\bullet}(W_{n})$ for different values of $r$.

######  Proposition 2.15.

There exists an exact sequence of cochain complexes

(2.23) $\displaystyle
0\to\tilde{C}^{\bullet}_{(0)}(W_{n})\otimes\Lambda^{n}\mathfrak{g}_{-1}$
$\displaystyle\to\tilde{C}^{\bullet}_{(1)}(W_{n})\otimes\Lambda^{n-1}\mathfrak{g}_{-1}\to\dots$
$\displaystyle\to\tilde{C}^{\bullet}_{(n)}(W_{n})\otimes\Lambda^{0}\mathfrak{g}_{-1}\to\tilde{C}^{\bullet}_{(n)}(W_{n})/\partial\tilde{C}^{\bullet}_{(n-1)}(W_{n})\to
0,$

where the differentials in every term are induced by the Chevalley-Eilenberg
differential of $C^{\bullet}(W_{n})$.

###### Proof.

Write $V:={\operatorname{\mathfrak{g}}}_{-1}$. Denote by $\vee$ the product of
symmetric tensors, and define for all $r\leq n-1$ the map

(2.24) $\displaystyle\sigma_{r}:S^{\bullet}V\otimes\Lambda^{r}V$
$\displaystyle\to S^{\bullet}V\otimes\Lambda^{r-1}V,$ $\displaystyle
u\otimes(\partial_{i_{1}}\wedge\dots\wedge\partial_{i_{r}})$
$\displaystyle\mapsto\sum_{j=1}^{r}(-1)^{j}(\partial_{i_{j}}\vee
u)\otimes(\partial_{i_{1}}\wedge\dotsb\widehat{\partial_{i_{j}}}\dotsb\wedge\partial_{i_{r}})\quad\forall
u\in S^{\bullet}V.$

These maps give rise to the well-known, acyclic Koszul complex

(2.25) $\displaystyle 0\to S^{\bullet}V\otimes\Lambda^{n}V$ $\displaystyle\to
S^{\bullet}V\otimes\Lambda^{n-1}V\to\dots$ $\displaystyle\to
S^{\bullet}V\otimes\Lambda^{1}V\to
S^{\bullet}V\otimes\Lambda^{0}V\to\mathbb{R}\to 0.$

Taking the tensor product of the above exact sequence with $S^{\bullet}V,$ and
using the canonical isomorphism $S^{\bullet}(V^{2})\cong S^{\bullet}(V)\otimes
S^{\bullet}(V)$, we get the exact sequence

(2.26) $\displaystyle 0\to S^{\bullet}V^{2}\otimes\Lambda^{n}V$
$\displaystyle\to S^{\bullet}V^{2}\otimes\Lambda^{n-1}V\to\dots$
$\displaystyle\to S^{\bullet}V^{2}\otimes\Lambda^{0}V\to S^{\bullet}V\to 0,$

and inductively, if we denote by $V_{\Delta}\subset V^{k}$ the diagonal
subspace, then we get an exact sequence

(2.27) $\displaystyle 0\to S^{\bullet}V^{k}\otimes\Lambda^{n}V$
$\displaystyle\to S^{\bullet}V^{k}\otimes\Lambda^{n-1}V\to\dots$
$\displaystyle\to S^{\bullet}V^{k}\otimes\Lambda^{0}V\to
S^{\bullet}(V^{k}/V_{\Delta})\to 0.$

Let $k\geq 1$ and denote by $\Sigma_{k}$ the permutation group in $k$
elements. The tensor product $S^{\bullet}(V^{k})\otimes(V^{*})^{\otimes k}$
admits a $\Sigma_{k}$ action by signed, simultaneous permutation of the tensor
factors in $(V^{*})^{\otimes k}$ and the direct summands in $V^{k}$. Taking
invariants with respect to this action, we find

(2.28) $\displaystyle\left(S^{\bullet}(V^{k})\otimes(V^{*})^{\otimes
k}\right)^{\Sigma_{k}}\cong C^{k}(W_{n}).$

Thus, taking the tensor product of the complex (2.27) with $(V^{*})^{\otimes
k}\leavevmode\nobreak\ $ and taking invariants under a finite group are exact
functors of $\mathbb{R}$-vector spaces, and as such we get for every $k\geq 1$
an exact sequence

(2.29) $\displaystyle 0\to C^{k}(W_{n})\otimes\Lambda^{n}V$ $\displaystyle\to
C^{k}(W_{n})\otimes\Lambda^{n-1}V\to\dots$ $\displaystyle\to
C^{k}(W_{n})\otimes\Lambda^{0}V\to
C^{k}(W_{n})/\partial\tilde{C}^{k}(W_{n})\to 0.$

The last nontrivial map in (2.29) is the canonical quotient projection, the
others are:

(2.30) $\begin{gathered}C^{k}(W_{n})\otimes\Lambda^{r}V\to
C^{k}(W_{n})\otimes\Lambda^{r-1}V,\\\
c\otimes(\partial_{i_{1}}\wedge\dots\wedge\partial_{i_{r}})\mapsto\sum_{j=1}^{r}(-1)^{j}(\partial_{i_{j}}\cdot
c)\otimes(\partial_{i_{1}}\wedge\dotsb\widehat{\partial_{i_{j}}}\dotsb\wedge\partial_{i_{r}})\quad\forall
c\in\tilde{C}^{\bullet}(W_{n}).\end{gathered}$

By Corollary 2.8, the differential of $C^{\bullet}(W_{n})$ commutes with the
action of $S^{\bullet}V$, and by Lemma 2.13, the differential commutes with
the projection

(2.31) $\displaystyle C^{\bullet}(W_{n})\to C^{\bullet}(W_{n})/\partial
C^{\bullet}(W_{n}).$

Hence, by taking the direct sum of the complexes (2.29) for all $k\geq 1$, we
receive the following exact sequence of chain complexes

(2.32) $\displaystyle 0\to\tilde{C}^{\bullet}(W_{n})\otimes\Lambda^{n}V$
$\displaystyle\to\tilde{C}^{\bullet}(W_{n})\otimes\Lambda^{n-1}V\to\dots$
$\displaystyle\to\tilde{C}^{\bullet}(W_{n})\otimes\Lambda^{0}V\to\tilde{C}^{\bullet}(W_{n})/\partial\tilde{C}^{\bullet}(W_{n})\to
0.$

With respect to the grading
$\tilde{C}^{\bullet}(W_{n})=\bigoplus_{r}\tilde{C}^{\bullet}_{(r)}(W_{n})$ the
maps (2.30) restrict to maps on the graded components

(2.33) $\displaystyle C^{\bullet}_{(k)}(W_{n})\otimes\Lambda^{r}(V)\to
C^{\bullet}_{(k+1)}(W_{n})\otimes\Lambda^{r-1}(V)\quad\forall
k,r\in\mathbb{Z},$

and the canonical quotient projection $C^{\bullet}(W_{n})\to
C^{\bullet}(W_{n})/\partial C^{\bullet}(W_{n})\leavevmode\nobreak\ $ restricts
to

(2.34) $\displaystyle C_{(k)}^{\bullet}(W_{n})\to
C_{(k)}^{\bullet}(W_{n})/\partial C_{(k-1)}^{\bullet}(W_{n})\quad\forall
k\in\mathbb{Z}.$

Considering the graded component of (2.32) whose leftmost term is
$C^{\bullet}_{(0)}(W_{n})\otimes\Lambda^{n}V$ yields the desired statement. ∎

######  Remark 2.16.

The construction of this sequence in [15, Thm 1] is carried out differently. A
detailed proof of their construction would require a study of Hopf algebra
theory, which we do not want to carry out here: they implicitly use that if
$M$ is a free module over a Hopf algebra $H$, then so is
$\Lambda^{k}_{\mathbb{K}}M$ with its induced, diagonal action for all $k>0$.
This can, for example, be shown with the methods of [16, Chapter 7.2].

######  Proposition 2.17 ([15], Corollary 1).

We have $H^{k}(W_{n})=0$ if $k=1,\dots,n$.

###### Proof.

Consider the exact sequence (2.23). We have established that
$\tilde{C}^{\bullet}_{(r)}(W_{n})$ is an acyclic subcomplex whenever $r\neq
0$, and as such, all terms in the exact sequence are acyclic except for the
leftmost and rightmost nontrivial ones. Now we can combine this with the
exactness of (2.23) and either apply a straightforward diagram chase, or by
view this sequence of cochain complexes as a double complex and comparing the
associated spectral sequences. Both methods allow one to deduce the following
isomorphisms:

(2.35) $H^{k}\left(\tilde{C}^{\bullet}_{(0)}(W_{n})\right)\cong
H^{k}\left(\tilde{C}^{\bullet}_{(0)}(W_{n})\otimes\Lambda^{n}\mathfrak{g}_{-1}\right)\cong
H^{k-n}\left(\tilde{C}^{\bullet}_{(n)}(W_{n})/\partial\tilde{C}^{\bullet}_{(n-1)}(W_{n})\right).$

But, as a relative complex, the complex on the right-hand side is zero in all
degrees smaller than $1$, hence so is its cohomology. Hence, for all
$k=1,\dots,n$ we have

(2.36) $\displaystyle H^{k}(W_{n})\cong
H^{k}_{(0)}(W_{n})=\tilde{H}^{k}_{(0)}(W_{n})=0.$

∎

###  2.3. A spectral sequence for the cohomology of formal vector fields

One can do even better than Proposition 2.17: We will formulate a spectral
sequence due to Gelfand and Fuks [3] which calculates the cohomology of
$W_{n}$, and fully specify its differentials. In other words, we will be able
to calculate the dimension of $H^{\bullet}(W_{n})$ in every degree and for
every $n\in\mathbb{N}$. The information from the previous section about low
degree cohomology will aid us for the analysis of this spectral sequence. To
this end, we begin with a short recollection of some representation and
cohomology theory of ${\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})$. We
largely follow the proof in [10], with certain adaptations that will be
indicated.

######  Theorem 2.18 ([17], Theorem 24.4).

Write $V=\mathbb{R}^{n}$, and consider $V$ and $V^{*}$ as
${\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})$-modules with the defining
representation and the dual thereof.
For every $\sigma\in\Sigma_{r}$, define

$\displaystyle\Psi_{\sigma}:V^{\otimes r}\otimes(V^{*})^{\otimes
r}\to\mathbb{R},$ $\displaystyle v_{1}\otimes\dots\otimes
v_{r}\otimes\alpha_{1}\otimes\dots\otimes\alpha_{r}\mapsto\alpha_{1}(v_{\sigma(1)})\dots\alpha_{r}(v_{\sigma(r)})\quad\forall\alpha_{i}\in
V^{*},v_{i}\in V.$

Then $\\{\Psi_{\sigma}\\}_{\sigma\in\Sigma_{r}}$ is a spanning set for

(2.37) $\displaystyle\operatorname{Hom}_{\mathbb{R}}\left(V^{\otimes
r}\otimes(V^{*})^{\otimes
r},\mathbb{R}\right)^{{\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})}\cong\left((V^{*})^{\otimes
r}\otimes V^{\otimes
r}\right)^{{\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})}$

For $r\leq n$, the set $\\{\Psi_{\sigma}\\}$ is linearly independent. Further,
if $r\neq s$, then

(2.38) $\displaystyle\operatorname{Hom}_{\mathbb{R}}\left(V^{\otimes
r}\otimes(V^{*})^{\otimes
s},\mathbb{R}\right)^{{\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})}=0.$

######  Theorem 2.19 ([10], Theorem 2.1.1).

The cohomology ring $H^{\bullet}(\mathfrak{gl}_{n}(\mathbb{R}))$ is isomorphic
to the exterior algebra

(2.39) $\displaystyle\Lambda^{\bullet}[\phi_{1},\dots,\phi_{2n-1}],$

where the $\phi_{i}$ are generators in degree $i$. The inclusion
$\mathfrak{gl}(n-1,\mathbb{R})\to\mathfrak{gl}(n,\mathbb{R})$ induces a
morphism

(2.40) $\displaystyle H^{q}(\mathfrak{gl}(n,\mathbb{R}))\to
H^{q}(\mathfrak{gl}(n-1,\mathbb{R}))$

which is an isomorphism for $q\leq 2n-3$.

######  Remark 2.20.

Note that our reference states the above theorem in an erroneous way: They
state the map induced by the inclusion has a one-dimensional kernel for $q=n$,
which, for example, cannot be true when $n=2$, since the second cohomology
vanishes for all $\mathfrak{gl}(n,\mathbb{R})$. They also write that the
inclusion only induces an isomorphism in degree $<n$, but their spectral
sequence argument actually shows the above, stronger property (see also [18,
Cor 4D.3].

######  Lemma 2.21.

Let $n\in\mathbb{N}$ be arbitrary, and consider the Hochschild-Serre spectral
sequence $\\{E^{p,q}_{r},d_{r}\\}$ of the Lie-algebra-subalgebra pair
$\mathfrak{g}_{0}\subset W_{n}$ in continuous cohomology (cf. Appendix B).
We have for all $p,q\geq 0$

(2.41) $\displaystyle
E^{p,q}_{2}=\begin{cases}H^{q}({\operatorname{\mathfrak{g}}}_{0})\otimes\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{{\operatorname{\mathfrak{g}}}_{0}}&\text{
if }p\text{ even and }p=2r,\\\ 0&\text{ if }p\text{ odd.}\end{cases}$

###### Proof.

By definition of the Hochschild-Serre spectral sequence, the first page takes
the following form:

(2.42) $\displaystyle E^{p,q}_{1}$
$\displaystyle=H^{q}\left(\mathfrak{g}_{0};\Lambda^{p}(W_{n}/\mathfrak{g}_{0})^{*}\right)=H^{q}\left(\mathfrak{g}_{0};\Lambda^{p}\left(\bigoplus_{j\neq
0}\mathfrak{g}_{j}\right)^{*}\right)$
$\displaystyle=\bigoplus_{p_{-1}+p_{1}+p_{2}+\dots=p}H^{q}\left(\mathfrak{g}_{0};\bigotimes_{j\neq
0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}\right)$

Note that, as Lie algebras, $\mathfrak{g}_{0}\subset W_{n}$ is isomorphic to
$\mathfrak{gl}_{n}(\mathbb{R})=\mathfrak{gl}(V)$ via

(2.43)
$\displaystyle\sum_{i,j=1}^{n}a_{ij}x_{i}\partial_{j}\mapsto(a_{ij})_{1\leq
i,j\leq n}.$

By Weyl’s complete reducibility theorem [19, Theorem 10.9], all coefficient
modules in (2.42) are completely reducible. Together with reducibility of the
Lie algebra ${\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R})$ and [20, Theorem
10], we may reduce the coefficient space in the above cohomologies to the
$\mathfrak{gl}_{n}(\mathbb{R})$-invariants. Hence,

(2.44) $\displaystyle E^{p,q}_{1}$
$\displaystyle=\bigoplus_{p_{-1}+p_{1}+p_{2}+\dots=p}H^{q}\left(\mathfrak{g}_{0};\left(\bigotimes_{j\neq
0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}\right)^{\mathfrak{gl}(V)}\right)$
$\displaystyle=H^{q}(\mathfrak{g}_{0})\otimes\left(\bigoplus_{p_{-1}+p_{1}+p_{2}+\dots=p}\left(\bigotimes_{j\neq
0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}\right)^{\mathfrak{gl}(V)}\right).$

By definition, ${\operatorname{\mathfrak{g}}}_{j}^{*}=(S^{j+1}V)\otimes V^{*}$
contains $j+1$ tensor factors that transform _covariantly_ (i.e. copies of
$V$) under the ${\operatorname{\mathfrak{gl}}}(V)$ action, and one tensor
factor that transforms _contravariantly_ (i.e. a copy of $V^{*}$), hence
$\Lambda^{p_{j}}{\operatorname{\mathfrak{g}}}_{j}^{*}$ contains $j\cdot p_{j}$
covariant and $p_{j}$ contravariant tensor factors. Hence

(2.45) $\displaystyle\bigotimes_{j\neq
0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}\subset(V^{*})^{\sum_{j\neq
0}p_{j}}\otimes V^{\sum_{j\neq 0}(j+1)p_{j}}$

The last part of Theorem 2.18 then implies that the space of
${\operatorname{\mathfrak{gl}}}(V)$-invariants of the space $\bigotimes_{j\neq
0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}$ is only nonzero if

(2.46) $\displaystyle\sum_{j\neq 0}p_{j}=\sum_{j\neq 0}(j+1)p_{j},\text{ or
equivalently }p_{-1}=\sum_{j=1}^{\infty}j\cdot p_{j}$

Simultaneously, again from Theorem 2.18, we know that
$\mathfrak{gl}_{n}(\mathbb{R})$-invariants in a tensor module

(2.47) $\displaystyle V^{\otimes r}\otimes(V^{*})^{\otimes
r}\cong\operatorname{Hom}((V^{*})^{\otimes r}\otimes V^{\otimes
r},\mathbb{R})$

can be described as the linear combinations of the functionals which contract
all covariant indices with permutations of the contravariant indices.
Correspondingly, the $\mathfrak{gl}(V)$ invariants in the subspaces
$\bigotimes_{j\neq 0}\Lambda^{p_{j}}\mathfrak{g}_{j}^{*}$ are given by
subjecting these functionals to the required (skew-)symmetrizations. Hence if

(2.48) $\displaystyle p_{-1}>\sum_{j=1}^{\infty}p_{j},$

then by the pidgeonhole principle, any invariant tensor contracts at least two
contravariant factors belonging to $\Lambda^{p_{-1}}\mathfrak{g}_{-1}$ with
two covariant factors, both belonging to a single copy of $\mathfrak{g}_{j}$
within
$\Lambda^{p_{j}}\mathfrak{g}_{j}=\mathfrak{g}_{j}\wedge\dots\wedge\mathfrak{g}_{j}$
for some $j\geq 1$. However, in such a contraction the contravariant factors
would behave skew-symmetrically and the covariant ones symmetrically under
permutation, hence their contraction is zero. Thus, we get the additional
requirement

(2.49) $\displaystyle p_{-1}\leq\sum_{j=1}^{\infty}p_{j}$

Combining (2.46) and (2.49) we end up with $p_{k}=0$ for $k\geq 2$ and
$p_{-1}=p_{1}=:r$. This implies that $p=2r$ is even whenever there are
nontrivial invariants, and thus every other column in the page $E^{p,q}_{1}$
vanishes. Hence all differentials on the first page are trivial, and
$E^{p,q}_{1}=E^{p,q}_{2}$. This concludes the proof. ∎

Since $H^{q}({\operatorname{\mathfrak{g}}}_{0})$,
$\Lambda^{r}{\operatorname{\mathfrak{g}}}_{-1}$, and
$\Lambda^{r}{\operatorname{\mathfrak{g}}}_{1}$ are nonzero only for finitely
many $q,r\geq 0$, and all involved spaces are finite-dimensional in every
degree, we have:

######  Corollary 2.22.

For all $q\geq 0$, the continuous cohomology $H^{q}(W_{n})$ is finite-
dimensional, and $H^{q}(W_{n})\neq 0$ only in finitely many degrees.

Let us further analyze the invariant space
$\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{{\operatorname{\mathfrak{g}}}_{0}}$.

######  Lemma 2.23.

Let $\\{E^{\bullet,\bullet}_{2},d_{r}\\}$ be the spectral sequence from Lemma
2.21 of the Lie-algebra-subalgebra pair
${\operatorname{\mathfrak{g}}}_{0}\subset W_{n}$.
For all $r=1,\dots,n$, there exist multiplicative generators $\Psi_{2r}\in
E^{2r,0}_{2}$ so that

(2.50) $\displaystyle E^{\bullet,0}_{2}$
$\displaystyle=\mathbb{R}[\Psi_{2},\Psi_{4}\dots,\Psi_{2n}]/\langle\Psi_{i_{1}}\dots\Psi_{i_{k}}:i_{1}+\dots+i_{k}>2n\rangle.$

###### Proof.

By Lemma 2.21 we have

(2.51) $\displaystyle E^{\bullet,0}_{2}=\bigoplus_{r\geq
0}\left(\Lambda^{r}{\operatorname{\mathfrak{g}}}_{-1}\otimes\Lambda^{r}{\operatorname{\mathfrak{g}}}_{1}\right)^{{\operatorname{\mathfrak{g}}}_{0}}.$

By Theorem 2.18 the elements in the invariant space
$\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{{\operatorname{\mathfrak{g}}}_{0}}$
arise by taking, for every permutation $\sigma\in\Sigma_{r}$, the functional

(2.52) $\displaystyle\bigotimes^{r}V\times\bigotimes^{r}\left(V^{*}\otimes
V^{*}\otimes V\right)\to\mathbb{R},$

with

(2.53)
$\displaystyle(\alpha_{1}^{1}\otimes\dots&\otimes\alpha_{r}^{1},(\beta_{1}^{1}\otimes\beta_{1}^{2}\otimes\alpha_{r+1}^{2})\otimes\dots\otimes(\beta_{r}^{1}\otimes\beta_{r}^{2}\otimes\alpha_{2r}^{2}))$
$\displaystyle\mapsto\beta_{1}^{1}(\alpha_{1}^{1})\dots\beta_{r}^{1}(\alpha_{r}^{1})\cdot\beta_{1}^{2}(\alpha_{\sigma(1)}^{2})\dots\beta_{r}^{2}(\alpha_{\sigma(r)}^{2}),\quad\forall\alpha_{i}\in
V,\,\beta_{i}^{1},\beta_{i}^{2}\in V^{*},$

skew-symmetrizing over the first $r$ and the last $r$ arguments, and
symmetrizing over the exchange $\beta_{i}^{1}\leftrightarrow\beta_{i}^{2}$.
Denote the arising functional by
$\Psi_{\sigma}\in\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}$.
From the skew-symmetry of the functionals in the first $r$ and last $r$
arguments, one deduces that

(2.54)
$\Psi_{\sigma}=\Psi_{\tau\sigma\tau^{-1}}\quad\forall\sigma,\tau\in\Sigma_{k}.$

Another straightforward calculation shows that for $\sigma\in\Sigma_{r}$ and
$\tau\in\Sigma_{l}$ we have

(2.55)
$\Psi_{\sigma}\wedge\Psi_{\tau}=\Psi_{\sigma\tau}\in\left(\Lambda^{r+l}\mathfrak{g}_{-1}\otimes\Lambda^{r+l}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}.$

Because $\mathfrak{g}_{-1}$ is $n$-dimensional,
$\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}=0$
if $r>n$, so in particular

(2.56) $\displaystyle\Psi_{\sigma}\wedge\Psi_{\tau}=0\text{ if
}\sigma\in\Sigma_{r},\tau\in\Sigma_{l},r+l>n.$

Denote by
$\Psi_{r}\in\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}$
the $\Psi_{\sigma}$ corresponding to an $r$-cycle $\sigma\in\Sigma_{r}$. This
is well-defined, since by (2.54) the functional $\Psi_{\sigma}$ only depends
on the conjugacy class of $\sigma$, and all $r$-cycles are conjugate to one
another.
Since every permutation can be uniquely (up to ordering) decomposed into a
composition of cycles, (2.55) shows that
$\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}$
is multiplicatively generated by the
$\Psi_{2r}\in\left(\Lambda^{r}\mathfrak{g}_{-1}\otimes\Lambda^{r}\mathfrak{g}_{1}\right)^{\mathfrak{gl}(V)}$
for every $r=1,\dots,n$. Hence every element in $E^{\bullet,0}_{2}$ is given
as a unique (up to ordering) product of elements in
$\\{\Psi_{2},\dots,\Psi_{2n}\\}$. Since all $\Psi_{\sigma}$ for
$\sigma\in\Sigma_{r}$ and $r\leq n$ are nonzero, the only relation between
these generators is that products $\Psi_{i_{1}}\dots\Psi_{i_{k}}$ are zero if
$i_{1}+\dots+i_{k}>2n$, and the lemma is proven. ∎

To understand this spectral sequence further, we will need the Borel
transgression theorem. To formulate it, let us first define some terminology.

######  Definition 2.24.

Let $\\{E^{p,q}_{r},d_{r}\\}_{r\geq 0}$ be a cohomological first-quadrant
spectral sequence. Denote by $\kappa_{r}^{r+1}:\ker d_{r}\to
E^{\bullet,\bullet}_{r+1}$ the natural quotient map from cocycles of the
$r$-th page differential $d_{r}$ to the $r+1$-th page, and

(2.57)
$\displaystyle\kappa_{r}^{s}=\kappa_{s-1}^{s}\circ\dots\circ\kappa_{r}^{r+1}\quad\forall
s>r,$

where the domain of $\kappa_{r}^{s}$ is defined inductively as all the $c\in
E^{\bullet,\bullet}_{r+1}$ in the domain of $\kappa_{r}^{s-1}$ so that
$\kappa_{r}^{s-1}c\in\ker d_{s-1}$. We call an element $c\in E^{p,0}_{2}$
_transgressive_ if, for all $r$ with $2\leq r\leq p$, we have that $c$ is in
the domain of $\kappa_{2}^{r}$.

Intuitively, the transgressive elements in $E^{p,0}_{2}$ are the ones which
“survive” until the very last moment: Only the differential
$d_{p+1}:E^{p,0}_{p+1}\to E^{0,p+1}_{p+1}$, also called the _transgression_ ,
can map it to something nontrivial. By abuse of notation, we often denote an
element in the domain of $\kappa_{r}^{s}$ by the same symbol as its image
under $\kappa_{r}^{s}$ in the higher page $E_{s}^{\bullet,\bullet}$ The
following theorem was originally proven in [21], but we cite a slightly
stronger version from [22, Thm 2.9].

######  Theorem 2.25 (Borel transgression theorem).

Consider two finite-dimensional, graded vector spaces
$B^{\bullet}:=\bigoplus_{p\in\mathbb{N}_{0}}B^{p}$ and
$F^{\bullet}:=\bigoplus_{q\in\mathbb{N}_{0}}F^{q}$. Assume there are elements
$x_{i}\in F^{\bullet}$ of odd degree such that

(2.58) $\displaystyle\Lambda^{\bullet}[x_{1},\dots,x_{l}]\to F^{\bullet}$

is bijective in degrees $\leq N$ and injective in degree $N+1$. Let further
$\\{E_{r}^{p,q},d_{r}\\}_{r\geq 0}$ be a cohomological spectral sequence whose
second page has the form

(2.59) $\displaystyle E_{2}^{p,q}=B^{p}\otimes F^{q},$

and which converges towards a graded vector space $H^{\bullet}$ with $H^{k}=0$
if $0<k\leq N+2$. Then we can choose the generators $x_{i}$ to be
transgressive, and if $y_{1},\dots,y_{l}\in B^{\bullet}$ denotes a collection
of elements with

(2.60) $\displaystyle d_{\deg x_{i}+1}x_{i}=y_{i}\quad i=1,\dots,l,$

then the map

(2.61) $\displaystyle\mathbb{R}[y_{1},\dots,y_{l}]\to B^{\bullet}$

is bijective for degrees $\leq N$ and injective for degree $N+1$.

Using this, we can fully describe the desired spectral sequence:

######  Theorem 2.26 ([10], Theorem 2.2.4).

Let $n\in\mathbb{N}$ be arbitrary, and consider the Hochschild-Serre spectral
sequence $\\{E^{p,q}_{r},d_{r}\\}$ of the pair $\mathfrak{g}_{0}\subset W_{n}$
in continuous cohomology. Its second page takes the form

(2.62) $\displaystyle E^{0,\bullet}_{2}$
$\displaystyle=\Lambda^{\bullet}[\phi_{1},\phi_{3},\dots,\phi_{2n-1}],$ (2.63)
$\displaystyle E^{\bullet,0}_{2}$
$\displaystyle=\mathbb{R}[\Psi_{2},\Psi_{4}\dots,\Psi_{2n}]/\langle\Psi_{i_{1}}\dots\Psi_{i_{k}}:i_{1}+\dots+i_{k}>2n\rangle,$
(2.64) $\displaystyle E^{p,q}_{2}$ $\displaystyle=E^{p,0}_{2}\otimes
E^{0,q}_{2},$

and all differentials of the spectral sequence are fully specified on the
generators by

(2.65) $\displaystyle d_{i+1}\phi_{i}$ $\displaystyle=\Psi_{i+1}\quad
i\in\\{1,3,\dots,2n-1\\}.$

###### Proof.

The form of the second page follows from Theorem 2.19, Lemma 2.21, and Lemma
2.23. It remains to show the statement about the differentials. Consider the
Hochschild-Serre spectral sequence for the pair of Lie algebras
$(W_{3n},{\operatorname{\mathfrak{gl}}}_{3n}(\mathbb{R}))$. By Proposition
2.17, we know that $H^{k}(W_{3n})=0$ in degrees $k=1,\dots,3n$, and up to
degree $2n$, the zeroeth column of the spectral sequence is equal to the
exterior algebra $\Lambda^{\bullet}[\phi_{1},\phi_{3},\dots,\phi_{2n-1}]$, the
$\phi_{i}$ being the generators of
$H^{\bullet}({\operatorname{\mathfrak{gl}}}_{3n}(\mathbb{R}))$. Hence we can
apply the Borel transgression theorem 2.25 with $N=2n-1$, implying that
generators $\tilde{\phi}_{1},\dots,\tilde{\phi}_{2n-1}$ of the zeroeth column
can be chosen so that

(2.66) $\displaystyle d_{2i}\tilde{\phi}_{i}=\Psi_{i+1},\quad
i\in\\{1,3,\dots,2n-1\\}.$

Now the inclusion $W_{n}\to W_{3n}$ induces a morphism from the Hochschild-
Serre spectral sequence for the Lie-algebra-subalgebra pair
$(W_{3n},{\operatorname{\mathfrak{gl}}}_{3n}(\mathbb{R}))$ to the spectral
sequence for $(W_{n},{\operatorname{\mathfrak{gl}}}_{n}(\mathbb{R}))$. Under
this morphism, the generators $\Psi_{i}$ for $W_{3n}$ restrict to equivalent
ones for $W_{n}$ by the explicit formula for them given in Lemma 2.23.
By functoriality of the Hochschild-Serre spectral sequence, the generators
$\tilde{\phi}_{i}$ must then restrict to something nonzero in the space
$E^{0,2i-1}_{2i}$ of the spectral sequence for $W_{n}$. By an inductive
argument, this vector space is one-dimensional and generated by the generator
$\phi_{i}$, so the image of $\tilde{\phi}_{i}$ under this morphism must be a
nonzero multiple of $\phi_{i}$. Hence, up to a nonzero constant, we have in
the spectral sequence for $W_{n}$ that

(2.67) $\displaystyle d_{2i}\phi_{i}=\Psi_{i+1},\quad
i\in\\{1,3,\dots,2n-1\\}.$

This proves that all generators $\phi_{1},\dots,\phi_{2n-1}$ in the spectral
sequence for $W_{n}$ map as desired. Since the differential of the Hochschild-
Serre spectral sequence is multiplicative and all pages are generated by the
$\phi_{i}$ and the $\Psi_{i}$, this fully specifies the differential on every
page. ∎

######  Remark 2.27.

In [10], the above argument is carried out with the Lie-algebra-subalgebra
pair $(W_{2n},{\operatorname{\mathfrak{gl}}}_{2n}(\mathbb{R}))$ rather than
$(W_{3n},{\operatorname{\mathfrak{gl}}}_{3n}(\mathbb{R}))$, which would not
fulfil the requirements of the version of the Borel transgression theorem we
use here.

[grid=chess,labelstep=1]0…20…1 R 0 1 2-1 0 1 [grid=chess,labelstep=1]0…40…4 R
0 1 2-1 0 1 2-1 b 0 1 0 3 a 0 1 2-1 0 1 2-1 0 1 a b [arrowto]

Figure 1. The spectral sequences for $W_{1}$ and $W_{2}$, with nonvanishing
differentials indicated. Every dot represents one basis element of the term in
the given position. The cohomology of $W_{1}$ is only nontrivial in degree $0$
and $3$, whereas the cohomology of $W_{2}$ is nontrivial in degree $0,5,7$ and
$8$, degree $5$ and $8$ having multiplicity 2.

This allows one to fully calculate the dimensions of $H^{\bullet}(W_{n})$ in
all degrees and even offers some insight into the behavior of representatives
of the cohomology classes. We are going to sumarize the most important
properties of $H^{\bullet}(W_{n})$ in the following corollary:

######  Corollary 2.28.

The space $H^{k}(W_{n})$ is trivial when $1\leq k\leq 2n$ or $k>n^{2}+2n$. The
wedge product of two cohomology classes of positive degree in
$H^{\bullet}(W_{n})$ is zero.

###### Proof.

Any element in $E^{p,q}_{2}$ with $(p,q)\neq(0,0)$ is a linear combination of
terms of the form

(2.68)
$\displaystyle\phi_{i_{1}}\dots\phi_{i_{s}}\Psi^{m_{1}}_{j_{1}}\dots\Psi^{m_{t}}_{j_{t}},$

where we have ordered the groups of indices so that $i_{1}<\dots<i_{s}$ and
$j_{1}<\dots<j_{t}$. We further have

(2.69) $\displaystyle i_{1}+\dots+i_{s}=q,\quad
m_{1}j_{1}+\dots+m_{t}j_{t}=p,$

and $s$ and $t$ are possibly zero, but not both at the same time. Theorem 2.26
shows:

  * i)

If $s=0$ or $i_{1}>j_{1}$, then
$\phi_{j_{1}-1}\phi_{i_{1}}\dots\phi_{i_{s}}\Psi^{m_{1}-1}_{j_{1}}\dots\Psi^{m_{t}}_{j_{t}}$
maps to the term (2.68) under the differential $d_{2(j_{1}-1)}$.

  * ii)

If $t=0$ or $i_{1}<j_{1}$, then the term (2.68) maps to
$\phi_{i_{2}}\dots\phi_{i_{s}}\Psi_{i_{1}+1}\Psi^{m_{1}}_{j_{1}}\dots\Psi^{m_{t}}_{j_{t}}$
under the differential $d_{2i_{1}}$.

Since necessarily $i_{1}$ is odd and $j_{1}$ is even, one of i) or ii) must
hold. Let us show that under the assumptions of ii), and if $p\leq n$ or
$p+q\leq 2n$, then the product
$\Psi_{i_{1}+1}\Psi^{m_{1}}_{j_{1}}\dots\Psi^{m_{t}}_{j_{t}}$ is nonzero. This
is equivalent to showing

(2.70) $\displaystyle i_{1}+1+m_{1}j_{1}+\dots+m_{t}j_{t}\leq 2n.$

If $t=0$, then this is trivial, so assume from here on $t>0$ and
$i_{1}<j_{1}$. If $p\leq n$, then we know that

(2.71) $\displaystyle i_{1}<j_{1}\leq m_{1}j_{1}+\dots+m_{t}j_{t}=p,$

thus

(2.72) $\displaystyle i_{1}+1+m_{1}j_{1}+\dots+m_{t}j_{t}\leq 2p\leq 2n.$

On the other hand, assume $p+q\leq 2n$. Note that $s=1$ is never the case,
since $i_{1}$ is always odd and $q$ is always even. Hence assume $s>1$, so
that $i_{1}+1\leq i_{1}+\dots+i_{s}$. But then

(2.73) $\displaystyle i_{1}+1+m_{1}j_{1}+\dots+m_{t}j_{t}\leq
i_{1}+\dots+i_{s}+m_{1}j_{1}+\dots+m_{t}j_{t}=q+p\leq 2n.$

With this, we have shown that $E^{p,q}_{\infty}$ is zero when $0<p+q\leq 2n$,
implying that $H^{k}(W_{n})$ vanishes in degree $0<k\leq 2n$. Further, we have
shown that $E^{p,q}_{\infty}$ is zero when $p\leq n$, so two cohomology
classes of positive degree in $H^{\bullet}(W_{n})$ correspond to two
equivalence classes in some spaces
$E^{p,q}_{\infty},E^{p^{\prime},q^{\prime}}_{\infty}$ with $p,p^{\prime}>n$.
By multiplicativity of the Hochschild-Serre spectral sequence, their product
must then correspond to an equivalence class in
$E^{p+p^{\prime},q+q^{\prime}}_{\infty}$, which must be zero because
$p+p^{\prime}>2n$. This proves the corollary. ∎

##  3\. Gelfand–Fuks cohomology on Euclidean space

In this section, we calculate the Gelfand–Fuks cohomology
$H^{\bullet}(\mathfrak{X}(M))$ for $M=\mathbb{R}^{n}$. We follow an elaborate
outline by Bott, see [13]. This approach will allow us to easily extend our
proof to the Gelfand–Fuks cohomology of finite disjoint unions
$\mathbb{R}^{n}\sqcup\dots\sqcup\mathbb{R}^{n}$ and also to certain _diagonal_
cohomologies thereof, a concept which we introduce in Section 5.

###  3.1. Definitions and calculation

The Lie algebra of smooth vector fields $\mathfrak{X}(\mathbb{R}^{n})$ on
Euclidean space is a locally convex Lie algebra with respect to the standard
Fréchet topology. We are interested in its continuous Chevalley-Eilenberg
cohomology with respect to this topology. We can express vector fields in the
canonical coordinates

(3.1)
$\displaystyle\mathfrak{X}(\mathbb{R}^{n})=\left\\{\sum_{i=1}^{n}f_{i}\partial_{i}:f_{i}\in
C^{\infty}(\mathbb{R}^{n})\right\\}$

Let us again identify some structures:

######  Definition 3.1.

Let $t>0$. We define the family of scaling operators $\\{T_{t}\\}_{t>0}$ as

(3.2) $\displaystyle T_{t}:\mathbb{R}^{n}\to\mathbb{R}^{n},\quad x\mapsto tx.$

We adopt the notation that if $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a
local diffeomorphism, then we denote by $\phi^{*}$ its pullback on vector
fields $\mathfrak{X}(\mathbb{R}^{n})$

(3.3) $\displaystyle\phi^{*}(X):=(d\phi)^{-1}(X\circ\phi).$

We overload our notation and also write $\phi^{*}$ for the pullback on
cochains $C^{k}(\mathfrak{X}(\mathbb{R}^{n}))$ for all $k\geq 0$:

(3.4)
$\displaystyle(\phi^{*}c)(X_{1},\dots,X_{k}):=c(\phi^{*}X_{1},\dots,\phi^{*}X_{k}),\quad\forall
k\geq 0.$

Note that for $c\in C^{0}(\mathfrak{X}(\mathbb{R}^{n}))$ this amounts to
$\phi^{*}c=c$.
For the scaling operators $\\{T_{t}\\}_{t>0}$, this translates to

$\displaystyle T_{t}^{*}X=\frac{1}{t}(X\circ T_{t})\quad\forall
X\in\mathfrak{X}(\mathbb{R}^{n}),$
$\displaystyle(T_{t}^{*}c)(X_{1},\dots,X_{k}):=\frac{1}{t^{k}}c(X_{1}\circ
T_{t},\dots,X_{k}\circ T_{t})\quad\forall c\in
C^{k}(\mathfrak{X}(\mathbb{R}^{n}).$

######  Definition 3.2.

We define the subspace
$\mathfrak{X}_{{\operatorname{\mathrm{pol}}}}(\mathbb{R}^{n})\subset\mathfrak{X}(\mathbb{R}^{n})$
of _polynomial vector fields_

(3.5)
$\displaystyle\mathfrak{X}_{{\operatorname{\mathrm{pol}}}}(\mathbb{R}^{n}):=\left\\{\sum_{i=1}^{n}f_{i}\partial_{i}:f_{i}\in\mathbb{R}[x_{1},\dots,x_{n}]\right\\}.$

It admits the structure of a graded Lie algebra

(3.6)
$\displaystyle\mathfrak{X}_{{\operatorname{\mathrm{pol}}}}(\mathbb{R}^{n})=\bigoplus_{k\in\mathbb{Z}}P_{k},\quad
P_{k}:=\\{X\in\mathfrak{X}_{\mathrm{pol}}(\mathbb{R}^{n}):T_{t}^{*}X=t^{k}X\quad\forall
t>0\\}.$

Elements of $P_{k}$ are called _homogeneous_ vector fields (of degree $k$).

######  Remark 3.3.

Compare this to the grading
$W_{n}=\bigoplus_{k\in\mathbb{Z}}{\operatorname{\mathfrak{g}}}_{k}$ on formal
vector fields. The sets $P_{k}$ are the images of the natural embeddings
$\mathfrak{g}_{k}\to\mathfrak{X}(\mathbb{R}^{n})$ that we get from considering
finite formal vector fields in $W_{n}$ as polynomial vector fields on
$\mathbb{R}^{n}$.

######  Definition 3.4.

For all $k\in\mathbb{Z},q\geq 0$, define

(3.7) $\displaystyle F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n})):=\left\\{c\in
C^{q}(\mathfrak{X}(\mathbb{R}^{n})):\lim_{t\to
0}t^{-k}T_{t}^{*}c(X_{1},\dots,X_{q})\text{ exists }\forall
X_{i}\in\mathfrak{X}(\mathbb{R}^{n})\right\\}.$

######  Lemma 3.5.

The spaces $F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ for
$k\in\mathbb{Z}$ fulfil the following properties:

  * i)

For every $k\in\mathbb{Z}$, the space
$F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is a subcomplex of
$C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$.

  * ii)

We have a descending chain

(3.8) $\displaystyle\dots\subset
F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\subset
F^{k-1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\subset
F^{k-2}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\subset\dots$

  * iii)

For all $k,l\in\mathbb{Z}$ we have

(3.9) $\displaystyle F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\wedge
F^{l}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\subset
F^{k+l}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})).$

  * iv)

For $k\leq-n$ we have
$F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))=C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$.

Summarizing,
$\\{F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\\}_{k\in\in\mathbb{Z}}$
constitutes a descending filtration of the Chevalley-Eilenberg complex
$C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ which is bounded from above.

###### Proof.

i) Pulling back vector fields along local diffeomorphisms is a Lie algebra
homomorphism, so

(3.10) $\displaystyle T_{t}^{*}[X,Y]=[T_{t}^{*}X,T_{t}^{*}Y]\quad\forall
X,Y\in\mathfrak{X}(\mathbb{R}^{n}).$

Hence, the pullback on cochains $T_{t}^{*}$ commutes with the Lie algebra
differential, so if the appropriate limits exist for a cochain $c$, they also
do for $dc$. Thus the $F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ are
indeed subcomplexes.
ii) This follows since if the limit $\lim_{t\to 0}t^{-k}f(t)$ exists for some
function $t\mapsto f(t)$, so does $\lim_{t\to 0}t^{-(k-1)}f(t)=0$.
iii) The compatibility with the wedge product follows from

(3.11) $\displaystyle T_{t}^{*}(c_{1}\wedge c_{2})=T_{t}^{*}c_{1}\wedge
T_{t}^{*}c_{2}\quad\forall c_{1},c_{2}\in
C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})).$

iv) In degree zero the statement is clear, since

(3.12) $\displaystyle
F^{k}C^{0}(\mathfrak{X}(\mathbb{R}^{n}))=C^{0}(\mathfrak{X}(\mathbb{R}^{n}))\quad\forall
k\leq 0.$

Assume now $q>0$. Fix $c\in C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$ and
$X_{1},\dots,X_{q}\in\mathfrak{X}(\mathbb{R}^{n})$. Applying the Hadamard
lemma to every one of the $X_{i}$ shows that there are vector fields
$X_{k}^{(i)}$ for all $k=1,\dots,q$ and $i=1,\dots,n$ so that

(3.13) $\displaystyle
X_{k}(x)=X_{k}(0)+\sum_{i=1}^{n}x_{i}X_{k}^{(i)}(x),\quad\forall
x\in\mathbb{R}^{n}.$

Then

(3.14) $\displaystyle
T_{t}^{*}X_{k}(x)=\frac{1}{t}X_{k}(0)+\sum_{i=1}^{n}x_{i}X_{k}^{(i)}(tx)\quad\forall
x\in\mathbb{R}^{n}.$

Hence we can rewrite

(3.15) $\displaystyle
T_{t}^{*}c(X_{1},\dots,X_{q})=c\left(\frac{1}{t}X_{1}(0)+\sum_{i=1}^{n}x_{i}X_{1}^{(i)}(tx),\dots,\frac{1}{t}X_{q}(0)+\sum_{i=1}^{n}x_{i}X_{q}^{(i)}(tx)\right).$

Decomposing this expression using multilinearity of $c$, we find that all the
terms whose order in $t$ is lower than $-n$ have to vanish, since, for any
collection of indices $\\{i_{1},\dots,i_{n+1}\\}\subset\\{1,\dots,q\\}$, the
set $\\{X_{i_{1}}(0),\dots,X_{i_{n+1}}(0)\\}$is necessarily linearly dependent
and $c$ is skew-symmetric. Note also that on any compact set in
$\mathbb{R}^{n}$, the vector fields $x\mapsto x_{i}X_{k}^{(i)}(tx)$ converge
uniformly to the vector field $x\mapsto x_{i}X_{k}^{(i)}(0)$ for $t\to 0$, and
the same holds for their derivatives. Combining the two previous facts, the
continuity of $c$ lets us conclude that the limit $\lim_{t\to
0}\tfrac{1}{t^{-n}}T_{t}^{*}c(X_{1},\dots,X_{q})$ exists. This proves the
statement. ∎

The analysis at the end of the previous proof motivates a different
characterization of the filtration:

######  Lemma 3.6.

Let $q>0$. A cochain $c\in C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$ lies in
$F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$ if and only if for all homogeneous
vector fields
$X_{1},\dots,X_{q}\in\mathfrak{X}_{{\operatorname{\mathrm{pol}}}}(\mathbb{R}^{n})$
we have

(3.16) $\displaystyle\sum_{i=1}^{q}\deg X_{i}<k\implies
c(X_{1},\dots,X_{k})=0.$

###### Proof.

$\Rightarrow:$ Let $c\in F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$, and let
$X_{1},\dots,X_{q}\in\mathfrak{X}_{{\operatorname{\mathrm{pol}}}}(\mathbb{R}^{n})$
be any homogeneous polynomial vector fields. Then

(3.17) $\displaystyle
t^{-k}T_{t}^{*}c(X_{1},\dots,X_{q})=t^{\left(\sum_{i=1}^{q}\deg
X_{i}\right)-k}c(X_{1},\dots,X_{q}).$

If $\sum_{i=1}^{q}\deg X_{i}<k$, this can only converge to a finite value in
the limit $t\to 0$ if $c$ vanishes on $(X_{1},\dots,X_{q})$. This proves the
first implication.
$\Leftarrow:$ Let $c\in C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$ be so that it
vanishes on all homogeneous vector fields whose degree adds to a value smaller
$k$. Set $r:=\max\\{q+k,1\\}$. Given any vector fields
$X_{1},\dots,X_{q}\in\mathfrak{X}(\mathbb{R}^{n})$, apply the Hadamard lemma
to each of them $r$ times to write, in multiindex notation,

(3.18) $\displaystyle
X_{k}(x)=Y_{k}(x)+\sum_{\begin{subarray}{c}\vec{\alpha}\in\mathbb{N}^{n}_{0}\\\
|\vec{\alpha}|=r+1\end{subarray}}x_{1}^{\alpha_{1}}\dots
x_{n}^{\alpha_{n}}Z_{k}^{\vec{\alpha}}(x)=:Y_{k}(x)+Z_{k}(x),$

where $Y_{k}$ is a polynomial vector field whose homogeneous components are of
degree $\leq r-1$, and $Z_{k}^{\vec{\alpha}}\in\mathfrak{X}(\mathbb{R}^{n})$.
Using multilinearity of $c$, decompose $T_{t}^{*}c(X_{1},\dots,X_{q})$ into
summands of the form $\tfrac{1}{t^{k}}T_{t}^{*}c$ with all arguments being
some $Y_{k}$ or some $Z_{k}$ for $k=1,\dots,q$. The limits $\lim_{t\to
0}tT_{t}^{*}Y_{k}$ and $\lim_{t\to 0}t^{-r}T_{t}^{*}Z_{k}$ in
$\mathfrak{X}(\mathbb{R}^{n})$ exist for all $k=1,\dots,q$. Thus any summand
in the decomposition of
$\tfrac{1}{t^{k}}\left(T_{t}^{*}c\right)(X_{1},\dots,X_{q})$ is of the
following form for a certain $s\geq 0$ and certain
$i_{1},\dots,i_{s},j_{1},\dots,j_{q-s}\in\\{1,\dots,q\\}$,:

(3.19) $\displaystyle\frac{1}{t^{k}}\left(T_{t}^{*}c\right)($ $\displaystyle
Z_{i_{1}},\dots,Z_{i_{s}},Y_{j_{1}},\dots,Y_{j_{q-s}})$
$\displaystyle=t^{rs-(q-s)-k}(T_{t}^{*}c)(t^{-r}Z_{i_{1}},\dots,t^{-r}Z_{i_{s}},tY_{j_{1}},\dots,tY_{j_{q-s}}).$

If $s\geq 1$, then we have due to $r\geq q+k$,

(3.20) $\displaystyle rs-(q-s)-k\geq s\geq 1,$

and the limit $t\to 0$ exists.
If $s=0$, then the summand is of the form
$\tfrac{1}{t^{k}}T_{t}^{*}c(Y_{1},\dots,Y_{q})$. The $Y_{k}$ are polynomial
vector fields, so we may use multilinearity to decompose this term so that we
get terms of $\tfrac{1}{t^{k}}T_{t}^{*}c\leavevmode\nobreak\ $ whose arguments
are homogeneous polynomial vector fields. In every such summand,
$\tfrac{1}{t^{k}}T_{t}^{*}c$ can be replaced by $t^{\Sigma-k}c$, where
$\Sigma$ is the sum of the degrees of inserted homogeneous vector fields. By
assumption on $c$, every summand where $\Sigma<k$ must vanish. This implies
that as $t\to 0$, the term $\tfrac{1}{t^{k}}T_{t}^{*}c(Y_{1},\dots,Y_{k})$
converges to a finite value. This concludes the proof. ∎

###  3.2. Gelfand–Fuks cohomology of Euclidean space

Let us now connect the Lie algebra $\mathfrak{X}(\mathbb{R}^{n})$ to the Lie
algebra of formal vector fields $W_{n}$. To this end, fix, for the rest of the
section some local frame of vector fields around $0\in\mathbb{R}^{n}$ to
induce an isomorphism
$J^{\infty}_{0}\mathfrak{X}(\mathbb{R}^{n})\stackrel{{\scriptstyle\sim}}{{\to}}W_{n}$,
cf. Remark 2.2, so that every elements of $W_{n}$ can be written as the
infinity-jet $j^{\infty}_{0}X$ at zero of some
$X\in\mathfrak{X}(\mathbb{R}^{n})$.

######  Definition 3.7.

If $X\in\mathfrak{X}(\mathbb{R}^{n})$, denote by $\tilde{X}^{(r)}$ the
polynomial vector field corresponding to the $r$-jet of $X$ at zero. Define
then for all $k\in\mathbb{Z}$ the maps

(3.23)
$\displaystyle\begin{gathered}\gamma_{k}:F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\to
C^{\bullet}_{(k)}(W_{n}),\\\
(\gamma_{k}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q}):=\lim_{r\to\infty}\lim_{t\to
0}t^{-k}(T_{t}^{*}c^{\prime})(\tilde{X}^{(r)}_{1},\dots,\tilde{X}^{(r)}_{q}),\end{gathered}$
(3.24) $\displaystyle\beta_{k}:C^{\bullet}_{(k)}(W_{n})\to
F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})),\quad(\beta_{k}c^{\prime})(Y_{1},\dots,Y_{k}):=c^{\prime}\left(j^{\infty}_{0}Y_{1},\dots,j^{\infty}_{0}Y_{k}\right).$

for all $X_{1},\dots,X_{q}\in W_{n},$
$Y_{1},\dots,Y_{q}\in\mathfrak{X}(\mathbb{R}^{n})$, $c\in
F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$ and $c^{\prime}\in
C^{q}_{(k)}(W_{n})$.

######  Lemma 3.8.

The maps $\beta_{k}$ and $\gamma_{k}$ are well-defined chain maps with
$\gamma_{k}\circ\beta_{k}=\operatorname{id}$.

###### Proof.

Note first that $\gamma_{k}$ is well-defined: By definition of the filtration,
for every $c\in F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ the pointwise
limit $\lim_{t\to 0}t^{-k}T_{t}^{*}c$ exists. Further, the sequence

(3.25) $\displaystyle\left(\lim_{t\to
0}t^{-k}(T_{t}^{*}c)(\tilde{X}^{(r)}_{1},\dots,\tilde{X}^{(r)}_{q})\right)_{r\in\mathbb{N}}$

is eventually constant in $r$, since the cochain $\lim_{t\to 0}T_{t}^{*}c$
vanishes on homogeneous vector fields whose sum of degrees is larger than $k$,
cf. (3.17). Further, if $c\in F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$, then
by Lemma 3.6, $c$ vanishes on polynomial vector fields the sum of whose
degrees is smaller than $k$. Hence $\gamma_{k}c\in C^{q}_{(k)}(W_{n})$.
Analogously, if $c\in C^{k}_{(r)}(W_{n})$, then $\beta_{k}c$ vanishes on
polynomial vector fields whose sum of degrees is smaller than $k$, hence Lemma
3.6 implies $\beta_{k}c\in F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$.
The identification of a finite formal vector field $X_{n}$ with its Taylor
polynomial in $\mathfrak{X}_{\mathrm{pol}}(\mathbb{R}^{n})$ is a Lie algebra
morphism, and so is the pullback of a vector field by the diffeomorphism
$T_{t}$. Hence $\gamma_{k}$ is a chain map.
The map $\beta_{k}$ is a chain map since taking the infinite jet of a vector
field at zero is a Lie algebra morphism $\mathfrak{X}(\mathbb{R}^{n})\to
W_{n}$.
It remains to check the composition of the two maps. Let us prove

(3.26)
$\displaystyle(\gamma_{k}\beta_{k}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q})=c(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q})$

for all $c\in C^{q}_{(k)}(W_{n})$ and homogeneous formal vector fields
$j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q}\in W_{n}$ whose degree sums up
to $k$. This suffices, since we have $c,\gamma_{k}\beta_{k}c\in
C^{q}_{(k)}(W_{n})$, so both cochains vanish on all other homogeneous vector
fields.
For a homogeneous vector field $X$ and sufficiently large $r$, we have

(3.27) $\displaystyle j^{\infty}_{0}\tilde{X}^{(r)}=j^{\infty}_{0}X\text{ and
}\deg\tilde{X}^{(r)}=\deg j^{\infty}_{0}X.$

Hence, since if $j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q}\in W_{n}$ are
homogeneous formal vector fields whose degree sums up to $k$, then

(3.28)
$\displaystyle(\gamma_{k}\beta_{k}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q})$
$\displaystyle=\lim_{r\to\infty}c(j^{\infty}_{0}\tilde{X}^{(r)}_{1},\dots,j^{\infty}_{0}\tilde{X}^{(r)}_{q})$
$\displaystyle=c(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{q}).$

This concludes the proof. ∎

######  Corollary 3.9.

For every $k\in\mathbb{Z}$,

(3.29) $\displaystyle 0\to
F^{k+1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\hookrightarrow
F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\stackrel{{\scriptstyle\gamma_{k}}}{{\to}}C^{\bullet}_{(k)}(W_{n})\to
0$

is a split exact sequence of cochain complexes.

###### Proof.

Because
$\\{F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\\}_{k\in\mathbb{Z}}$
constitutes a filtration of $C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$, the
inclusion $F^{k+1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\to
F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is a chain map. The map
$\gamma_{k}$ is a chain map due to Lemma 3.8. Hence, the sequence is a
sequence of chain complexes. The injectivity of the first map is clear, and
the second map is surjective, since it is split by $\beta_{k}$. Exactness at
the middle term follows from the characterization of the filtration in Lemma
3.6. This concludes the proof. ∎

For the following lemma, recall the action of a Lie algebra on its Chevalley-
Eilenberg cochains and the interior product defined in Definition 2.6.

######  Lemma 3.10.

Let $M$ be a smooth manifold, let $\\{\phi_{t}\\}_{t>0}$ a one-parameter
semigroup of diffeomorphisms on $M$, and $\\{X_{t}\in\mathfrak{X}(M)\\}_{t>0}$
its time-dependent generator. Then, for all $t_{0},t_{1}>0$ and $c\in
C^{\bullet}(\mathfrak{X}(M))$ we have

(3.30)
$\displaystyle\phi_{t_{1}}^{*}c-\phi_{t_{0}}^{*}c=K_{t_{1},t_{0}}dc+dK_{t_{1},t_{0}}c,$

where $K_{t_{1},t_{0}}:=-\int_{t_{0}}^{t_{1}}\phi_{t}^{*}(X_{t}\intprod c)dt$.

###### Proof.

By definition of the generator $\\{X_{t}\\}_{t\in\mathbb{R}}$, we have for
every $Y\in\mathfrak{X}(\mathbb{R}^{n})$

(3.31)
$\displaystyle\frac{d}{dt}\phi_{t}^{*}Y=\phi_{t}^{*}[X_{t},Y]\quad\forall
t\in\mathbb{R}.$

By standard differentiation rules and the homotopy formula from Lemma 2.7 we
have for $c\in C^{q}(\mathfrak{X}(M))$ and
$Y_{1},\dots,Y_{q}\in\mathfrak{X}(\mathbb{R}^{n})$:

(3.32) $\displaystyle\frac{d}{dt}\phi^{*}_{t}c(X_{1},\dots,X_{q})$
$\displaystyle=\sum_{k=1}^{q}(\phi_{t}^{*}c)(X_{1},\dots,[X_{t},X_{k}],\dots,X_{q})$
$\displaystyle=-(X_{t}\cdot\phi_{t}^{*}c)(X_{1},\dots,X_{q})=-d(X_{t}\intprod\phi_{t}^{*}c)-X_{t}\intprod(\phi_{t}^{*}dc).$

An application of the fundamental theorem of calculus now gives the desired
statement. ∎

######  Corollary 3.11.

The complex $F^{1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is acyclic.

###### Proof.

Consider for every $t_{0},t_{1}>0$ the operator
$K_{t_{0},t_{1}}:C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\to
C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ from Lemma 3.10 associated to the
one-parameter semigroup $\\{T_{t}\\}_{t>0}$. By definition, for $c\in
F^{1}C^{k}(\mathfrak{X}(\mathbb{R}^{n}))$ and all all
$X_{1},\dots,X_{k}\in\mathfrak{X}(\mathbb{R}^{n})$, we have

(3.33) $\displaystyle\lim_{t\to 0}(T_{t}^{*}c)(X_{1},\dots,X_{k})=0.$

But then

(3.34) $\displaystyle c(X_{1},\dots,X_{k})$ $\displaystyle=\lim_{t\to
0}(T_{1}^{*}c-T_{t}^{*}c)(X_{1},\dots,X_{k})$ $\displaystyle=\lim_{t\to
0}\left(K_{1,t}dc+dK_{1,t}c\right)(X_{1},\dots,X_{k})=:(Kdc+dKc)(X_{1},\dots,X_{k}).$

In the above, we defined the operator
$K:C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})\to
C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ as the pointwise limit of the
operators $K_{1,t}$:

(3.35) $\displaystyle Kc=\lim_{t\to
0}K_{1,t}c=\int_{0}^{1}T_{t}^{*}(X_{t}\intprod c)\mathop{}\\!\mathrm{d}t,$

where $X_{t}$ is the generator of the semigroup of diffeomorphisms
$\\{T_{t}\\}_{t>0}$. Hence $K$ is a chain homotopy between the identity map
and zero on $F^{1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$, which proves the
statement. ∎

Finally we can state a variation of Lemma 1 in Section 2.4.B. of [10]:

######  Theorem 3.12.

The inclusion

(3.36) $\displaystyle F^{0}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\hookrightarrow C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$

and the maps

(3.37) $\displaystyle\gamma_{0}:F^{0}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))\to
C^{q}_{(0)}(W_{n}),\quad\beta_{0}:C^{q}_{(0)}(W_{n})\to
F^{0}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$

from Definition 3.7 are quasi-isomorphisms and unital algebra morphisms.
In particular, there is an isomorphism of algebras

(3.38) $\displaystyle H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\cong
H^{\bullet}(W_{n}),$

and the wedge product of two elements of positive degree in
$H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is zero.

###### Proof.

The compatibility of the maps with the algebra structure is immediate from the
multiplicativity of the filtration $F^{k}C^{q}(\mathfrak{X}(\mathbb{R}^{n}))$
and the formulas for $\gamma_{0},\beta_{0}$. By the split exact sequence of
Corollary 3.9, we have for every $k\in\mathbb{Z}$ an isomorphism of cochain
complexes

(3.39) $\displaystyle F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\cong
F^{k+1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\oplus
C^{\bullet}_{(k)}(W_{n}).$

Inserting $k=0$ into (3.39), the acyclicity of
$F^{1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ by Corollary 3.11 shows that
we have algebra isomorphisms

(3.40) $\displaystyle
H^{\bullet}(F^{0}\tilde{C}^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})))\cong
H^{\bullet}_{(0)}(W_{n})\stackrel{{\scriptstyle\text{Prop.\leavevmode\nobreak\
\ref{PropositionWnCohmologyInZeroethDegree}}}}{{\cong}}H^{\bullet}(W_{n}),$

hence $\gamma_{0}$ and its splitting $\beta_{0}$ must be quasi-isomorphisms.
Let further $k=-1,\dots,-n$ in (3.39). By Proposition v), the complexes
$\tilde{C}^{\bullet}_{(k)}(W_{n})$ are acyclic and we have isomorphisms

(3.41) $\displaystyle
H^{\bullet}(F^{0}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})))$
$\displaystyle\cong
H^{\bullet}(F^{-1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})))$
$\displaystyle\cong\dots$ $\displaystyle\cong
H^{\bullet}(F^{-n}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})))$
$\displaystyle\stackrel{{\scriptstyle\text{Lem.\leavevmode\nobreak\
\ref{LemmaPropertiesOfRnFiltration}}}}{{=}}H^{\bullet}(C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})))=H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})).$

Hence the inclusion
$F^{0}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\hookrightarrow
C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is a quasi-isomorphism.
The product of positive degree elements in
$H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is zero since this is true for
$H^{\bullet}(W_{n})$ by Corollary 2.28. ∎

We end this section by an extension of our proof of
$H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\cong H^{\bullet}(W_{n})$ to the
setting of Gelfand–Fuks cohomology of a disjoint union of finitely many copies
of Euclidean space.

######  Remark 3.13.

The isomorphism of topological Lie algebras

(3.42)
$\displaystyle\mathfrak{X}(\mathbb{R}^{n}\sqcup\mathbb{R}^{n})\cong\mathfrak{X}(\mathbb{R}^{n})\oplus\mathfrak{X}(\mathbb{R}^{n}),$

insinuates that the Gelfand–Fuks cohomology of such a disjoint union may be
calculated by the use of a _Künneth formula_ : For finite-dimensional Lie
algebras ${\operatorname{\mathfrak{g}}},{\operatorname{\mathfrak{h}}}$ over
$\mathbb{R}$, the Künneth theorem implies

(3.43) $\displaystyle
H^{\bullet}({\operatorname{\mathfrak{g}}}\oplus{\operatorname{\mathfrak{h}}})\cong
H^{\bullet}({\operatorname{\mathfrak{g}}})\otimes
H^{\bullet}({\operatorname{\mathfrak{h}}}).$

And indeed, such Künneth theorems in Lie algebra cohomology are well-known in
the purely algebraic setting, but extending them to _continuous_ Lie algebra
cohomology relies on nontrivial topological assumptions in order to deal with
the arising topological tensor products and their completion. For such
formulas, the reader may, for example, consult [23]. We avoid this approach
here.

######  Proposition 3.14.

Let $M:=\bigsqcup_{i=1}^{r}\mathbb{R}^{n}$ be a disjoint collection of copies
of $\mathbb{R}^{n}$. Then every choice of order on the copies of
$\mathbb{R}^{n}$ induces an algebra isomorphism

(3.44) $\displaystyle H^{\bullet}\left(\mathfrak{X}\left(M\right)\right)\cong
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))^{\otimes^{r}}.$

###### Proof.

We mimic the proof for the $r=1$ situation, but we expand the scaling of
$\mathbb{R}^{n}$ to the same scaling in every copy of $\mathbb{R}^{n}$:

(3.45) $\displaystyle
T_{t}:\bigsqcup_{i=1}^{r}\mathbb{R}^{n}\to\bigsqcup_{i=1}^{r}\mathbb{R}^{n},\quad
x\mapsto tx.$

The definition of the corresponding spaces $F^{k}C^{q}(\mathfrak{X}(M))$ is
identical to in the $r=1$ case, and by the same proofs, they constitute a
descending, filtration that is bounded from below with

(3.46) $\displaystyle
F^{k}C^{\bullet}(\mathfrak{X}(M))=C^{\bullet}(\mathfrak{X}(M))\quad\forall
k\leq-rn.$

In analogy to Definition 3.7, we can define a map $\gamma_{k}^{(r)}$, which,
together with some choice of ordering on the components of $M$ gives rise to
an exact sequence for every $k\in\mathbb{Z}$:

(3.47) $\displaystyle 0\to F^{k+1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\to F^{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\stackrel{{\scriptstyle\gamma_{k}^{(r)}}}{{\to}}\bigoplus_{k_{1}+\dots+k_{r}=k}C^{\bullet}_{(k_{1})}(W_{n})\otimes\dots\otimes
C^{\bullet}_{(k_{r})}(W_{n})\to 0.$

For the tensor product complex on the right hand side, we can use the Künneth
theorem to calculate its cohomology, and due to acyclicity of
$C^{\bullet}_{(k)}(W_{n})$ for $k\neq 0$, the only one of the complexes with
nontrivial cohomology is the one with the condition $k_{1}+\dots+k_{r}=0$. By
the same steps as in Corollary 3.11 and Theorem 3.12 we arrive at the desired
isomorphism of vector spaces. This isomorphism respects the wedge product, as
we see with the arising quasi-isomorphism

(3.48) $\beta_{0}^{(r)}:C^{\bullet}_{(0)}(W_{n})^{\otimes^{r}}\to
C^{\bullet}(\mathfrak{X}(M)),\quad c_{1}\otimes\dots\otimes
c_{r}\mapsto\beta_{0}^{1}c_{1}\wedge\dots\wedge\beta_{0}^{r}c_{r},$

where $\beta_{0}^{k}$ maps formal cochains exactly like the map $\beta_{0}$
from the Definition 3.7, but all jets of vector fields are evaluated at the
zero in the $k$-th copy of $\mathbb{R}^{n}$. Because the $\beta_{0}$ in the
$r=1$ case respect the wedge product, so does $\beta_{0}^{(r)}$. ∎

The formula for the quasi-isomorphism $\beta_{0}^{(r)}$ from the previous
proof implies:

######  Corollary 3.15.

Let $B_{1},\dots,B_{r}\subset\mathbb{R}^{n}$ be pairwise disjoint sets
diffeomorphic to $\mathbb{R}^{n}$. Assume their union is contained in another
set $C\subset\mathbb{R}^{n}$ diffeomorphic to $\mathbb{R}^{n}$. The extension
map

(3.49) $\displaystyle[\iota_{B_{1}\cup\dots\cup
B_{r}}^{C}]:\bigotimes_{i=1}^{r}H^{\bullet}(\mathfrak{X}(B_{i}))\cong
H^{\bullet}\left(\mathfrak{X}\left(B_{1}\cup\dots\cup B_{r}\right)\right)\to
H^{\bullet}(\mathfrak{X}(C))$

is given by

(3.50)
$\displaystyle[c_{1}]\otimes\dots\otimes[c_{r}]\mapsto[\iota_{B_{i}}^{C}c_{1}\wedge\dots\wedge\iota_{B_{r}}^{C}c_{r}].$

##  4\. Cosheaf-theoretic aspects of Gelfand-Fuks cohomology

The previous section concludes the analysis of the cohomology of
$\mathfrak{X}(\mathbb{R}^{n})$. This constitutes an important building block
to understand the Gelfand–Fuks cohomology for arbitrary smooth manifolds $M$.
We will explore some properties related to _extension_ of the cochains from a
smaller to a larger open set of $M$. The following presentation of these
results, especially the framing in terms of cosheaf-theoretic data, is a novel
contribution to the literature, though the results themselves are implicitly
used in [1, 2, 10]. We also remark a close similarity of these methods to the
standard constructions in the theory of factorization algebras [24].

Recall that we identify $J^{\infty}_{0}\mathfrak{X}(\mathbb{R}^{n})\cong
W_{n}$ by a choice of local frame of vector fields around 0. The group of
local diffeomorphisms $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ that fix zero
admits a right action on infinity-jets $j^{\infty}_{0}X\in
J^{\infty}_{0}\mathfrak{X}(\mathbb{R}^{n})$ via the pullback of vector fields:

(4.1) $\displaystyle\phi^{*}j^{\infty}_{0}X:=j^{\infty}_{0}(\phi^{*}X).$

This action factors through to an action of the group of infinity-jets of
diffeomorphisms that fix zero, denoted
$J^{\infty}_{0}\operatorname{Diff}(\mathbb{R}^{n})$. Hence,
$J^{\infty}_{0}\operatorname{Diff}(\mathbb{R}^{n})$ acts on $W_{n}$ and by
pullback on the complex $C^{\bullet}(W_{n})$, and we write for all
$\phi\in\operatorname{Diff}(\mathbb{R}^{n})$, all $c\in C^{k}(W_{n})$ and
$X_{1},\dots,X_{k}\in\mathfrak{X}(\mathbb{R}^{n})$:

(4.2)
$\displaystyle((j^{\infty}_{0}\phi)^{*}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{k}):=c(j^{\infty}_{0}(\phi^{*}X_{1}),\dots,j^{\infty}_{0}(\phi^{*}X_{k})).$

######  Lemma 4.1.

Let $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a local diffeomorphism.

  * i)

The induced map $[\phi^{*}]:H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\to
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is a unital algebra isomorphism.

  * ii)

Assume $\phi$ is additionally orientation-preserving with respect to some
fixed orientation on $\mathbb{R}^{n}$. Then $[\phi^{*}]=\operatorname{id}$.

###### Proof.

Lemma 3.10 shows that the maps induced on
$C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ by translations

(4.3) $\displaystyle\tau_{a}:\mathbb{R}^{n}\to\mathbb{R}^{n},\quad x\mapsto
x+a,\quad a\in\mathbb{R}^{n}$

are homotopic to the identity $\operatorname{id}=\tau_{0}^{*}$. Hence on
cohomology $\tau_{a}$ acts as the identity for all $a\in\mathbb{R}^{n}$. Since
all $\tau_{a}$ are orientation-preserving, we may without loss of generality
assume that, in all that follows, $\phi$ fixes zero.
i) Recall the map $\beta_{0}:C^{\bullet}(W_{n})\to
C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ from Definition 3.7. By Theorem
3.12, $\beta_{0}$ is a quasi-isomorphism, so every cohomology class in
$H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ has a representative of the form
$\beta_{0}c$ for some $c\in C^{\bullet}(W_{n})$. By definition of $\beta_{0}$,
it intertwines the action of the group of local diffeomorphisms that fix zero
on $C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})$ with the action of
$J^{\infty}_{0}\operatorname{Diff}(\mathbb{R}^{n})$ on $C^{\bullet}(W_{n})$,
meaning

(4.4)
$\displaystyle\phi^{*}(\beta_{0})c=\beta_{0}((j^{\infty}_{0}\phi)^{*}c).$

Since $\phi$ is a local diffeomorphism, it admits a local inverse around 0, so
the action of $j^{\infty}_{0}\phi$ on $c$ is invertible. Thus
$\phi^{*}:H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))\to
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is an isomorphism of vector spaces.
The conditions $\phi^{*}(c_{1}\wedge c_{2})=\phi^{*}c_{1}\wedge\phi^{*}c_{2}$
for all $c_{1},c_{2}\in C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ and
$\phi^{*}(1)=1$ follow directly from the definition of the pullback of local
diffeomorphisms. Hence $\phi$ induces a unital algebra isomorphism.
ii) Fix an arbitrary cocycle $c\in C^{\bullet}(W_{n})$. Since $c$ is a
continuous cochain, it is zero on formal vector fields of sufficiently high
degree. Hence, for some $N\in\mathbb{N}$, it factors through to a multilinear
map

(4.5) $\displaystyle
c_{N}:\left(\frac{W_{n}}{\bigoplus_{r=N+1}^{\infty}{\operatorname{\mathfrak{g}}}_{r}}\right)^{k}\to\mathbb{R}.$

The action of $J^{\infty}_{0}\operatorname{Diff}(\mathbb{R}^{n})$ on $W_{n}$
descends to an action of the Lie group
$J^{N}_{0}\operatorname{Diff}(\mathbb{R}^{n})$ on the quotient
$\frac{W_{n}}{\bigoplus_{r=N+1}^{\infty}{\operatorname{\mathfrak{g}}}_{r}}$,
so that so that for all $X_{1},\dots,X_{k}\in\mathfrak{X}(\mathbb{R}^{n})$

(4.6)
$\displaystyle(j^{\infty}_{0}\phi^{*}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{k})=(j^{N}_{0}\phi^{*}c_{N})(j^{N}_{0}X_{1},\dots,j^{N}_{0}X_{k}),$

where the action on the right-hand side is defined analogously to the action
of the infinity-jets. The Lie algebra of
$J^{N}_{0}\operatorname{Diff}(\mathbb{R}^{n})$ is given by

(4.7) $\displaystyle
J^{N}_{0}\mathfrak{X}(\mathbb{R}^{n})\cong\frac{\bigoplus_{r=0}^{\infty}{\operatorname{\mathfrak{g}}}_{r}}{\bigoplus_{r=N+1}^{\infty}{\operatorname{\mathfrak{g}}}_{r}}\subset\frac{W_{n}}{\bigoplus_{r=N+1}^{\infty}{\operatorname{\mathfrak{g}}}_{r}}.$

Since $\phi$ is orientation-preserving, its $N$-jet at zero lies in the
identity component of the Lie group
$J_{0}^{N}\operatorname{Diff}(\mathbb{R}^{n})$. This component is generated by
the image of the exponential map, there exist vector fields
$Y_{1},\dots,Y_{r}\in\mathfrak{X}(\mathbb{R}^{n})$ so that for all
$X_{1},\dots,X_{k}\in\mathfrak{X}(\mathbb{R}^{n})$

(4.8)
$\displaystyle(j^{\infty}_{0}\phi^{*}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{k})$
$\displaystyle=(j^{N}_{0}\phi^{*}c_{N})(j^{N}_{0}X_{1},\dots,j^{N}_{0}X_{k})$
$\displaystyle=(\exp(j^{N}_{0}Y_{1})^{*}\dotsm\exp(j^{N}_{0}Y_{r})^{*}c_{N})(j^{N}_{0}X_{1},\dots,j^{N}_{0}X_{k})$
$\displaystyle=(\exp(j^{\infty}_{0}Y_{1})^{*}\dotsm\exp(j^{\infty}_{0}Y_{r})^{*}c)(j^{\infty}_{0}X_{1},\dots,j^{\infty}_{0}X_{k}).$

The action of $J^{\infty}_{0}\mathfrak{X}(\mathbb{R}^{n})\cong W_{n}$ on
$H^{\bullet}(W_{n})$ is trivial by Corollary 2.8, and as a consequence

(4.9)
$\displaystyle[\phi^{*}]([\beta_{0}c])=[\beta_{0}(\exp(j^{\infty}_{0}Y_{1})^{*}\dotsm\exp(j^{\infty}_{0}Y_{r})^{*}c)]=[\beta_{0}c].$

Hence $[\phi^{*}]=\operatorname{id}$ as a map on cohomology, and the statement
is shown. ∎

We will now make use of the language of _cosheaves_ to describe the extension
of Gelfand-Fuks cochains between open sets of a smooth manifold. While sheaf
theory is well known, the dual concept of cosheaves is less commonly
considered. For self-containedness, we direct the reader to Appendix A, or
[25] for a more detailed study of both sheaf and cosheaf theory.

######  Definition 4.2.

Let $M$ be a smooth manifold, and $U\subset V$ open subsets of $M$.

  * i)

Define the _extension of cochains_

(4.10) $\displaystyle\iota_{U}^{V}:C^{\bullet}(\mathfrak{X}(U))$
$\displaystyle\to C^{\bullet}(\mathfrak{X}(V))$

on cochains of degree $k>0$ as

(4.11) $\displaystyle(\iota_{U}^{V}c)(X_{1},\dots,X_{k})$
$\displaystyle:=c(X_{1}\big{|}_{U},\dots,X_{k}\big{|}_{U})$

for all $c\in C^{k}(\mathfrak{X}(U))$, $X_{1},\dots,X_{k}\in\mathfrak{X}(V)$.
On cochains of degree 0, we set $\iota_{U}^{V}:C^{0}(\mathfrak{X}(U))\to
C^{0}(\mathfrak{X}(V))$ to be the identity.

  * ii)

The extension of cochains induces an _extension of cohomology classes_

(4.12) $\displaystyle\iota_{U}^{V}:H^{\bullet}(\mathfrak{X}(U))\to
H^{\bullet}(\mathfrak{X}(V))$

which we will denote with the same symbol $\iota_{U}^{V}$ by an abuse of
notation.

  * iii)

We define a precosheaf $\mathcal{H}^{\bullet}$ of algebras over $M$, assigning
to an open set $U\subset M$ the algebra $H^{\bullet}(\mathfrak{X}(U))$, and to
an inclusion of open sets $U\subset V$ the extension map
$\iota_{U}^{V}:H^{\bullet}(\mathfrak{X}(U))\to H^{\bullet}(\mathfrak{X}(V))$.

An alternative perspective on the extension maps $\iota_{U}^{V}$ is that they
are the pullback along the inclusion map $U\to V$, as defined in (3.4). This
map is a local diffeomorphism, hence this pullback is well-defined.
For the next corollary, recall that a $\mathrm{C}$-valued precosheaf on $M$ is
simply a covariant functor from the category $\mathrm{Open}(M)$ of open sets
of $M$ to the category $\mathrm{C}$.

######  Definition 4.3.

Let $A$ be a associative $\mathbb{R}$\- algebra and $X$ a locally connected
topological space. We define the _constant precosheaf of algebras_ associated
to $A$ as the assignment

(4.13) $\displaystyle U\mapsto A^{\otimes\pi_{0}(U)}.$

Here, if $\pi_{0}(U)=\infty$, we set $A^{\otimes\pi_{0}(U)}$ to be the
infinite coproduct of algebras, i.e.

(4.14) $\displaystyle A^{\otimes\pi_{0}(U)}:=\varinjlim
A_{1}\otimes\dots\otimes A_{n},$

and the inclusion maps $A_{1}\otimes\dots\otimes A_{n}\to
A_{1}\otimes\dots\otimes A_{m}$ for $n\leq m$ of this colimit are given by

(4.15) $\displaystyle a_{1}\otimes\dots\otimes a_{n}\mapsto
a_{1}\otimes\dots\otimes a_{n}\otimes 1\otimes\dots\otimes 1.$

The extension maps of the precosheaf are given by taking products along
connected components.

######  Remark 4.4.

This is the categorical generalization of a constant cosheaf from the category
of vector spaces, replacing the coproduct $\oplus$ of vector spaces by the
coproduct $\otimes$ of algebras. However, the category of
$\mathbb{R}$-algebras is not even preadditive. Hence, we cannot define exact
sequences, and hence cosheaves, in the way we did in Appendix A. This is why
we refrain from calling this construction a cosheaf.

######  Corollary 4.5.

Let $M$ be a smooth, orientable manifold of dimension $n$. Consider the full
subcategory of $\mathrm{Open}(M)$ defined by finite unions of pairwise
disjoint sets diffeomorphic to $\mathbb{R}^{n}$:

(4.16) $\displaystyle\mathcal{B}_{M}:=\\{U_{1}\cup\dots\cup
U_{k}:k\in\mathbb{N},U_{i}\cong\mathbb{R}^{n},U_{i}\cap U_{j}=\emptyset\text{
for }i\neq j\\}.$

The functor associated to the restriction of the precosheaf
$\mathcal{H}^{\bullet}$ to this subcategory is naturally isomorphic to the
analogous restriction of the functor associated to the constant cosheaf of
algebras $U\mapsto
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))^{\otimes^{\pi_{0}(U)}}$.

###### Proof.

Fix some orientation on $\mathbb{R}^{n}$. Using orientability of $M$, we can
construct for every connected $U\in\mathcal{B}_{M}$ a diffeomorphism
$\phi_{U}:U\to\mathbb{R}^{n}$ so that if $V\in\mathcal{B}_{M}$ is any other
connected set with $U\cap V\neq\emptyset$, the transition map
$(\phi_{U}\circ\phi_{V}^{-1})|_{\phi_{V}(V\cap U)}$ is orientation-preserving.
Fix now some set $U\in\mathcal{B}_{M}$ with connected components
$U_{1},\dots,U_{k}$. In Proposition 3.14 we defined an isomorphism

(4.17) $\displaystyle H^{\bullet}(\mathfrak{X}(U_{1}))\otimes\dots\otimes
H^{\bullet}(\mathfrak{X}(U_{k}))\cong H^{\bullet}(\mathfrak{X}(U)).$

Thus, under the identification (4.17) we can express every element in
$H^{\bullet}(\mathfrak{X}(U))$ as a span of elements of the form

(4.18)
$\displaystyle[\phi^{*}_{U_{1}}c_{1}]\otimes\dots\otimes[\phi^{*}_{U_{k}}c_{k}],\quad
c_{1},\dots,c_{k}\in C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})).$

Let us show that the collection of the maps

(4.19)
$\displaystyle[\phi^{*}_{U_{1}}]\otimes\dots\otimes[\phi^{*}_{U_{k}}]:H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))^{\pi_{0}(U)}\to
H^{\bullet}(\mathfrak{X}(U))$

for all $U\in\mathcal{B}$ induce the desired natural isomorphism of functors.
By Lemma 4.1, every tensor factor is an isomorphism of vector spaces, and by
an explicit calculation with the wedge product, the maps are compatible with
the algebra structures. It remains to show that the precosheaf extension maps
are respected.
Let $V\in\mathcal{B}_{M}$ contain $U$. Assume first that two connected
components of $U$, say $U_{1},U_{2}$ lie in a single connected component of
$V$, say $V_{1}$. By Corollary 2.28 the wedge product of $H^{\bullet}(W_{n})$
is zero in nonzero degree, so the extension map

(4.20) $\displaystyle H^{q_{1}}(\mathfrak{X}(U_{1}))\otimes
H^{q_{2}}(\mathfrak{X}(U_{2}))\to H^{q_{1}+q_{2}}(\mathfrak{X}(V_{1}))$

is zero if $q_{1}$ and $q_{2}$ are simultaneously nonzero.
If one of the $q_{1},q_{2}$ is equal to zero, say $q_{1}=0$, then the
extension map $H^{q_{1}}(\mathfrak{X}(U_{1}))\otimes
H^{q_{2}}(\mathfrak{X}(U_{2}))\to H^{q_{1}+q_{2}}(\mathfrak{X}(V_{1}))$ is
simply the isomorphism

(4.21) $\displaystyle\mathbb{R}\otimes
H^{q_{2}}(\mathfrak{X}(U_{2}))\xrightarrow{\sim}H^{q_{2}}(\mathfrak{X}(U_{2})),\quad
1\otimes[c]\mapsto[c].$

For the constant precosheaf of algebras $U\mapsto
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))^{\otimes\pi_{0}(U)}$ analogous
properties hold, and as a consequence the maps (4.19) commute with inclusions
where two connected components of $U$ are contained in a single connected
component of $V$.
Hence it remains to consider the case in which every connected component
$V_{i}$ of $V$ contains at most one connected component of $U$, so by
reordering we may assume $U_{i}\subset V_{i}$ for all $i=1,\dots,k$. It
suffices to show that
$(\phi_{V_{i}}^{-1})^{*}\circ\iota_{U_{i}}^{V_{i}}\circ\phi_{U_{i}}^{*}$
induces the identity map on $H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ for
all $i=1,\dots,k$. As maps on cochains
$C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})$ we have the identity

(4.22)
$\displaystyle(\phi_{V_{i}}^{-1})^{*}\circ\iota_{U_{i}}^{V_{i}}\circ\phi_{U_{i}}^{*}=\left(\phi_{V_{i}}^{-1}|_{U_{i}}\circ\phi_{U_{i}}\right)^{*}.$

Now $\phi_{V_{i}}^{-1}|_{U_{i}}\circ\phi_{U_{i}}$ is an orientation-preserving
local diffeomorphism, so by Lemma 4.1 the map
$(\phi_{V_{i}}^{-1})^{*}\circ\iota_{U_{i}}^{V_{i}}\circ\phi_{U_{i}}^{*}$
equals the identity on cohomology and
$[\iota_{U_{i}}^{V_{i}}\circ\phi_{U_{i}}]=[\phi_{V_{i}}]$. Hence, the
statement is shown. ∎

######  Remark 4.6.

In the non-orientable case, the previous proof shows that the restriction of
$\mathcal{H}^{\bullet}$ is naturally isomorphic to the restriction of a
_locally constant cosheaf_ $U\mapsto S(U)$, i.e. every point $x\in M$ has an
open neighborhood $U_{x}$ so that the restriction of $S$ to $U_{x}$ is a
constant cosheaf.

##  5\. Gelfand–Fuks cohomology for smooth manifolds

In this section, we construct a spectral sequence due to Gelfand and Fuks that
calculates the continuous Lie algebra cohomology for smooth manifolds,
following a local-to-global principle using sheaf theoretic ideas. The
spectral sequences were originally constructed in [1], by an involved global
analysis of the cochain spaces $C^{\bullet}(\mathfrak{X}(M))$ in terms of
explicit distributions.
The proposed local-to-global principle has originally been outlined in [13]
and [26], and, according to the last reference, was initially suggested by
Segal. However, in these latter two references, there are some unaddressed
subtleties: It is (indirectly) claimed that the assignment of open sets $U$ to
$C^{\bullet}(\mathfrak{X}(U))$ is a _cosheaf_ of graded vector spaces, i.e.
its Čech homology (see Appendix A) vanishes with respect to every good cover
$\mathcal{U}$ of $M$. We show in Proposition 5.1 that this is false. We
present a novel proof that works around this problem by using so-called
$k$-_good_ covers, an adaptation to the concept of a good cover originating
from [9]. This was inspired by the recent preprint [11], treating Gelfand–Fuks
cohomology in the setting of factorization algebras.
However, we want to emphasize that this subtlety does not influence the
validity of the final results of Bott and Segal. The mistake is not repeated
in [4] and [10], where similar, but more sophisticated Čech-theoretic methods
are used.
Regardless, our proof gives a more elementary way to calculate Gelfand–Fuks
spectral sequences for _$k$ -diagonal cohomology_, an approximation of
Gelfand–Fuks cohomology which we will introduce in the following section. The
expression for the spectral sequences have been given in [10] without an
explicit proof for $k\neq 1$; the proof in [13] is only a sketch, with
previously mentioned issues, and the proof in [1] uses explicit, distribution-
theoretic formulas which may be difficult for non-experts to retrace.
The construction we propose in the following section is inspired by a preprint
by Kapranov and Hennion [11], in which they use the theory of factorization
algebras to prove that Gelfand–Fuks cohomology can be identified with singular
cohomology of a certain section space. We borrow from this theory a notion of
generalized good covers which makes it possible to construct the desired
spectral sequences from a local-to-global principle. This strategy is easily
generalizable to construct local-to-global spectral sequences in other
cohomology theories, as we plan to show in future work [12].

###  5.1. Diagonal Filtration

Fix a smooth manifold $M$ of dimension $n$. The previously established
precosheaf structure (see Definition 4.2) of the cochains
$C^{\bullet}(\mathfrak{X}(M))$ does _not_ extend to a cosheaf structure.

######  Proposition 5.1.

Let $M$ be a smooth manifold. If $k>1$ and $\dim M>0$, then the precosheaf
$U\mapsto C^{k}(\mathfrak{X}(U))$ from Definition 4.2 is not a cosheaf.

###### Proof.

Because $\dim M>0$, there are smooth, nonzero
$X_{1},\dots,X_{k}\in\mathfrak{X}(M)$ whose supports are pairwise disjoint,
and some $c\in C^{k}(\mathfrak{X}(M))$ with $c(X_{1},\dots,X_{k})\neq 0$. Then
the sets

(5.1) $\displaystyle U_{i}:=M\setminus\left(\bigcup_{j\neq
i}\operatorname{supp}X_{j}\right),\quad i=1,\dots,k$

define an open cover $\\{U_{i}\\}_{i=1,\dots,k}$ of $M$. If the assignment of
an open set $U\subset M$ to cochains $C^{k}(\mathfrak{X}(U))$ was a cosheaf,
then there would exist $c_{i}\in C^{\bullet}(\mathfrak{X}(U_{i}))$ for
$i=1,\dots,k$ with

(5.2) $\displaystyle c=\sum_{i=1}^{k}\iota_{U_{i}}^{M}c_{i}.$

But then, because $X_{i}\big{|}_{U_{j}}=0$ for $i\neq j$, it follows that

(5.3) $\displaystyle 0\neq
c(X_{1},\dots,X_{k})=\sum_{i=1}^{k}c_{i}(X_{1}|_{U_{i}},\dots,X_{k}|_{U_{i}})=0.$

A clear contradiction, hence, the precosheaf $U\mapsto C^{k}(\mathfrak{X}(U))$
is not a cosheaf for $k>1$. ∎

Hence, as we increase the number of arguments in our cochains, we may get
_locality_ or _diagonality problems_ as in the above proof. It will be
valuable to replace these spaces by certain diagonal replacements:

######  Definition 5.2.

Let $U$ be an open subset of $M$.

  * i)

Define the graded vector space $B^{\bullet}(\mathfrak{X}(U)):=\bigoplus_{q\geq
0}B^{q}(\mathfrak{X}(U)),$ where

(5.4) $\displaystyle
B^{q}(\mathfrak{X}(U)):=\left\\{c:\mathfrak{X}(U)^{q}\to\mathbb{R}\,\mid\,c\text{
multilinear and jointly continuous}\right\\}.$

  * ii)

Let $\\{X_{1},\dots,X_{q}\\}\subset\mathfrak{X}(U)$ be a finite collection of
vector fields and $k\geq 1$ arbitrary. We say that this collection _has the
property $\Delta_{k}$_ if for every finite set $\Gamma\subset U$ of $k$
arbitrary points, there is an $X_{i}$ that vanishes in a neighborhood of
$\Gamma$.

  * iii)

For $q,k\geq 1$ integers, we define the _$k$ -diagonal distributions_ as those
$c\in B^{q}(\mathfrak{X}(U))$ with

(5.5) $\displaystyle\\{X_{1},\dots,X_{q}\\}\text{ has property
}\Delta_{k}\implies c(X_{1},\dots,X_{q})=0.$

Their collection is denoted $\Delta_{k}B^{q}(\mathfrak{X}(U))$.
If $q=0$, set $\Delta_{k}B^{0}(\mathfrak{X}(U))=B^{0}(\mathfrak{X}(U))$ for
all $k\geq 1$.

  * iv)

Define the _$k$ -diagonal cochains_ $\Delta_{k}C^{q}(\mathfrak{X}(U))\subset
C^{q}(\mathfrak{X}(M))$ as the skew-symmetric cochains which are contained in
$\Delta_{k}B^{q}(\mathfrak{X}(U))$.

######  Proposition 5.3.

For all $q\geq 0$ and all open $U\subset M$, we have the ascending chain

(5.6) $\displaystyle 0=:\Delta_{0}C^{q}(\mathfrak{X}(U))$
$\displaystyle\subset\Delta_{1}C^{q}(\mathfrak{X}(U))\subset\dots$
$\displaystyle\subset\Delta_{q-1}C^{q}(\mathfrak{X}(U))\subset\Delta_{q}C^{q}(\mathfrak{X}(U))=C^{k}(\mathfrak{X}(U)).$

Further, the $\Delta_{k}C^{\bullet}(\mathfrak{X}(U))$ constitute a
multiplicative filtration of $C^{\bullet}(\mathfrak{X}(U))$.

###### Proof.

The property $\Delta_{k}$ for a set $\\{X_{1},\dots,X_{q}\\}$ implies the
property $\Delta_{k-1}$. This proves the ascending chain of inclusions.
Further, a set $\\{X_{1},\dots,X_{k}\\}$ of $k$ vector fields can only have
the property $\Delta_{k}$ if one of the $X_{i}$ is zero everywhere. Hence
$\Delta_{k}C^{k}(\mathfrak{X}(M))=C^{k}(\mathfrak{X}(M))$. Further, notice
that if $\\{X_{1},\dots,X_{q+1}\\}$ has the property $\Delta_{k}$, so does
$\\{[X_{1},X_{2}],X_{3},\dots,X_{q}\\}$. From this it follows that

(5.7) $\displaystyle
d(\Delta_{k}C^{q}(\mathfrak{X}(M))\subseteq\Delta_{k}C^{q+1}\mathfrak{X}(M).$

Lastly, if $\\{X_{1},\dots,X_{q+r}\\}$ has the property $\Delta_{k+l}$, then
the set $\\{X_{1},\dots,X_{q}\\}$ must have the property $\Delta_{k}$ or the
set $\\{X_{q+1},\dots,X_{q+r}\\}$ must have the property $\Delta_{l}$. Hence

(5.8)
$\displaystyle\Delta_{k}C^{\bullet}(\mathfrak{X}(M))\wedge\Delta_{l}C^{\bullet}(\mathfrak{X}(M))\subseteq\Delta_{k+l}C^{\bullet}(\mathfrak{X}(M)).$

∎

######  Example 5.4.

A set $\\{X_{1},\dots,X_{k}\\}\subset\mathfrak{X}(M)$ has the property
$\Delta_{1}$ if and only if

(5.9) $\displaystyle\bigcap_{i=1}^{k}\operatorname{supp}X_{i}=\emptyset.$

Note that $\Delta_{q}C^{q}(\mathfrak{X}(U))=C^{q}(\mathfrak{X}(U))$, hence

(5.10)
$\displaystyle\Delta_{k}H^{q}(\mathfrak{X}(U))=H^{q}(\mathfrak{X}(U))\quad\forall
k\geq q+1.$

To put this in terms of more sheaflike data, let us view these cochains
through a different lens.

######  Definition 5.5.

Given $q\geq 1$ and the canonical projections
${\operatorname{\mathsf{pr}}}_{1},\dots,{\operatorname{\mathsf{pr}}}_{q}:M^{q}\to
M$, consider the vector bundle

(5.11)
$\displaystyle\boxtimes^{q}TM:=\bigotimes^{q}_{i=1}{\operatorname{\mathsf{pr}}}_{i}^{*}TM\to
M^{q}.$

Equipping the space of sections $\mathfrak{X}(M)$ with its standard Fréchet
topology, it is a well-known consequence of the Schwartz kernel theorem that
there is a natural vector space isomorphism

(5.12) $\displaystyle B^{q}(\mathfrak{X}(M))\cong\Gamma(\boxtimes^{q}TM)^{*},$

the star denoting the continuous dual with respect to the Fréchet topology
(cf. [27] for the distributional statement). This isomorphism is dual to the
map

(5.13)
$\begin{gathered}\mathfrak{X}(M)\otimes\dots\otimes\mathfrak{X}(M)\to\Gamma(\boxtimes^{q}TM),\quad(X_{1},\dots,X_{q})\mapsto
X_{1}\boxtimes\dots\boxtimes X_{q},\\\ (X_{1}\boxtimes\dots\boxtimes
X_{q})(x_{1},\dots,x_{q}):=X_{1}(x_{1})\otimes\dots\otimes
X_{q}(x_{q})\quad\forall x_{1},\dots,x_{q}\in M.\end{gathered}$

######  Definition 5.6.

Let $X$ be a topological space and $k,q\in\mathbb{N}$. The _$k$ -th diagonal_
of $X$ in $X^{q}$ is the subspace

(5.14) $\displaystyle X^{q}_{k}:=\\{(x_{1},\dots,x_{q})\in
X^{q}:|\\{x_{1},\dots,x_{q}\\}|\leq k\\}.$

Special examples are the _thin diagonal_ $M^{q}_{1}$ and _fat diagonal_
$M^{q}_{q-1}$, which take the following form:

$\displaystyle M_{1}^{q}:=\\{(x,\dots,x)\in M^{q}\\},$ $\displaystyle
M_{q-1}^{q}=\\{(x_{1},\dots,x_{q})\in M^{q}\mid\exists i,j:i\neq j\text{ and
}x_{i}=x_{j}\\}.$

Clearly $M_{1}^{q}\subset M_{2}^{q}\subset\dots\subset M_{q}^{q}=M^{q}$. Note
that a set of vector fields $\\{X_{1},\dots,X_{q}\\}$ has the property
$\Delta_{k}$ if and only if the support of $X_{1}\boxtimes\dots\boxtimes
X_{q}$ is contained in $M^{q}\setminus M^{q}_{k}$. This proves the following:

######  Lemma 5.7.

An element $c\in B^{q}(\mathfrak{X}(M))$ is $k$-diagonal if and only if the
support of its image under the Schwartz kernel map in
$\operatorname{Hom}_{\mathbb{R}}(\Gamma(\boxtimes^{q}TM),\mathbb{R})$ is
contained in $M^{q}_{k}$.

With this perspective, we can deduce:

######  Lemma 5.8.

For $U\subset M^{q}$, the assignments

(5.15) $\displaystyle M^{q}\supset U\mapsto\mathcal{B}^{q}(U):=$
$\displaystyle\operatorname{Hom}_{\mathbb{R}}(\Gamma(\boxtimes^{q}TM)\big{|}_{U},\mathbb{R}),$
(5.16) $\displaystyle M^{q}_{k}\supset U\mapsto\mathcal{B}^{q}_{k}(U):=$
$\displaystyle\left\\{c\in\operatorname{Hom}_{\mathbb{R}}(\Gamma(\boxtimes^{q}TM),\mathbb{R}):\operatorname{supp}c\subset
U\right\\},$

constitute flabby cosheaves on $M^{q}$ and $M^{q}_{k}$, respectively, where
the extension maps are induced by the restriction maps of the section spaces.

###### Proof.

The sheaves of distributions

(5.17) $\displaystyle M^{q}\supset U\mapsto\mathcal{D}^{q}(U)$
$\displaystyle:=\operatorname{Hom}_{\mathbb{R}}\left(\Gamma_{c}(\boxtimes^{q}TM)\big{|}_{U},\mathbb{R}\right),$
(5.18) $\displaystyle M^{q}_{k}\supset U\mapsto\mathcal{D}^{q}_{k}(U)$
$\displaystyle:=\left\\{c\in\operatorname{Hom}_{\mathbb{R}}(\Gamma_{c}(\boxtimes^{q}TM),\mathbb{R}):\operatorname{supp}c\subset
U\right\\}$

are shown to be soft using standard partition of unity arguments.
Alternatively, this follows since the first sheaf is a module over a soft
sheaf of rings, namely the sheaf of smooth functions on $M^{q}$, and the
second one is a restriction of the first sheaf to a closed subspace, hence
soft (see [25, Chapter II, Thm 9.2, Thm 9.16]). But $\mathcal{B}^{q}$ is
exactly the precosheaf of compactly supported sections of the sheaf
$\mathcal{D}^{q}$, and analogously for $\mathcal{B}^{q}_{k}$ and
$\mathcal{D}^{q}_{k}$. By Proposition A.5, this implies that these
precosheaves are flabby cosheaves. ∎

###  5.2. Generalized good covers

We have seen that the precosheaf $U\mapsto C^{k}(\mathfrak{X}(U))$ for open
sets $U$ of a smooth manifold $M$ does not define a cosheaf for $k\geq 2$.
However, Lemma 5.8 gives some hope that we can meaningfully study them over
the Cartesian power $M^{k}$. As such, we will need methods to compare
different Cartesian powers $M,M^{2},M^{3},\dots$ of $M$. One such tool we can
use is the notion of a $k$-good cover in the sense of [9, Def 2.9]:

######  Definition 5.9.

Let $k\geq 1$. An open cover $\mathcal{U}$ of $M$ is _$k$ -good_ if:

  * i)

Given $k$ points $x_{1},\dots,x_{k}\in M$, there is a $U\in\mathcal{U}$ with
$x_{1},\dots,x_{k}\in U$.

  * ii)

Intersections of elements of $\mathcal{U}$ are diffeomorphic to a disjoint
union of at most $k$ copies of $\mathbb{R}^{n}$.

For $k=1$ this agrees with the usual notion of a good cover.

######  Remark 5.10.

The $k$-good covers are, in a sense, finite approximations to so-called _Weiss
covers_ , which have property i) of the previous definition with no
restriction on the number $k$, but without any replacement for property ii),
so the sets in the cover may, a priori, be homologically wild. Weiss covers
are heavily used in the theory of _factorization algebras,_ which have strong
ties to our setting, see [24, 11, 28].

######  Definition 5.11.

Let $X$ be a topological space, $\mathcal{U}$ an open cover of $X$, and
$k,q\geq 1$ integers. Define

$\displaystyle\mathcal{U}^{q}:=\\{V\subset M^{q}:V\text{ a connected component
of }U^{q}\text{ for some }U\in\mathcal{U}\\},$
$\displaystyle\mathcal{U}^{q}_{k}:=\\{V\cap M^{q}_{k}:V\in\mathcal{U}^{q}\\}.$

Property i) of a $k$-good cover $\mathcal{U}$ implies that the sets
$\mathcal{U}^{1},\mathcal{U}^{2},\dots,\mathcal{U}^{k}$ are open covers of
$M,M^{2},\dots,M^{k}$, making $k$-good covers useful tools in comparing data
between Cartesian powers of $M$.
If $\mathcal{U}$ is a $k$-good cover, then the equation

(5.19) $\displaystyle(U\cap V)^{q}=U^{q}\cap V^{q}$

shows that all intersections of the open cover $\mathcal{U}^{q}$ are
diffeomorphic to finite disjoint unions of at most $k$-copies of
$\mathbb{R}^{qn}$. To show a similar result for the open cover
$\mathcal{U}^{q}_{k}$ of $M^{q}_{k}$, let us prepare an auxiliary statement:

######  Lemma 5.12.

Let $X$ be a topological space and $q,k\geq 1$ integers. If $X$ has finitely
many connected components and all of them are contractible, then the same
holds for $X^{q}_{k}$.

###### Proof.

Let $X_{1},\dots,X_{s}$ be the connected components of $X$. Every connected
component $C$ of $X^{q}$ is then of the form

(5.20) $\displaystyle C=X_{i_{1}}\times\dots\times X_{i_{q}}$

for some $i_{1},\dots,i_{q}\in\\{1,\dots,s\\}$, not necessarily all different.
By assumption, the $X_{1},\dots,X_{s}$ are contractible, hence, for all
$j=1,\dots,s$, there are deformation retracts $F_{j}:X_{j}\times[0,1]\to
X_{j}$ of $X_{j}$ onto a point. Then the map

(5.21) $\displaystyle F:C\times[0,1]\to
C,\quad(x_{1},\dots,x_{q},t)\mapsto(F_{i_{1}}(x_{1},t),\dots,F_{i_{q}}(x_{q},t))$

is a deformation retract of $C$ onto a point.
If $C\cap X^{q}_{k}$ is nonempty, this map restricts to a deformation retract
of $C\cap X^{q}_{k}$ to a point. Hence, if $C$ is a connected component of
$U^{q}$, and $C\cap X^{q}_{k}\neq\emptyset$, then $C\cap X^{q}_{k}$ is
contractible. But the connected components of $X^{q}_{k}$ are exactly the
nonempty sets $C\cap X^{q}_{k}$ for connected components $C$ of $X^{q}$. Hence
all connected components of $X^{q}_{k}$ are contractible. Lastly, since
$X^{q}$ only has finitely many connected components, so does $X^{q}_{k}$. This
finishes the proof. ∎

######  Lemma 5.13.

Let $X$ be a locally connected topological space, $q,k\geq 1$ integers, and
$\mathcal{U}$ a k-good open cover of $X$. Then $\mathcal{U}^{q}_{k}$ is an
open cover of $X^{q}_{k}$. Further, all nonempty, finite intersections of
elements in $\mathcal{U}^{q}_{k}$ have finitely many connected components, and
all these connected components are contractible.

###### Proof.

If $(x_{1},\dots,x_{q})\in X^{q}_{k}$, then $|\\{x_{1},\dots,x_{q}\\}|\leq k$.
But since $\mathcal{U}$ is $k$-good, there is some $U\in\mathcal{U}$
containing all $x_{1},\dots,x_{k}$. Hence $x\in U^{q}\cap X^{q}_{k}$. Since
$x$ was arbitrary, this shows that $\mathcal{U}_{k}^{q}$ is an open cover of
$X^{q}_{k}$.
Let us now show that if $V,V^{\prime}\in\mathcal{U}^{q}_{k}$ have nonempty
intersection, then $V\cap V^{\prime}$ is a finite disjoint union of
contractible sets. By definition there exist $U,U^{\prime}\in\mathcal{U}$ so
that $V$ and $V^{\prime}$ are connected components of $U^{q}_{k}$ and
$U^{\prime q}_{k}$, respectively. Since $\mathcal{U}$ is a $k$-good cover,
$U\cap U^{\prime}$ has only finitely many connected components, and all these
connected components are contractible. Thus by Lemma 5.12 the same holds for
$(U\cap U^{\prime})^{q}_{k}=U^{q}_{k}\cap(U^{\prime})^{q}_{k}$.
Since $X$ is locally connected, so are $X^{q}_{k}$ and its open subsets
$U^{q}_{k}$ and $(U^{\prime})^{q}_{k}$. In a locally connected space, all
connected components are closed and open, hence $V$ and $V^{\prime}$ are
closed and open in $U^{q}_{k}$ and $(U^{\prime})^{q}_{k}$, respectively. Thus
$V\cap V^{\prime}$ is closed and open in $U^{q}_{k}\cap(U^{\prime})^{q}_{k}$,
and hence must be a union of connected components of
$U^{q}_{k}\cap(U^{\prime})^{q}_{k}$. But we have seen that there are only
finitely many connected components of $U^{q}_{k}\cap(U^{\prime})^{q}_{k}$, and
that they are all contractible. This shows that the same holds for $V\cap
V^{\prime}$.
By induction this extends to arbitrary finite intersections of sets
$V_{1},\dots,V_{q}\in\mathcal{U}^{q}_{k}$, and the lemma is shown. ∎

The first part of the following theorem is Proposition 2.10 in [9]:

######  Theorem 5.14.

For every smooth manifold $M$, a $k$-good open cover exists. Further, if $M$
is compact, then $M$ admits finite $k$-good open covers.

###### Proof.

The existence of $k$-good open covers is shown in [9]. If $M$ is compact,
choose any $k$-good cover $\mathcal{U}$, then $\mathcal{U}^{k}$ is a cover of
$M^{k}$, and since $M^{k}$ is compact, there is a finite subcover
$\tilde{\mathcal{U}}\subset\mathcal{U}$ so that $\tilde{\mathcal{U}}^{k}$ is a
cover of $M^{k}$. Hence the set $\tilde{\mathcal{U}}$ fulfils property i) of
being a $k$-good cover, and as a subset of a $k$-good cover, it also fulfils
property ii). ∎

###  5.3. The Čech-Bott-Segal double complex

Finally, let us define a double complex which intertwines Čech complexes with
the Chevalley-Eilenberg complex structure. Analyzing this double complex will
provide us with spectral sequences that calculate the Gelfand–Fuks cohomology
of $M$. We name this double complex after Bott and Segal, in reference to
their spirtually similar local-to-global analysis in [4], though we emphasize
that our double complex differs from theirs.

######  Definition 5.15.

Let $\mathcal{U}:=\\{U_{i}\\}_{i\in I}$ be an open cover of a smooth manifold
$M$, and $k\geq 1$. For $U_{i_{1}},\dots,U_{i_{q}}\in\mathcal{U}$, set

(5.22) $\displaystyle U_{i_{1}\dots i_{q}}:=U_{i_{1}}\cap\dots\cap U_{i_{q}}.$

We define the _$k$ -th Čech-Bott-Segal (CBS) double complex_ for the cover
$\mathcal{U}$ as the following double complex:

$\bigoplus_{i}\Delta_{k}C^{1}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}\Delta_{k}C^{1}(\Gamma(\mathfrak{X}(U_{ij}))$$\
\dotsc\ $$\ \dotsc\
$$\dotsc$$\dotsc$$\bigoplus_{i}\Delta_{k}C^{2}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}\Delta_{k}C^{2}(\Gamma(\mathfrak{X}(U_{ij}))$$\bigoplus_{i}\Delta_{k}C^{3}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}\Delta_{k}C^{3}(\mathfrak{X}(U_{ij}))$$\
\dotsc\ $

The horizontal maps are given by the Čech differentials associated to the
precosheaf structure, the vertical maps by the direct sum of Chevalley-
Eilenberg differentials for the complexes
$C^{\bullet}\left(\mathfrak{X}\left(U_{i_{1}\dots i_{p}}\right)\right)$. The
grading is defined so that the term
$\bigoplus_{i_{1},\dots,i_{p}}\Delta_{k}C^{q}(\mathfrak{X}(U_{i_{1}\dots
i_{p}}))$ lies in degree $(p,q)$.
The _$k$ -th skew-symmetrized CBS double complex_ is the CBS double complex
with all horizontal Čech complexes replaced by their skew-symmetrized
versions, see Remark A.3.

######  Remark 5.16.

Many remarks on the form of this double complex are in order:

  * i)

Note that the zeroeth row of the CBS double complex is zero, and not, as one
might expect, the Čech complex associated to
$\Delta_{k}C^{0}(\mathfrak{X}(M))$. Since the zeroeth degree is connected to
the rest of the complex by a zero differential, we do not lose any information
by leaving it out.

  * ii)

To deduce the ring structure on Gelfand–Fuks cohomology, it would be helpful
if we could define a product structure on the CBS double complex in a way
which extends the wedge product of cochains. At this point in time, it is
unclear to the author how to accomplish this.

  * iii)

The CBS double complex is not a first-quadrant double complex: It mixes a
cohomological and a homological differential. A priori, this means there is an
ambiguity in defining the associated total complex, given by the choice of
taking either direct sums or direct products on the relevant diagonals, since
there may now be infinitely many nonzero terms on each such diagonal. The
usual convergence theorems for the spectral sequences arising from horizontal
and vertical filtration will, in general, not apply.

Especially part iii) of the previous remark poses a significant problem. We
borrow an argument from [4] to circument this:

######  Lemma 5.17.

Let $k\in\mathbb{N}$ and $\mathcal{U}$ a finite open cover of a smooth
manifold $M$. The $k$-th skew-symmetrized CBS double complex associated to
$\mathcal{U}$ has only finitely many nonzero columns. In particular, it is
bounded as a double complex.

###### Proof.

By finiteness of $\mathcal{U}$, there is a largest $n$ such that there is a
nonempty intersection $U_{1}\cap\dots\cap U_{n}$ with $U_{i}\neq U_{j}$ for
$i\neq j$. Hence all columns in degree $>n$ vanish in the skew-symmetrized
double complex. This concludes the proof. ∎

We begin with horizontal cohomology.

######  Proposition 5.18.

Let $k,q\geq 1$ be integers and $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ a $k$-good
cover of a smooth manifold $M$. The Čech complex

(5.23)
$\displaystyle\bigoplus_{i}\Delta_{k}B^{q}(\mathfrak{X}(U_{i}))\leftarrow\bigoplus_{i,j}\Delta_{k}B^{q}(\mathfrak{X}(U_{ij}))\leftarrow\dots$

is isomorphic to the Čech complex associated to the flabby cosheaf
$\mathcal{B}^{q}_{k}$ on $M^{q}_{k}$ defined in Lemma 5.8) with respect to the
open cover $\\{U^{q}_{k}\subset M^{q}_{k}:U\in\mathcal{U}\\}$ of $M^{q}_{k}$.
The same statement holds for the skew-symmetrized Čech complex, and the
isomorphism is equivariant under the natural permutation action of the
symmetric group $\Sigma_{q}$.

###### Proof.

Note first that $\mathcal{U}^{q}_{k}$ is a cover of $M^{q}_{k}$ by Lemma 5.13.
The (restriction of the) Schwartz kernel maps (5.12) give us a family of
isomorphisms $\\{\phi_{U}:U\subset M\text{ open}\\}$ as in Lemma 5.7, so that
for all open $U\subset V$ the following diagram commutes:

${\Delta_{k}B^{q}(\mathfrak{X}(U))}$${\Delta_{k}B^{q}(\mathfrak{X}(V))}$${\mathcal{B}^{q}_{k}(U^{q}_{k})}$${\mathcal{B}^{q}_{k}(V^{q}_{k})}$$\scriptstyle{\phi_{U}}$$\scriptstyle{\phi_{V}}$

Hence, we have isomorphisms on the precosheaf data; this lifts to an
isomorphism of the two Čech complexes. This argument is independent of the
choice of the standard or the skew-symmetrized Čech complex. Since the sets
$U^{q}_{k}$ are invariant under the natural $\Sigma_{q}$-action on $M^{q}$,
both of the terms $\Delta_{k}B^{q}(\mathfrak{X}(U))$ and
$\mathcal{B}^{q}_{k}(U^{q}_{k})$ admit a $\Sigma_{k}$-action by permutation of
vector fields. The Schwartz kernel map is equivariant with respect to this
permutation, as one finds from the explicit formula of its dual map 5.13. ∎

######  Theorem 5.19.

Let $q,k\geq 1$ be integers, and consider the $k$-th (skew-symmetrized) CBS
double complex for a $k$-good cover $\mathcal{U}$ of $M$. The cohomology of
the $q$-th row is equal to $\Delta_{k}C^{q}(\mathfrak{X}(M))$ in degree zero,
and trivial in all other degrees.

###### Proof.

By Proposition 5.18, the Čech complex in this row, associated to the cover
$\mathcal{U}$ and the presheaf $U\mapsto\Delta_{k}B^{q}(\mathfrak{X}(U))$ over
$M$, has the same homology as the Čech complex of the flabby cosheaf
$U\mapsto\mathcal{B}^{q}_{k}(U)$ over $M^{q}_{k}$ with respect to the cover
$\mathcal{U}^{q}_{k}$. Flabby cosheaves have trivial Čech homology independent
of the chosen cover by Proposition A.6, hence the homology is equal to
$\Delta_{k}B^{q}(\mathfrak{X}(M))$ in zeroeth degree and zero in higher
degree.
The isomorphism identifying the two complexes is equivariant with respect to
the $\Sigma_{q}$-action on both spaces. The functor taking the complexes to
its $\Sigma_{q}$-invariants is exact, as it arises from the action of a finite
group in characteristic zero. Hence, the skew-symmetrized complex also has
trivial cohomology in nonzero degree, and in degree zero
$\left(\Delta_{k}B^{q}(\mathfrak{X}(M))\right)^{\Sigma_{q}}=\Delta_{k}C^{q}(\mathfrak{X}(M))$.
Since the skew-symmetrized complex is exactly the $q$-th row of the skew-
symmetrized CBS complex, this concludes the proof. ∎

######  Corollary 5.20.

Let $k\geq 1$ and assume there exists a finite, $k$-good cover $\mathcal{U}$
of $M$, e.g. when $M$ is compact. The spectral sequences
$\\{E^{p,q}_{r},d_{r}\\}$ associated to the skew-symmetrized $k$-th CBS double
complex for $\mathcal{U}$ by filtering horizontally or vertically converges to
$\Delta_{k}\tilde{H}^{\bullet}(\mathfrak{X}(M))$. The tilde denotes reduced
cohomology, cf. Definition 2.14.

###### Proof.

By Theorem 5.19, filtering by rows makes the spectral sequence collapse on the
second page, with the indicated limit term
$\Delta_{k}\tilde{H}^{\bullet}(\mathfrak{X}(M))$. Due to finiteness of
$\mathcal{U}$ and Lemma 5.17, the skew-symmetrized double complex has bounded
rows, and for such double complexes both filtrations yield spectral sequences
which converge to the same cohomology, see [29, Chapter XV]. This shows the
statement. ∎

###  5.4. Spectral sequences for diagonal cohomology

0$A\subset\mathbb{R}$$A^{2}\subset\mathbb{R}^{2}$ Figure 2. An illustration of
Lemma 5.22: the set $A$ has three connected components in $\mathbb{R}$, so its
square $A^{2}\subset\mathbb{R}^{2}$ has $9=3^{2}$, all arising by taking
products of connected components of $A$. The products of a connected component
of $A$ with itself are exactly the connected components of $A^{2}$ which
intersect the diagonal in $\mathbb{R}^{2}$.

To arrive at Corollary 5.20 we filtered the CBS double complex by rows, so
now, let us study its filtration by rows. The cohomology among the arising
vertical complexes amounts to calculating $k$-diagonal Lie algebra cohomology
of $\mathfrak{X}(U)$, where the sets $U$ are finite disjoint unions of
$\mathbb{R}^{n}$. We begin by showing that our methods of Section 3 allow us
to calculate $k$-diagonal cohomology for such $U$.

######  Proposition 5.21.

Let $1\leq r\leq k$ and $U=\bigsqcup_{i=1}^{r}\mathbb{R}^{n}$. The inclusion

(5.24) $\displaystyle\Delta_{k}C^{\bullet}(\mathfrak{X}(U))\subset
C^{\bullet}(\mathfrak{X}(U))$

induces an isomorphism

(5.25) $\displaystyle\Delta_{k}H^{\bullet}(\mathfrak{X}(U))\cong
H^{\bullet}(\mathfrak{X}(U)).$

###### Proof.

The construction in the proof of Proposition 3.14 restricts without change to
the diagonally filtered complex. Specifically, the filtration
$F^{q}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ restricts to a filtration
$F^{q}(\Delta_{k}C^{\bullet})(\mathfrak{X}(\mathbb{R}^{n}))$, it is
straightforward to check that the exact sequence

(5.26) $\displaystyle 0\to F^{q+1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\to F^{q}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\to\bigoplus_{k_{1}+\dots+k_{r}=q}C^{\bullet}_{(k_{1})}(W_{n})\otimes\dots\otimes
C^{\bullet}_{(k_{r})}(W_{n})\to 0$

restricts to an exact sequence

(5.27) $\displaystyle 0\to
F^{q+1}(\Delta_{k}C^{\bullet})(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\to F^{q}(\Delta_{k}C^{\bullet})(\mathfrak{X}(\mathbb{R}^{n}))$
$\displaystyle\to\bigoplus_{k_{1}+\dots+k_{r}=q}C^{\bullet}_{(k_{1})}(W_{n})\otimes\dots\otimes
C^{\bullet}_{(k_{r})}(W_{n})\to 0,$

and the image of the splitting
$\beta_{0}^{(r)}:C^{\bullet}_{(0)}(W_{n})^{\otimes^{r}}\to
C^{\bullet}(\mathfrak{X}(M))$ from the proof of Proposition 3.14 is contained
in $\Delta_{k}C^{\bullet}(\mathfrak{X}(M))$. Hence,

(5.28) $\displaystyle\Delta_{k}H^{\bullet}(\mathfrak{X}(U))\cong
H^{\bullet}(W_{n})^{\otimes^{k}}\cong H^{\bullet}(\mathfrak{X}(U)),$

so that all nontrivial cohomology classes in $H^{\bullet}(\mathfrak{X}(U))$
have representatives contained in $\Delta_{k}H^{\bullet}(\mathfrak{X}(U))$.
This shows that the inclusion of complexes induces an isomorphism and the
proposition is shown. ∎

Consider now the CBS double complex for some $k$-good cover $\mathcal{U}$, and
the spectral sequence with respect to the filtration by columns. Every
intersection in the cover $\mathcal{U}$ is diffeomorphic to a disjoint union
of at most $k$ copies of $\mathbb{R}^{n}$. Hence, Proposition 5.21 applies in
every column and we can replace diagonal cohomology with standard Lie algebra
cohomology. Hence, the first page of the spectral sequence assumes the form in
Figure 3.

$\bigoplus_{i}H^{1}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}H^{1}(\Gamma(\mathfrak{X}(U_{ij}))$$\
\dotsc\ $$\ \dotsc\
$$\dotsc$$\dotsc$$\bigoplus_{i}H^{2}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}H^{2}(\Gamma(\mathfrak{X}(U_{ij}))$$\bigoplus_{i}H^{3}(\mathfrak{X}(U_{i}))$$\bigoplus_{i,j}H^{3}(\mathfrak{X}(U_{ij}))$$\
\dotsc\ $ Figure 3. The first page of the spectral sequence associated to the
CBS double complex by beginning with taking the cohomology along the vertical,
Chevalley-Eilenberg differential.

We state the following simple lemma without proof (cf. Figure 2):

######  Lemma 5.22.

Let $X$ be a topological space. The connected components of $X^{q}$ that do
not intersect $X^{q}_{q-1}$ are exactly the Cartesian products of $q$ pairwise
different connected components of $X$.

The following proposition makes use of _relative_ Čech homology of a pair of
topological spaces $(X,A)$ with respect to a cover $\mathcal{U}$ of $X$, cf.
[30, Chapter IX]. In this situation, the space

(5.29) $\displaystyle\mathcal{U}_{A}:=\\{U\cap A:U\in\mathcal{U}\\}$

is a cover of $A$, and the relative complex is the quotient by the inclusion
of Čech complexes $\check{C}(\mathcal{U}_{A})\to\check{C}(\mathcal{U}_{X})$.
We denote the relative complex by
$\check{C}_{\bullet}(\mathcal{U}_{X},\mathcal{U}_{A})$ and its homology by
$\check{H}_{\bullet}(\mathcal{U}_{X},\mathcal{U}_{A})$. If $\mathcal{U}$ and
$\mathcal{U}_{A}$ are good covers of $X$ and $A$, respectively, it is well
known that the Čech homologies are isomorphic to singular homology111see [31,
Chapter VI.D, Theorem 4] for a proof in Čech cohomology. This easily dualizes
to our setting.

(5.30) $\displaystyle\check{H}^{\bullet}(\mathcal{U}_{X})\cong
H^{\bullet}(X),\quad\check{H}^{\bullet}(\mathcal{U}_{A})\cong H^{\bullet}(A).$

Hence, by an argument on long exact sequences in homology, one has an
isomorphism to relative singular homology.

(5.31)
$\displaystyle\check{H}_{\bullet}(\mathcal{U}_{M},\mathcal{U}_{A})\cong\check{H}_{\bullet}(M,A)$

######  Proposition 5.23.

Let $q,k\geq 1$ be integers, $M$ be a smooth, orientable manifold, and
$\mathcal{U}$ a $k$-good cover of $M$. Denote by $\\{E^{p,q}_{r},d_{r}\\}$ the
spectral sequence associated to the $k$-th CBS double complex with respect to
$\mathcal{U}$, arising from the horizontal filtration by columns.
The $q$-th row of the first page $E_{1}^{\bullet,\bullet}$ is naturally
isomorphic to a direct sum of relative Čech complexes with respect to covers
$(\mathcal{U}^{r},\mathcal{U}^{r}_{r-1}$) of $(M^{r},M^{r}_{r-1})$ for
$r=1,\dots k$ (cf. Definition 5.11):

(5.32) $\displaystyle E^{\bullet,q}_{1}\cong$
$\displaystyle\check{C}_{\bullet}(\mathcal{U}_{M})\otimes H^{q}(W_{n})$
$\displaystyle\,\oplus$
$\displaystyle\left(\bigoplus_{\begin{subarray}{c}q_{1}+q_{2}=q\\\
q_{1},q_{2}>0\end{subarray}}\check{C}_{\bullet}(\mathcal{U}^{2}_{M^{2}},\mathcal{U}^{2}_{M^{2}_{1}})\otimes
H^{q_{1}}(W_{n})\otimes H^{q_{2}}(W_{n})\right)^{\Sigma_{2}}$
$\displaystyle\oplus$ $\displaystyle\dots$ $\displaystyle\oplus$
$\displaystyle\left(\bigoplus_{\begin{subarray}{c}q_{1}+\dots+q_{k}=q\\\
q_{1},\dots,q_{k}>0\end{subarray}}\check{C}_{\bullet}(\mathcal{U}^{k}_{M^{k}},\mathcal{U}^{k}_{M^{k}_{k-1}})\otimes
H^{q_{1}}(W_{n})\otimes\dots\otimes H^{q_{k}}(W_{n})\right)^{\Sigma_{k}}.$

Here, the symmetric groups $\Sigma_{2},\dots,\Sigma_{k}$ act by simultaneous,
skew-symmetric permutation of the Cartesian factors $U_{1}\times\dots\times
U_{k}$ of any set in the covers $(\mathcal{U}^{k},\mathcal{U}^{k}_{k-1})$ and
the tensor factors of $H^{q_{1}}(W_{n})\otimes\dots\otimes H^{q_{k}}(W_{n})$.
The same statement holds for the skew-symmetrized CBS complex, when the Čech
complexes in (5.32) are replaced by their skew-symmetrized versions.

###### Proof.

Set $n:=\dim M$. Since $\mathcal{U}$ is $k$-good, all intersections of
elements in $\mathcal{U}$ are diffeomorphic to a finite disjoint union of open
balls in $\mathbb{R}^{n}$. Hence we can apply Corollary 4.5 to find that the
Čech complex in $q$-th row of $E_{1}^{\bullet,\bullet}$ is naturally
isomorphic to the Čech complex of the $q$-th degree component of the constant
precosheaf of algebras $U\mapsto
H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))^{\pi_{0}(U)}$, i.e.

(5.33) $\displaystyle
U\mapsto\bigoplus_{q_{1}+\dots+q_{\pi_{0}(U)}=q}H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{\pi_{0}(U)}}(\mathfrak{X}(\mathbb{R}^{n})).$

Denote this precosheaf by $H^{q}_{M}$. The multiplication of
$H^{\bullet}(\mathfrak{X}(\mathbb{R}^{n}))$ is trivial on two elements of
positive degree. Hence, if $H^{q}_{M}(U)\to H^{q}_{M}(V)$ is an extension map
associated to an inclusion $U\subset V$, then the associated map

(5.34) $\displaystyle
H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{\pi_{0}(U)}}(\mathfrak{X}(\mathbb{R}^{n}))\to
H^{q_{1}^{\prime}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{\pi_{0}(V)}^{\prime}}(\mathfrak{X}(\mathbb{R}^{n}))$

is only nonzero if

(5.35)
$\displaystyle|\\{r:q_{r}>0,r=1,\dots,\pi_{0}(U)\\}|=|\\{r:q_{r}^{\prime}>0,r=1,\dots,\pi_{0}(V)\\}|.$

Hence the Čech complex associated to (5.33) decomposes into a direct sum of
complexes $C_{1},C_{2},\dots$, where $C_{r}$ is defined as the subcomplex on
which the number of tensor factors of nonzero degree in every term equals $r$.
Since $\mathcal{U}$ is a $k$-good cover, $C_{r}=0$ if $r>k$, so

(5.36)
$\displaystyle\check{C}(\mathcal{U}_{M},H^{q}_{M})=C_{1}\oplus\dots\oplus
C_{k}.$

We are done if we can show that we have the following isomorphism of chain
complexes:

(5.37) $\displaystyle
C_{r}\cong\left(\bigoplus_{\begin{subarray}{c}q_{1}+\dots+q_{r}=q\\\
q_{1},\dots,q_{r}>0\end{subarray}}\check{C}_{\bullet}(\mathcal{U}^{r}_{M^{r}},\mathcal{U}^{r}_{M^{r}_{r-1}})\otimes
H^{q_{1}}(W_{n})\otimes\dots\otimes H^{q_{r}}(W_{n})\right)^{\Sigma_{r}}.$

Let now $U$ be an intersection of elements of $\mathcal{U}$, and write
$U_{1},\dots,U_{s}\subset U$ for the connected components of $U$,
respectively. If $s\geq r$, then there is a nontrivial direct summand in the
complex $C_{r}$ associated to $U$, specifically

(5.38) $\displaystyle\bigoplus_{\begin{subarray}{c}q_{1}+\dots+q_{s}=q\\\
|\\{i:q_{i}>0,i=1,\dots,s\\}|=r\end{subarray}}$ $\displaystyle
H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{s}}(\mathfrak{X}(\mathbb{R}^{n}))$ (5.39)
$\displaystyle=\bigoplus_{\begin{subarray}{c}q_{1}+\dots+q_{r}=q\\\
q_{1},\dots,q_{r}>0\end{subarray}}H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{r}}(\mathfrak{X}(\mathbb{R}^{n})).$

Every term $H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{r}}(\mathfrak{X}(\mathbb{R}^{n}))$ in the latter direct sum is
associated to a a subset of $r$ pairwise different connected components
$U_{i_{1}},\dots,U_{i_{r}}$ of $U$. The product
$U_{i_{\sigma(1)}}\times\dots\times U_{i_{\sigma(r)}}$ is a connected
component of the set $U^{r}$, and $U$ is an intersection of elements in
$\mathcal{U}$. Hence $U_{i_{\sigma(1)}}\times\dots\times U_{i_{\sigma(r)}}$ is
an intersection of elements of the cover $\mathcal{U}^{r}$, and we can write
$c_{i_{1}\dots i_{r}}\in\check{C}_{\bullet}(\mathcal{U}^{r}_{M^{r}})$ for the
Čech simplex associated to it.
Now we can identify the term
$H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{r}}(\mathfrak{X}(\mathbb{R}^{n}))$ with a direct summand of the right-
hand side of (5.37):

$\displaystyle H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{r}}(\mathfrak{X}(\mathbb{R}^{n})$
$\displaystyle\cong\mathbb{R}c_{i_{1}\dots i_{r}}\otimes
H^{q_{1}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{r}}(\mathfrak{X}(\mathbb{R}^{n})$
$\displaystyle\cong\left(\bigoplus_{\sigma\in\Sigma_{r}}\mathbb{R}c_{i_{\sigma(1)}\dots
i_{\sigma(r)}}\otimes
H^{q_{\sigma(1)}}(\mathfrak{X}(\mathbb{R}^{n}))\otimes\dots\otimes
H^{q_{\sigma(r)}}(\mathfrak{X}(\mathbb{R}^{n}))\right)^{\Sigma_{r}}.$

Lemma 5.22 shows that this isomorphism induces the isomorphism (5.37):
Firstly, the lemma implies that the Cartesian product of the pairwise
different connected components $U_{i_{1}},\dots,U_{i_{r}}$ does not intersect
the diagonal $M^{r}_{r-1}$. Hence the Čech simplices $c_{i_{1}\dots i_{r}}$ do
not vanish in the relative Čech complex.
Secondly, the lemma implies that every direct summand in the relative Čech is
associated to a product of $r$ pairwise different connected components. Thus,
this construction exhausts the right-hand side of (5.37).
Hence the isomorphism (5.37) holds on a level of vector spaces, and it is
straightforward to see that this identification respects extension maps. Hence
it is even an isomorphism of chain complexes, and the statement is shown. ∎

By the argument before Proposition 5.23, the cohomology of the relative Čech
complexes in the previous proposition is isomorphic to relative singular
homology of the associated spaces. Together with Corollary 5.20 and a degree
reflection $p\mapsto-p$ to bring the spectral sequence into a cohomological
form, we end up with the following corollary:

######  Corollary 5.24.

Let $M$ be an orientable manifold which admits a finite, $k$-good open cover
(e.g. if $M$ is compact). There exists a cohomological spectral sequence
$\\{E^{\bullet,\bullet}_{r},d_{r}\\}$ which converges to reduced $k$-diagonal
cohomology $\Delta_{k}\tilde{H}^{\bullet}(\mathfrak{X}(M))$, and the entries
$E_{2}^{p,q}$ of its second page are, for $q\geq 1$, of the following form:

(5.40) $\displaystyle E^{p,q}_{2}$ $\displaystyle\cong H_{-p}(M)\otimes
H^{q}(W_{n})$
$\displaystyle\oplus\;\,\,\bigoplus_{\begin{subarray}{c}q_{1}+q_{2}=q\\\
q_{i}>0\end{subarray}}\;\;\left(H_{-p}(M^{2},M^{2}_{1})\otimes
H^{q_{1}}(W_{n})\otimes H^{q_{2}}(W_{n})\right)^{\Sigma_{2}}$
$\displaystyle\oplus\quad\;\,\,\dots$
$\displaystyle\oplus\bigoplus_{\begin{subarray}{c}q_{1}+\dots+q_{k}=q\\\
q_{i}>0\end{subarray}}\left(H_{-p}(M^{k},M^{k}_{k-1})\otimes
H^{q_{1}}(W_{n})\otimes\dots\otimes H^{q_{k}}(W_{n})\right)^{\Sigma_{k}}.$

Here, a permutation $\sigma\in\Sigma_{r}$ acts by simultaneous permutation of
the Cartesian factors of $M^{k}$ and the tensor factors $H^{q}(W_{n})$.

######  Remark 5.25.

These spectral sequences differ from the ones stated in [10], but only insofar
as [10] considers the quotient complexes
$\Delta_{k}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})/\Delta_{k-1}C^{\bullet}(\mathfrak{X}(\mathbb{R}^{n})$
rather than the diagonal complexes themselves. This essentially gives one
spectral sequence for every row in (5.40).

For $k\geq q+1$, we have
$\Delta_{k}H^{q}(\mathfrak{X}(M))=H^{q}(\mathfrak{X}(M))$ so in principle,
these spectral sequences can be used to calculate the full Lie algebra
cohomology of $\mathfrak{X}(M)$, degree by degree. In particular, since we
know that the nontrivial cohomology of $W_{n}$ is contained within the degrees
$q=2n+1,\dots,2n+n^{2}$ and the relative cohomology of $(M^{k},M^{k}_{k-1})$
in degrees $\leq nk$, we have the following:

######  Corollary 5.26.

For all smooth manifolds $M$ that admits $k$-good open covers for all
$k\in\mathbb{N}$, and all $n\geq 0$, the Gelfand–Fuks cohomology
$H^{n}(\mathfrak{X}(M))$ is finite-dimensional. Further, if $1\leq k\leq\dim
M$ then $H^{k}(\mathfrak{X}(M))=0$.

######  Example 5.27.

[grid=chess,labelstep=2]-4…00…6 0 3 R -10 R 0 3 R -10 R

Figure 4. The spectral sequence for 2-diagonal Lie algebra cohomology for
$\mathfrak{X}(S^{1})$. For the $k$-diagonal spectral sequences for $k\geq 2$,
this pattern continues into the upper-left direction.

We sketch here how one now can calculate the well-known Gelfand–Fuks
cohomology of $M=S^{1}$. By using excision and Poincaré duality222We thank
Moishe Kohan for communicating to us a proof idea for this., we can find

(5.41) $\displaystyle
H_{r}\left((S^{1})^{k},(S^{1})^{k}_{k-1}\right)=\tilde{H}_{r}\left((S^{1})^{k}\setminus(S^{1})^{k}_{k-1}\right)=\begin{cases}\mathbb{R}^{(k-1)!}&\text{
if }r=k,k-1,\\\ 0&\text{ else,}\end{cases}$

where the copies of $\mathbb{R}$ in $\mathbb{R}^{(k-1)!}$ are enumerated by
permutations of the $(k-1)$-th symmetric group, and the invariant space under
the action of the $k$-th symmetric group $\Sigma_{k}$ is one-
dimensional.333Note that in [1] it is incorrectly claimed that the nontrivial
degrees of $H_{\bullet}\left((S^{1})^{k},(S^{1})^{k}_{k-1}\right)$ are
$k(k-1)/2$-dimensional. Already $(S^{1})^{3}\setminus(S^{1})^{3}_{2}$ has only
2, not 3 connected components. Using this, we find that in the spectral
sequence for $k$-diagonal cohomology, there is only ever at most a single
nontrivial term on every diagonal $p+q=\text{const}$, and those only exist on
the diagonals $p+q=0,2,3,5,6,8,9,\dots$. From lacunary arguments, one
concludes that all differentials beyond the second page must be trivial.
By retracing the construction of the spectral sequence, one can verify that
the following cocycles $c_{2},c_{3}\in C^{\bullet}(\mathfrak{X}(S^{1}))$
indeed are representatives of the nontrivial cohomology classes in degree 2
and 3: For $x_{0}\in S^{1}$ an arbitrary point, $\partial_{\phi}$ the standard
coordinate vector field on $S^{1}$, and $f,g\in C^{\infty}(S^{1})$, we set

(5.42) $\displaystyle
c_{2}(f\partial_{\phi},g\partial_{\phi})=\int_{S^{1}}\left(f(\phi)g^{\prime}(\phi)-f^{\prime}(\phi)g(\phi)\right)\mathop{}\\!\mathrm{d}\phi,$
(5.43) $\displaystyle
c_{3}(f\partial_{\phi},g\partial_{\phi},h\partial_{\phi})=\det\begin{pmatrix}f(x_{0})&g(x_{0})&h(x_{0})\\\
f^{\prime}(x_{0})&g^{\prime}(x_{0})&h^{\prime}(x_{0})\\\
f^{\prime\prime}(x_{0})&g^{\prime\prime}(x_{0})&h^{\prime\prime}(x_{0})\\\
\end{pmatrix}.$

We note that higher cohomology classes arise as the wedge products of these
diagonal generators, but due to the lack of an obvious product on our spectral
sequence, this is difficult to see directly. In any case, as a ring, we have

(5.44) $\displaystyle H^{\bullet}(\mathfrak{X}(S^{1}))\cong
S^{\bullet}(\mathbb{R}c_{2})\otimes\Lambda^{\bullet}(\mathbb{R}c_{3}).$

##  Appendix A Cosheaves and Čech homology

In this appendix, we will recall the definition of (pre-)cosheaves, Čech
homology of precosheaves and simple properties thereof from [25]. For the
remainder of this section, fix a topological space $X$.

######  Definition A.1.

[25, Chapter V.1]

  * i)

A _precosheaf_ (of $\mathbb{R}$-vector spaces) $\mathcal{P}$ on $X$ is a
covariant functor from the category of open sets of $X$, morphisms given by
inclusions, into the category of $\mathbb{R}$-vector spaces. Given an
inclusion $U\subset V$ of open sets, we denote the associated mapping
$\mathcal{P}(U)\to\mathcal{P}(V)$ by $\iota_{U}^{V}$, called the _extension
map_ from $U$ to $V$ of the precosheaf $\mathcal{P}$.

  * ii)

A _cosheaf_ is a precosheaf $\mathcal{P}$ with the property that for every