On the cohomology of vector fields on parallelizable manifolds

Yuly Billig, Karl-Hermann Neeb

Abstract. In the present paper we determine for each parallelizable smooth compact manifold M the cohomology spaces H^{2}(V_{M} ,\overline{Ω}_{M}^{p}) of the Lie algebra V_{M} of smooth vector fields on M with values in the module \overline{Ω}_{M}^{p} =Ω_{M}^{p} /dΩ_{M}^{p−1} . The case of p=1 is of particular interest since the gauge algebra C^{∞}(M,𝔨) has the universal central extension with center \overline{Ω}_{M}^{1} , generalizing affine Kac-Moody algebras. The second cohomology H^{2}(V_{M} ,\overline{Ω}_{M}^{1} ) classifies twists of the semidirect product of V_{M} with the universal central extension C^{∞}(M,𝔨)⊕\overline{Ω}_{M}^{1} . Keywords: Lie algebra of vector fields, Lie algebra cohomology, Gelfand-Fuks cohomology, extended affine Lie algebra MSC 2000: 17B56, 17B65, 17B68

One of the most important insights in the theory of affine Kac–Moody Lie algebras is that they can all be realized as twisted or untwisted loop algebras. In the untwisted case, this realization starts with the Lie algebra of maps from a circle 𝕊^{1} to a finite-dimensional complex simple Lie algebra 𝔨,

L(𝔨) := ℂ[t, t^{−1}] ⊗ 𝔨,

then proceeds with the universal central extension

ℂ ↪ L̂(𝔨) → → L(𝔨),

and completes the picture by adding a derivation d acting by t\frac{d}{dt} on L(𝔨) and, accordingly, on the central extension. In the representation theory of affine algebras, an important role is played by the Virasoro algebra, which emerges via the Sugawara construction. At the Lie algebra level, the Witt algebra of vector fields on a circle, 𝔡 := Der(ℂ[t, t^{−1}]) acts on L̂(𝔨), so that we may form the semidirect product 𝔤 := L̂(𝔨) ⋊ 𝔡. To get the Virasoro algebra, we need to twist the above semidirect product by a 2-cocycle τ ∈ Z^{2}(𝔡, ℂ), which leads to the affine-Virasoro Lie algebra 𝔤_{τ} with the bracket

[(x, d), (x′, d′)] = ([x, x′] + d.x′ − d′.x + τ(d, d′), [d, d′]),

which contains both affine Kac-Moody algebra and the Virasoro algebra as subalgebras.

The theory of affine Kac–Moody algebras has an analytic side, where one replaces the algebra ℂ[t, t^{−1}] of Laurent polynomials by the Fréchet algebra C^{∞}(𝕊^{1}, ℂ) of complex valued functions on the circle. In this context one also obtains a one-dimensional central extension and the role of the Witt algebra is played by the Fréchet–Lie algebra V_{𝕊^{1}} of smooth vector fields on the circle, which also has a one-dimensional central extension. For an exposition of these ideas we refer to the monograph of Pressley and Segal [PS86].

Let us discuss the generalization of this construction from 𝕊^{1} to the case of an arbitrary C^{∞} -manifold M. The gauge Lie algebra of 𝔨-valued functions on M,

L_{M}(𝔨) = C^{∞}(M, 𝔨),

endowed with the pointwise defined Lie bracket and the natural Fréchet topology, has the universal central extension with the central space

\overline{Ω}^{1}(M, ℂ) := Ω^{1}(M, ℂ)/dΩ^{0}(M, ℂ)

(cf. [Ka84], [Ma02]). The Lie bracket on the Lie algebra

L̂_{M}(𝔨) = C^{∞}(M, 𝔨) ⊕ \overline{Ω}^{1}(M, ℂ)

is given by the formula

[f_{1} ⊗ g_{1}, f_{2} ⊗ g_{2}] = f_{1}f_{2} ⊗ [g_{1}, g_{2}] + (g_{1}|g_{2})[f_{2}df_{1}],

where f_{1}, f_{2} ∈ C^{∞}(M, ℂ), g_{1}, g_{2} ∈ 𝔨, (·|·) is the Cartan–Killing form on 𝔨 and [α] denotes the class of a 1-form α in \overline{Ω}^{1}(M, ℂ).

The Lie algebra V_{M} of smooth vector fields on M acts on both the gauge algebra C^{∞}(M, 𝔨) and on \overline{Ω}^{1}(M, ℂ) by the Lie derivative. This action is compatible with the above central cocycle, allowing us to consider the semidirect product of V_{M} with L̂_{M}(𝔨):

𝔤 = (C^{∞}(M, 𝔨) ⊕ \overline{Ω}^{1}(M, ℂ)) ⋊ V_{M}.

However the interplay between affine Lie algebras and the Virasoro algebra suggests that it will be natural here to twist the Lie bracket in 𝔤 by means of a 2-cocycle τ ∈ Z_{c}^{2}(V_{M}, \overline{Ω}^{1}(M, ℂ)), resulting in the family of Lie algebras

𝔤_{τ} = (C^{∞}(M, 𝔨) ⊕ \overline{Ω}^{1}(M, ℂ)) ⊕_{τ} V_{M},

so that the equivalence classes of twistings are classified by the cohomology space

H_{c}^{2}(V_{M}, \overline{Ω}^{1}(M, ℂ)) ≅ H_{c}^{2}(V_{M}, \overline{Ω}^{1}(M, ℝ)) ⊗_{ℝ} ℂ.

In the present paper we calculate this space for parallelizable manifolds M, relying heavily on results on the cohomology of Lie algebras of vector fields with values in differential forms by Gelfand–Fuks, Haefliger and Tsujishita. We calculate this space using the short exact sequences

0 → H_{dR}^{1}(M, ℝ) ↪ \overline{Ω}^{1}(M, ℝ)) → B_{dR}^{2}(M, ℝ) → 0

and

0 → B_{dR}^{1}(M, ℝ) ↪ Ω^{1}(M, ℝ) → \overline{Ω}^{1}(M, ℝ) → 0,

and then applying the corresponding long exact sequence in cohomology. Hence we need detailed knowledge on the cohomology of V_{M} with values in differential forms.

In the following M denotes an N -dimensional parallelizable compact manifold. The main goal of this paper is to describe for each p ∈ ℕ the second cohomology spaces

H_{c}^{2}(V_{M}, \overline{Ω}^{p}(M, ℝ)) for p ∈ ℕ, \overline{Ω}^{p}(M, ℝ) := Ω^{p}(M, ℝ)/dΩ^{p−1}(M, ℝ)

with respect to the natural action of the Lie algebra V_{M} of smooth vector fields. For reasons described above, our main interest lies in the case p = 1. For N = 1 and connected M we have M ≅ 𝕊^{1} and \overline{Ω}^{1}(M, ℝ) ≅ ℝ is a trivial module. Then H_{c}^{2}(V_{𝕊^{1}}, ℝ) ≅ ℝ describes the central extensions of V_{𝕊^{1}} , and a generator corresponds to the Virasoro algebra. For N ≥ 2, we find that

H_{c}^{2}(V_{M}, \overline{Ω}^{1}(M, ℝ)) ≅ H_{dR}^{3}(M, ℝ) ⊕ ℝ^{2}.

Here we use that each closed 3-form ω yields an \overline{Ω}^{1}(M, ℝ)-valued 2-cocycle by ω^{[2]}(X, Y) := [i_{Y} i_{X}ω] (cf. [Ne06a]). In terms of a trivializing 1-form κ ∈ Ω^{1}(M, ℝ^{N}), the other two cocycles can be described as follows. Define θ: V_{M} → C^{∞}(M, 𝔤𝔩_{N}(ℝ)) by L_{X}κ = −θ(X) · κ,

\overline{Ψ}_{1}(X) := Tr(θ(X)) ∈ Ω^{0}(M, ℝ) = C^{∞}(M, ℝ), Ψ_{1}(X) := Tr(dθ(X)) ∈ Ω^{1}(M, ℝ),

and

\overline{Ψ}_{2}(X, Y) := [Tr(dθ(X) ∧ θ(Y) − dθ(Y) ∧ θ(X))] ∈ \overline{Ω}^{1}(M, ℝ).

Then \overline{Ψ}_{1} ∧ Ψ_{1} and \overline{Ψ}_{2} provide the two additional generators of H_{c}^{2}(V_{M}, \overline{Ω}^{1}(M, ℝ)).

In the case when M is a torus 𝕋^{N}, we give explicit formulas for these cocycles in coordinates. For the toroidal Lie algebras, the cocycles \overline{Ψ}_{2} and \overline{Ψ}_{1} ∧ Ψ_{1} were discovered in the representation theory of these algebras ([EM94], [L99], see also [BB99], [Bi06]). It turns out here that it is easier to construct representations of 𝔤_{τ} for a non-trivial cocycle τ, rather than for the semidirect product 𝔤.

For the toroidal Lie algebras it would be natural to work in the algebraic setting, considering the algebra A of Laurent polynomials in N variables as the algebra of functions on 𝕋^{N} (Fourier polynomials), instead of C^{∞}(𝕋^{N}, ℂ). Although the cocycles that we get here are well-defined in this algebraic setting, our results do not solve the problem of describing the algebraic cohomology H^{2}(Der(A), Ω^{1}(A)/dA). The reason for this is that our proof is based on the results of Tsujishita and Haefliger, while the algebraic counterparts of these have not been established.

Toroidal Lie algebras are also closely related to the class of extended affine Lie algebras ([Neh04], [BN06]). According to a result of [ABFP05], most of the extended affine Lie algebras are twisted versions of toroidal Lie algebras. However, instead of the Lie algebra of vector fields V_{𝕋^{N}} , in the theory of extended affine Lie algebras one must use the Lie algebra of divergence zero vector fields V_{𝕋^{N}}^{div} (or a subalgebra). Thus it would be also interesting to describe the space H^{2}(V_{𝕋^{N}}^{div}, \overline{Ω}^{1}(𝕋^{N}, ℝ)). This question remains open. We point out here that the restriction of the cocycle \overline{Ψ}_{1} ∧ Ψ_{1} to V_{𝕋^{N}}^{div} vanishes, while \overline{Ψ}_{2} does not. Also we note that the cocycles corresponding to H_{dR}^{3}(M, ℝ) are not relevant for the theory of extended affine Lie algebras either, because of the additional restriction that the cocycle should vanish on the subalgebra of degree zero derivations, so that this subalgebra remains abelian.

The cohomology of the subcomplex C_{loc}^{p}(V_{M}, Ω^{p}) of local cochains, i.e., cochains f ∈ C_{c}^{p}(V_{M}, Ω_{M}^{p}) that are differential operators in each argument is easier accessible than the full complex. In view of [Tsu81], the difference fully comes from the case p = 0. In this case, the description of H_{loc}^{•}(V_{M}, F_{M}) is facilitated significantly by the results of de Wilde and Lecomte ([dWL83]) who show that the cohomology of the differential graded algebra C_{loc}^{•}(V_{M}, F_{M}) coincides with the cohomology of the subalgebra generated by the image of Ω_{M}^{•}, the image of the Chern–Weil homomorphism

χ_{1}: Sym^{•}(𝔤𝔩_{N}(ℝ), ℝ)^{𝔤𝔩_{N} (ℝ)} → C_{loc}^{2•}(V(M), F_{M})

corresponding to a connection on the frame bundle J^{1}(M) of M (which yields cocycles depending on first order derivatives), and the image of an algebra homomorphism

χ_{2}: C^{•}(𝔤𝔩_{N}(ℝ), ℝ) → C_{loc}^{•}(V(M), F_{M})

whose image consists of cocycles if the connection is flat. If all Prontrjagin classes of M vanish, then this results in an isomorphism

H_{loc}^{•}(V_{M}, F_{M}) ≅ H_{dR}^{•}(M, ℝ) ⊗ H^{•}(𝔤𝔩_{N}(ℝ), ℝ) ≅ H_{dR}^{•}(M, ℝ) ⊗ C^{•}(𝔤𝔩_{N}(ℝ), ℝ)^{𝔤𝔩_{N} (ℝ)}.

As our results show, for N > 1, all classes in H_{c}^{2}(V_{M}, \overline{Ω}^{p}(M, ℝ)) can be represented by local cochains, but the generator of H_{c}^{2}(V_{𝕊^{1}}, ℝ) ⊆ H_{c}^{2}(V_{𝕊^{1}}, C^{∞}(M, ℝ)) is non-local because it involves an integration.

Some of Tsujishita’s results have been generalized by Rosenfeld in [Ro71] to other classes of irreducible primitive Lie subalgebras of V_{M}. It would be of some interest to see if these results could be used to obtain the cohomology of these Lie algebras with values in \overline{Ω}_{M}^{1} .

Notation and conventions

If V is a module of the Lie algebra 𝔤, we write C^{p}(𝔤, V) for the corresponding space p-cochains, d_{𝔤}: C^{p}(𝔤, V) → C^{p+1}(𝔤, V) for the Chevalley–Eilenberg differential, B^{p}(𝔤, V) = d_{𝔤}(C^{p−1}(𝔤, V)) for the space of p-coboundaries, Z^{p}(𝔤, V) = ker(d_{𝔤}: C^{p}(𝔤, V) → C^{p+1}(𝔤, V)) for the space of p-cocycles and H^{p}(𝔤, V) for the Lie algebra cohomology. If 𝔤 is a topological Lie algebra and V a continuous 𝔤-module, then C_{c}^{p}(𝔤, V) etc. stands for the space of continuous cochains.

For x ∈ 𝔤 we have the insertion operator i_{x}: C^{p+1}(𝔤, V) → C^{p}(𝔤, V) and there is a natural action of 𝔤 on C^{p}(𝔤, V), which is denoted by the operators L_{x}, satisfying the Cartan relation

L_{x} = d_{𝔤} ∘ i_{x} + i_{x} ∘ d_{𝔤}.

For a subalgebra 𝔥 ≤ 𝔤 we write

C^{p}(𝔤, 𝔥, V) := {α ∈ C^{p}(𝔤, V): (∀x ∈ 𝔥) i_{x}α = 0, i_{x}(d_{𝔤}α) = 0}

for the relative cochains modulo 𝔥, and accordingly B^{p}(𝔤, 𝔥, V), Z^{p}(𝔤, 𝔥, V) and H^{p}(𝔤, 𝔥, V).

If G is a group, we denote the identity element by 1, and for g ∈ G, we write λ_{g}: G → G, x ↦ gx for the left multiplication by g, ρ_{g}: G → G, x ↦ xg for the right multiplication by g, m_{G}: G × G → G, (x, y) ↦ xy for the multiplication map, and η_{G}: G → G, x ↦ x^{−1} for the inversion.

If M is a smooth manifold, we write V_{M} for the Lie algebra of smooth vector fields on M, F_{M} := C^{∞}(M, ℝ) for the algebra of smooth real-valued functions on M, Ω_{M}^{k} := Ω^{k}(M, ℝ), the space of smooth real-valued p-forms on M, and

Z_{M}^{k} := ker(d|_{Ω_{M}^{k}}), B_{M}^{k} := dΩ_{M}^{k−1}, H_{M}^{k} := H_{dR}^{k}(M, ℝ), and \overline{Ω}_{M}^{k} := Ω_{M}^{k}/B_{M}^{k}.

We write T^{(p,q)}(M) := T (M)^{⊗p} ⊗ T^{∗}(M)^{⊗q} for the tensor bundles over M and Γ(T^{(p,q)}(M)) for their spaces of smooth sections, i.e., the (p, q)-tensor fields.

I. Crossed homomorphisms of Lie algebras and pull-backs

In this short first section, we collect some basic facts on crossed homomorphisms of Lie algebras which provide the tools used throughout the forthcoming sections. The main point is Theorem I.7 on maps in Lie algebra cohomology defined by crossed homomorphisms.

Let 𝔥 and 𝔫 be two Lie algebras, with 𝔥 acting on 𝔫 by derivations, i.e., we are given a Lie algebra homomorphism τ : 𝔥 → Der(𝔫).

Example I.1. (a) Our primary example would be 𝔥 = V_{M} and 𝔫 = C^{∞}(M, 𝔤), where 𝔤 is a finite-dimensional Lie algebra and the bracket on 𝔫 is defined pointwise. Then

(τ(v).f)(m) := (v.f)(m) := df(m)v(m)

defines a Lie algebra homomorphism from V_{M} to Der(C^{∞}(M, 𝔤)).

(b) If A is a commutative associative algebra and 𝔤 a finite-dimensional Lie algebra, then 𝔫 := A ⊗ 𝔤 carries a Lie algebra structure defined by

[a ⊗ x, a′ ⊗ x′] := aa′ ⊗ [x, x′].

Then we obtain an action of 𝔥 := Der(A) on 𝔤 by τ(D).(a ⊗ x) := (D.a) ⊗ x.

Definition I.2. A linear map θ : 𝔥 → 𝔫 is called a crossed homomorphism if

θ([x, y]) = [θ(x), θ(y)] + τ(x)θ(y) − τ(y)θ(x) for all x, y ∈ 𝔥.

Remark I.3. (a) Note that θ is a crossed homomorphism if and only if the map (θ, id_{𝔥}): 𝔥 → 𝔫 ⋊ 𝔥 is a homomorphism of Lie algebras.

In terms of the Lie algebra cohomology of 𝔥 with values in the 𝔥-module 𝔫, the condition that θ is a crossed homomorphism can be written as (d_{𝔥}θ)(x, y)+[θ(x), θ(y)] = 0, i.e., θ satisfies the Maurer–Cartan equation

d_{𝔥}θ + \frac{1}{2}[θ, θ] = 0.

(b) If 𝔥 ≅ 𝔫 ⋊_{η} 𝔰 is a semidirect product and the action of 𝔥 on 𝔫 is defined by τ(n, s) := η(s), then

[(n, s), (n′, s′)] = ([n, n′] + η(s).n′ − η(s′).n, [s, s′])

implies that θ: 𝔥 → 𝔫, (n, s) ↦ n is a crossed homomorphism.

(c) The kernel of any crossed homomorphism is a subalgebra.

Remark I.4. Let ρ: 𝔥 → 𝔫 be a homomorphism of Lie algebras. Then we obtain on 𝔫 an action of 𝔥 by τ(x).y := [ρ(x), y]. Now a linear map θ: 𝔤 → 𝔫 is a crossed homomorphism if and only if ρ′ := ρ + θ is a homomorphism of Lie algebras.

This follows directly by comparing

ρ′([x, y]) = ρ([x, y]) + θ([x, y]),

with

[ρ′(x), ρ′(y)] = [ρ(x), ρ(y)] + [θ(x), ρ(y)] + [ρ(x), θ(y)] + [θ(x), θ(y)].

It follows in particular that θ := −ρ is a crossed homomorphism.

Let us show how a crossed homomorphism may be used to pull back cocycles on 𝔫 to cocycles on 𝔥. First, we need to define the notion of an equivariant cochain.

Definition I.5. Let V be a module for the Lie algebra 𝔥 and a trivial module for 𝔫. A cochain φ ∈ C^{k}(𝔫, V) is called equivariant if

xφ(y_{1}, …, y_{k}) = ∑_{j=1}^{k} φ(y_{1}, …, τ(x)y_{j}, …, y_{k}) for all x ∈ 𝔥, y_{1}, …, y_{k} ∈ 𝔫,

which is equivalent to L_{x}φ = 0 for each x ∈ 𝔥.

Since [L_{x}, d_{𝔫}] = 0 holds for each x ∈ 𝔥 on C^{•}(𝔫, V), we have:

Lemma I.6. Equivariant cochains form a subcomplex C_{eq}^{•}(𝔫, V) in C^{•}(𝔫, V).

By definition, the subcomplex of equivariant cochains yields the equivariant cohomology H_{eq}^{•}(𝔫, V). Identifying 𝔫 with a subalgebra of the semidirect product 𝔫 ⋊ 𝔥, we may identify C^{p}(𝔫, V) with the subspace

{α ∈ C^{p}(𝔫 ⋊ 𝔥, V): (∀x ∈ 𝔥) i_{x}α = 0}.

Hence the relative cochain space

C^{p}(𝔥 ⋊ 𝔫, 𝔥, V) := {α ∈ C^{p}(𝔫 ⋊ 𝔥, V): (∀x ∈ 𝔥) i_{x}α = 0, i_{x}(dα) = 0}

can be identified with C_{eq}^{p}(𝔫, V) because if i_{x}α = 0 holds for each x ∈ 𝔥, then L_{x}α = i_{x}dα, so that invariance is equivalent to α ∈ C^{p}(𝔫 ⋊ 𝔥, 𝔥, V) (cf. [Fu86, p. 16]).

Theorem I.7. Let V be an 𝔥-module and consider it as a trivial 𝔫-module. Further let θ be a crossed homomorphism of Lie algebras θ : 𝔥 → 𝔫. Then the map

θ^{∗} : C_{eq}^{•}(𝔫, V) → C^{•}(𝔥, V), φ ↦ φ ∘ (θ × … × θ)

is a morphism of cochain complexes, hence induces a linear map of cohomology spaces θ^{∗} : H_{eq}^{•}(𝔫, V) → H^{•}(𝔥, V).

Proof. We have seen above that we may identify the complex (C_{eq}^{•}(𝔫, V), d_{𝔫}) with the relative Lie algebra subcomplex

(C^{•}(𝔫 ⋊ 𝔥, 𝔥, V), d_{𝔫⋊𝔥}) ⊆ (C^{•}(𝔫 ⋊ 𝔥), d_{𝔫⋊𝔥}).

For φ ∈ C_{eq}^{p}(𝔫, V), we write φ̃ for the corresponding element of C^{p}(𝔫 ⋊ 𝔥, V). Then we have

θ^{∗}φ = (θ, id_{𝔥})^{∗}φ̃,

and the assertion follows from the fact that (θ, id_{𝔥}): 𝔥 → 𝔫 ⋊ 𝔥 is a morphism of Lie algebras, hence induces a morphism of cochain complexes C^{•}(𝔫 ⋊ 𝔥, V) → C^{•}(𝔥, V).

Some applications to the Lie algebra of vector fields

In this subsection M denotes a finite-dimensional smooth manifold.

Proposition I.8. A 𝔤-valued differential form θ ∈ Ω^{1}(M, 𝔤) defines an F_{M} -linear crossed homomorphism

V_{M} → C^{∞}(M, 𝔤), X ↦ i_{X}θ

if and only if θ satisfies the Maurer–Cartan equation dθ + \frac{1}{2}[θ, θ] = 0.

Proof. Since the exterior differential on Ω^{•}(M, 𝔤) coincides with the V_{M} -Lie algebra differential, this is an immediate consequence of Remark I.3(a).

Recall that the left Maurer–Cartan form κ_{G} ∈ Ω^{1}(G, 𝔤) of a Lie group G with Lie algebra 𝔤 is defined by κ_{G}(v) := T (λ_{g}^{−1})v for v ∈ T_{g}(G).

Corollary I.9. (cf. [BR73], Th. 3.1) If M is a smooth manifold, 𝔤 a Lie algebra and κ ∈ Ω^{1}(M, 𝔤) satisfies the Maurer–Cartan equation, then we have a map

ζ: C^{•}(𝔤, ℝ) → C_{c}^{•}(V_{M}, F_{M}), ζ(ω)(X_{1}, …, X_{p}) := ω(κ(X_{1}), …, κ(X_{p})),

whose range consists of F_{M} -multilinear cocycles represented by differential forms. In particular, we obtain an algebra homomorphism

H^{•}(𝔤, ℝ) → H_{c}^{•}(V_{M}, F_{M}).

Proof. In view of Theorem I.7, it only remains to verify that ζ is compatible with the multiplication of cochains, but this is an immediate consequence of the definition.

Corollary I.10. If G is a Lie group with Lie algebra 𝔤 and κ_{G} ∈ Ω^{1}(G, 𝔤) the left Maurer– Cartan form, then

θ: V_{G} → C^{∞}(G, 𝔤), X ↦ i_{X}κ_{G}

is a F_{G} -linear crossed homomorphism which is bijective.

Proof. In view of Proposition I.8, we only have to recall that κ_{G} satisfies the Maurer–Cartan equation.

II. Cocycles with values in differential forms

Let M be a smooth paracompact N -dimensional manifold. In this section we first explain how affine connections can be used to define for each k ∈ ℕ cocycles Ψ_{k} ∈ Z_{c}^{k}(V_{M}, Ω_{M}^{k}). If, in addition, the tangent bundle of M is trivial, we obtain two more families of cocycles \overline{Ψ}_{k} ∈ Z_{c}^{k}(V_{M}, \overline{Ω}_{M}^{k−1}) and Φ_{k} ∈ Z_{c}^{2k−1}(V_{M}, F_{M}). These cocycles satisfy the relation d ∘ \overline{Ψ}_{k} = Ψ_{k} that will play a crucial role in our determination of the spaces H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{p}) in Section IV.

The set of affine connections ∇ on M is an affine space whose tangent space is the space Γ(T^{(1,2)}(M)) of tensors of type (2, 1). The elements of this space are F_{M} -bilinear maps V_{M} × V_{M} → V_{M} and any such map can also be considered as an F_{M} -linear map V_{M} → Γ(End(T M)), i.e., as a 1-form with values in the endomorphism bundle End(T M) ≅ T^{(1,1)}(M) of the tangent bundle T (M). In this sense we identify Γ(T^{(1,2)}(M)) with the space Ω^{1}(M, End(T M)) of 1-forms with values in the endomorphism bundle End(T M). Note that this space carries a natural module structure for the Lie algebra V_{M}, given by the Lie derivative.

Lemma II.1. ([Ko74]) Any affine connection ∇ on M defines a 1-cocycle

ζ: V_{M} → Ω^{1}(M, End(T M)), X ↦ L_{X}∇,

where

(L_{X}∇)(Y)(Z) := [X, ∇_{Y} Z] − ∇_{[X,Y]}Z − ∇_{Y} [X, Z].

For any other affine connection ∇′ the corresponding cocycle ζ′ has the same cohomology class.

To understand the cocycle ζ associated to an affine connection, we first describe it in a local chart.

Remark II.2. If U is an open subset of the vector space V , then any affine connection on U is given by

∇_{X}Y = dY · X + Γ(X, Y)

with a (2, 1)-tensor Γ. We then have

L_{X}(∇ − Γ)(Y, Z) = [X, dZ(Y)] − dZ([X, Y]) − d[X, Z](Y)
= d^{2}Z(X, Y) + dZ(dY (X)) − dX(dZ(Y))
− dZ(dY (X)) + dZ(dX(Y)) − d(dZ(X) − dX(Z))(Y)
= d^{2}Z(X, Y) − dX(dZ(Y)) + dZ(dX(Y))
− d^{2}Z(Y, X) − dZ(dX(Y)) + d^{2}X(Y, Z) + dX(dZ(Y))
= d^{2}X(Y, Z).

This means that

L_{X}∇ = d^{2}X + L_{X}Γ.

Since the Lie derivative of any symmetric tensor is symmetric, we see that L_{X}∇ is symmetric if and only if L_{X}Γ is symmetric, which is the case if Γ is symmetric, and this in turn means that ∇ is torsion free.

As a consequence of the preceding discussion, we can associate to any smooth manifold a canonical cohomology class [ζ] ∈ H_{c}^{1}(V_{M}, Ω^{1}(M, End(T M))) (cf. [Ko74]). That this class is always non-zero can be seen in local coordinates by showing that there is no (1, 2)-tensor Γ with d^{2}X = L_{X}Γ for all smooth vector fields on a 0-neighborhood in ℝ^{N}. For constant vector fields X, the preceding relation means that Γ is constant and for linear vector fields X we see that Γ: ℝ^{N} ⊗ ℝ^{N} → ℝ^{N} should be GL_{N}(ℝ)-equivariant, so that λ^{2}Γ(v, w) = Γ(λv, λw) = λΓ(v, w) for each λ ∈ ℝ yields Γ = 0.

Example II.3. Let V := ℝ^{N} and assume that M is parallelizable of dimension N. Then there exists some κ ∈ Ω^{1}(M, V) such that each κ_{m} is invertible, i.e., κ defines a trivialization of the tangent bundle of M via M × V → T M, (m, v) ↦ κ_{m}^{−1}v. Then, for each X ∈ V_{M}, L_{X}κ also in an element of Ω^{1}(M, V), and since all the linear maps κ_{m} are invertible, it can be written as L_{X}.κ = −θ(X) · κ for some smooth function θ(X) ∈ C^{∞}(M, 𝔤𝔩(V)). Clearly, C^{∞}(M, 𝔤𝔩(V)) ≅ C^{∞}(M, ℝ) ⊗ 𝔤𝔩(V) is a Lie algebra with respect to the pointwise bracket and V_{M} acts naturally by derivations.

We claim that θ is a crossed homomorphism:

θ([X, Y]) · κ = −L_{[X,Y]}κ = −L_{X}(L_{Y} κ) + L_{Y} (L_{X}κ)
= L_{X}(θ(Y) · κ) − L_{Y} (θ(X) · κ)
= (L_{X}θ(Y) − L_{Y} θ(X))κ + θ(Y)L_{X}κ − θ(X)L_{Y} κ
= (L_{X}θ(Y) − L_{Y} θ(X))κ + (θ(Y)θ(X) − θ(X)θ(Y))κ
= (L_{X}θ(Y) − L_{Y} θ(X) + [θ(X), θ(Y)])κ.

Now let

∇_{X}Y := κ^{−1}(X.κ(Y))

denote the corresponding affine connection. Then

(2.1)  κ(∇_{X}Y) = X.κ(Y) = −θ(X)κ(Y) + κ([X, Y])

is an immediate consequence of the definition of θ. Moreover, the map

κ̃: Γ(End(T M)) → C^{∞}(M, 𝔤𝔩(V)), κ̃(φ)(m) = κ_{m} ∘ φ_{m} ∘ κ_{m}^{−1}

satisfies

(2.2)  κ̃ ∘ L_{X}∇ = −dθ(X) ∈ Ω^{1}(M, 𝔤𝔩(V)).

In fact, we have

κ((L_{X}∇)(Y)(Z)) = κ([X, ∇_{Y} Z] − ∇_{[X,Y]}Z − ∇_{Y} [X, Z])
= κ(∇_{X}∇_{Y} Z) + θ(X)κ(∇_{Y} Z) − [X, Y].κ(Z) − Y.κ([X, Z])
= XY.κ(Z) + θ(X)(Y.κ(Z)) − [X, Y].κ(Z) − Y.(X.κ(Z) + θ(X) · κ(Z))
= θ(X)(Y.κ(Z)) − Y.(θ(X).κ(Z))
= −Y.(θ(X)) · κ(Z) = −dθ(X)(Y) · κ(Z),

and this calculation shows that κ̃((L_{X}∇)(Y)) = −dθ(X)(Y) for each vector field Y , which is (2.2).

Remark II.4. For any affine connection ∇, the operator η(X) := ∇_{X} − ad X on V_{M} is F_{M} -linear, hence defines a section of End(T M). We thus obtain a map

η: V_{M} → Γ(End(T M)), X ↦ ∇_{X} − ad X.

The space Γ(End(T M)) carries a natural associative algebra structure, hence in particular the structure of a Lie algebra (the gauge Lie algebra of the vector bundle T M) and V_{M} acts via the Lie derivative by derivations. The map η is a crossed homomorphism for this structure if and only if X ↦ ∇_{X} defines a representation of V_{M} on itself, i.e., if ∇ is a flat connection (Remark I.4). This is in particular the case if ∇ is obtained from a trivialization κ ∈ Ω^{1}(M, V) of the tangent bundle. In the latter case, (2.1) implies that

κ̃(η(X)) = −θ(X).

In view of κ̃(∇_{X}) = L_{X}, we now see that

κ̃(ad X) = κ̃(L_{X}) = L_{X} + θ(X),

which is a representation of V_{M} on C^{∞}(M, V) if θ is a crossed homomorphism (Remark I.4).

We likewise see for the action of V_{M} on Γ(End(T M)) and C^{∞}(M, 𝔤𝔩(V)) that κ̃(φ) := κ ∘ φ ∘ κ^{−1} leads to

κ̃ ∘ L_{X} = (L_{X} + ad(θ(X)) ∘ κ̃,

i.e., the representation of V_{M} on Γ(End(T M)) by the Lie derivative is transformed by κ̃ into the representation given by L_{X} + ad(θ(X)) on C^{∞}(M, 𝔤𝔩(V)). Similarly, the map

κ̃_{1}: Ω^{1}(M, End(T M)) → Ω^{1}(M, 𝔤𝔩(V)), ω ↦ κ̃ ∘ ω

intertwines the Lie derivative on the left hand side with the representation on the right hand side given by L_{X} + ad(θ(X)).

Definition II.5. Next we use the cocycle ζ, associated to the affine connection ∇, to define k -cocycles Ψ_{k} ∈ Z_{c}^{k}(V_{M}, Ω_{M}^{k}) depending on second order partial derivatives of vector fields. Let V := ℝ^{N}. For each p ∈ ℕ, we have a polynomial of degree p on 𝔤𝔩(V), invariant under conjugation, given by A ↦ Tr(A^{p}). The corresponding invariant symmetric p-linear map is given by

β(A_{1}, …, A_{p}) = ∑_{σ∈S_{p}} Tr(A_{σ(1)} ⋯ A_{σ(p)}),

and we consider it as a linear GL(V)-equivariant map End(V)^{⊗p} → ℝ, where GL(V) acts trivially on ℝ. This GL(V)-equivariant map leads to a vector bundle map

β_{M}: End(T M)^{⊗p} → M × ℝ,

where M × ℝ stands for the trivial vector bundle with fiber ℝ. On the level of bundle-valued differential forms, this in turn yields an alternating p-linear map

β_{M}^{1}: Ω^{1}(M, End(T M))^{p} → Ω^{p}(M, ℝ) = Ω_{M}^{p}.

To see that this map is V_{M} -equivariant, we note that in local coordinates we have on an open subset U ⊆ V ≅ ℝ^{N} the corresponding linear map

Ω^{1}(U, End(T U))^{⊗p} ≅ Ω^{1}(U, ℝ)^{⊗p} ⊗ End(V)^{⊗p} → Ω^{p}(U, ℝ),

acting on Ω^{1}(U, ℝ)^{⊗p} as the p-fold exterior product and on End(V)^{⊗p} as β. If φ: U_{1} → U_{2} is a local diffeomorphism, and α ∈ Ω^{1}(U_{2}, End(T (U_{2}))), then φ^{∗}α ∈ Ω^{1}(U_{1}, End(T (U_{1}))) is given by

(φ^{∗}α)_{m}(v) := T_{m}(φ)^{−1} ∘ α_{φ(m)}(T_{m}(φ)v) ∘ T_{m}(φ).

From this it follows that β_{M}^{1} is equivariant under diffeomorphisms, and hence with respect to the infinitesimal action of vector fields.

We conclude that we can use β_{M}^{1} to multiply Lie algebra cocycles (cf. [Fu86], App. F in [Ne04]). In particular, this leads with the cocycle ζ from Lemma II.1 for each k ∈ ℕ to a Lie algebra cocycle

Ψ_{k} ∈ Z_{c}^{k}(V_{M}, Ω_{M}^{k}),

defined by

Ψ_{k}(X_{1}, …, X_{k}) := (−1)^{k}β_{M}^{1}(ζ(X_{σ(1)}), …, ζ(X_{σ(k)}))

(cf. [Fu86]).

For any other affine connection ∇′, the corresponding cocycle ζ′, and the associated cocycles Ψ′_{k}, we first note that the difference ζ′ − ζ is a coboundary, and since products of cocycles and coboundaries are coboundaries, Ψ′_{k} − Ψ_{k} is a coboundary. Hence its cohomology class in H_{c}^{k}(V_{M}, Ω_{M}^{k}) does not depend on the choice of the affine connection.

Remark II.6. If the connection ∇ is defined by a trivialization of T M as in Example II.3, then ζ(X) is transformed into −dθ(X) ∈ Ω^{1}(M, 𝔤𝔩(V)), which leads to

Ψ_{k}(X_{1}, …, X_{k}) = ∑_{σ∈S_{k}} sgn(σ) Tr (dθ(X_{σ(1)}) ∧ … ∧ dθ(X_{σ(k)})) .

Next we construct several equivariant cocycles on a gauge Lie algebra.

Theorem II.7. Let 𝔤 be a finite-dimensional Lie algebra, F_{M} ⊗ 𝔤 be the corresponding gauge Lie algebra, consider Ω_{M}^{p} as a trivial module of F_{M} ⊗ 𝔤, and let ρ: 𝔤 → 𝔤𝔩(V) be a finitedimensional representation.

(a) By F_{M} -linear extension, we get a morphism of cochain complexes, C^{•}(𝔤, ℝ) → C_{eq}^{•}(F_{M} ⊗ 𝔤, F_{M}), where equivariance refers to the action of V_{M}, which gives a homomorphism of cohomology algebras

H^{•}(𝔤, ℝ) → H_{eq}^{•}(F_{M} ⊗ 𝔤, F_{M}).

(b) The following expression is a V_{M} -equivariant cocycle with values in k -forms, ψ_{k} ∈ Z_{eq}^{k}(F_{M} ⊗ 𝔤, Ω_{M}^{k}):

ψ_{k}(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k}) = ( ∑_{σ∈S_{k}} Tr (ρ(x_{σ(1)}) ⋯ ρ(x_{σ(k)})) )df_{1} ∧ ⋯ ∧ df_{k},

where f_{1}, …, f_{k} ∈ F_{M}, x_{1}, …, x_{k} ∈ 𝔤.

(c) The following expression is an equivariant cocycle with values in \overline{Ω}_{M}^{k−1} , k ≥ 1:

\overline{ψ}_{k}(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k}) = ( ∑_{σ∈S_{k}} Tr (ρ(x_{σ(1)}) ⋯ ρ(x_{σ(k)})) )[f_{1}df_{2} ∧ … ∧ df_{k}],

where f_{1}, …, f_{k} ∈ F_{M}, x_{1}, …, x_{k} ∈ 𝔤.

Proof. Verification of part (a) is straightforward. Assertions (b) and (c) easily follow from Proposition A.3, which is first used with Ω^{p} = Ω_{M}^{p} to obtain the cocycle property on F_{M} ⊗𝔤𝔩(V), and then we pull it back by the homomorphism id_{F_{M}} ⊗ρ: F_{M} ⊗ 𝔤 → F_{M} ⊗ 𝔤𝔩(V).

Definition II.7. Combining Theorems I.7 and II.7 with Renark II.4, we get for each trivialization of the tangent bundle T (M) of an N -dimensional manifold M by pulling back with the corresponding crossed homomorphism θ: V_{M} → F_{M} ⊗ 𝔤𝔩(V), V = ℝ^{N}, the following cocycles

Φ_{k} ∈ Z_{c}^{2k−1}(V_{M}, F_{M}), Φ_{k}(X_{1}, …, X_{2k−1}) = ∑_{σ∈S_{2k−1}} sgn(σ) Tr (θ(X_{σ(1)}) ⋯ θ(X_{σ(2k−1)}))

and \overline{Ψ}_{k} ∈ Z_{c}^{k}(V_{M}, \overline{Ω}_{M}^{k−1}), defined by

\overline{Ψ}_{k}(X_{1}, …, X_{k}) = ∑_{σ∈S_{k}} sgn(σ)[Tr (θ(X_{σ(1)})dθ(X_{σ(2)}) ∧ … ∧ dθ(X_{σ(k)}))].

Note that we have for each k ≥ 1 the relation

(2.3)  d ∘ \overline{Ψ}_{k} = Ψ_{k}.

III. Cohomology of smooth vector fields with values in differential forms

In this section, we first recall Tsujishita’s Theorem ([Tsu81], Thm. 5.1.6, see also 3.3.4), describing the continuous cohomology of V_{M} with the values in differential forms.

To explain its statement, we recall that each compact manifold M satisfies Tsujishita’s condition (F): H_{M}^{•} is finite-dimensional and, for each p ∈ ℕ, the subspace Z_{M}^{p} of closed p-forms in Ω_{M}^{p} has a closed complement. For compact oriented manifolds, the latter assertion is a direct consequence of the Hodge Decomposition Theorem (cf. [AMR83, Thm. 7.5.3]) and the general case can be reduced to this one via the orientable 2-sheeted covering. Moreover, Beggs shows in [Beg87] (Theorems 4.4 and 6.6) that for all finite-dimensional paracompact smooth manifolds the space Z_{M}^{p} of closed p-forms in Ω_{M}^{p} has a closed complement, so that condition (F) is equivalent to all cohomology spaces H_{M}^{p}, resp., the cohomology algebra H_{M}^{•}, being finite-dimensional.

The following theorem is a key result of this paper because it provides a bridge between Tsujishita’s results an the cocycles Ψ_{k}. It will be proved in Appendix C.

Theorem III.1. Let M be an N -dimensional manifold for which H_{M}^{•} is finite-dimensional. Then H_{c}^{q}(V_{M}, Ω_{M}^{p}) vanishes if p > q and for p, n ∈ ℕ_{0} , we have

H_{c}^{p+n}(V_{M}, Ω_{M}^{p}) ≅ H_{c}^{n}(V_{M}, F_{M}) ⊗ E_{p},

where

E_{p} := span {[Ψ_{1}]^{m_{1}}[Ψ_{2}]^{m_{2}} ⋯ [Ψ_{p}]^{m_{p}}: ∑_{j=1}^{p} jm_{j} = p} ⊆ H_{c}^{p}(V_{M}, Ω_{M}^{p}).

In particular, H_{c}^{•}(V_{M}, Ω_{M}^{•}) is a free H^{•}(V_{M}, F_{M})-module, generated by the non-zero products of the classes [Ψ_{k}], k = 1, …, N.

From the preceding theorem, we immediately derive with H^{0}(V_{M}, F_{M}) = H_{M}^{0} ≅ ℝ:

Corollary III.2.

(a) If p < k, then H^{p}(V_{M}, Ω_{M}^{k}) = 0.

(b) If M is connected, then H^{p}(V_{M}, Ω_{M}^{p}) ≅ E_{p}.

Lemma III.3. Let M be a compact connected orientable manifold and μ a volume form on M. Then

div: V_{M} → F_{M}, L_{X}μ = div(X)μ

defines a continuous Lie algebra cocycle and furthermore, each closed 1-form α ∈ Z_{M}^{1} defines a Lie algebra cocycle V_{M} → F_{M}, X ↦ α(X). These cocycles span the cohomology space

H_{c}^{1}(V_{M}, F_{M}) ≅ H_{M}^{1} ⊕ ℝ[div].

If, in addition, the volume form is defined by a trivialization of the tangent bundle T (M), then − div = Φ_{1} = \overline{Ψ}_{1} . Proof. In view of [Fu86, Th. II.4.11], we only have to verify the last part (cf. also [FL80] for the local cohomology of V_{M} with values in Ω_{M}^{p}). So let us assume that the volume form can be written as

μ = κ_{1} ∧ … ∧ κ_{N},

where the κ_{i} are the components of a trivializing 1-form κ ∈ Ω^{1}(M, ℝ^{N}). From L_{X}κ = −θ(X)κ we derive L_{X}κ_{j} = − ∑_{i=1}^{N} θ(X)_{ji}κ_{i} and further

L_{X}μ = ∑_{i=1}^{N} κ_{1} ∧ … ∧ L_{X}κ_{i} ∧ … ∧ κ_{N} = −( ∑_{i=1}^{N} θ(X)_{ii})κ_{1} ∧ … ∧ κ_{N} = − Tr(θ(X)) · μ.

This proves that div X = −Φ_{1}(X) = −\overline{Ψ}_{1}(X).

Remark III.4. We take a closer look at the second cohomology of the V_{M} -modules Ω_{M}^{p}.

From Corollary III.2(a) we know that H_{c}^{2}(V_{M}, Ω_{M}^{p}) vanishes for p > 2, and for p = 2 the space

H_{c}^{2}(V_{M}, Ω_{M}^{2}) = span{[Ψ_{1}^{2}], [Ψ_{2}]}

is 2-dimensional. For p = 1 we also know that

H_{c}^{2}(V_{M}, Ω_{M}^{1}) = H_{c}^{1}(V_{M}, F_{M}).[Ψ_{1}] = H_{M}^{1} · [Ψ_{1}] ⊕ ℝ[\overline{Ψ}_{1} ∧ Ψ_{1}]

is of dimension b_{1}(M) + 1 (Lemma III.3). The most intricate case is p = 0, i.e., H_{c}^{2}(V_{M}, F_{M}) (cf. Theorem IV.8 below.

Example III.5. Let M = G be an abelian Lie group of dimension N and κ_{1}, …, κ_{N} a basis of the space of invariant ℝ-valued 1-forms on G. Then κ := (κ_{1}, …, κ_{N}) is a closed trivializing 1form (the Maurer–Cartan form of G). Let X_{1}, …, X_{N} ∈ V(G) be the dual basis of left invariant vector fields. Writing X ∈ V(G) as X = ∑_{i=1}^{N} f_{i}X_{i}, we obtain

L_{X}κ_{i} = d(i_{X}κ_{i}) = df_{i}.

Therefore θ(X) ∈ C^{∞}(G, 𝔤𝔩_{N}(ℝ)) is given by θ(X)_{ij} = −df_{i}(X_{j}), i.e., −θ(X) is the Jacobian matrix of X with respect to the basis X_{1}, …, X_{N}.

IV. Cohomology with values in differential forms modulo exact forms

Let M be a parallelizable connected compact manifold of dimension N. Since the subspace H_{M}^{k} of \overline{Ω}_{M}^{k} is a trivial V_{M} -module, we derive from Corollary D.5 that

(4.1)  H_{c}^{k}(V_{M}, H_{M}^{m}) = 0 for 0 < k ≤ N, m ∈ ℕ_{0}.

Therefore the short exact sequence of V_{M} -modules

0 → H_{M}^{m} → \overline{Ω}_{M}^{m} ⟶^{d}B_{M}^{m+1} → 0

induces a long exact sequence in cohomology, hence leads to isomorphisms

(4.2a)  d_{∗}: H_{c}^{k}(V_{M}, \overline{Ω}_{M}^{m}) → H_{c}^{k}(V_{M}, B_{M}^{m+1}) for m ∈ ℕ_{0}, 0 < k ≤ N − 1

and an exact sequence

(4.2b) 0 → H_{c}^{N}(V_{M}, \overline{Ω}_{M}^{m})⟶^{d_{∗}}H_{c}^{N}(V_{M}, B_{M}^{m+1}) → H_{c}^{N+1}(V_{M}, H_{M}^{m}) ≅ H_{c}^{N+1}(V_{M}, ℝ) ⊗ H_{M}^{m}.

In particular, we get

dim H_{c}^{N}(V_{M}, \overline{Ω}_{M}^{m}) ≤ dim H_{c}^{N}(V_{M}, B_{M}^{m+1}).

Lemma IV.1. ([Ne06a, Lemma 23], [HS53]) For any smoothly paracompact manifold M we have

H_{c}^{0}(V_{M}, \overline{Ω}_{M}^{p}) = (\overline{Ω}_{M}^{p})^{V_{M}} = H_{M}^{p}.

Lemma IV.2. If M is parallelizable, then for each k ∈ ℕ the exterior differential d induces a surjective map

d_{∗}: H_{c}^{k}(V_{M}, \overline{Ω}_{M}^{k−1}) → H_{c}^{k}(V_{M}, Ω_{M}^{k}), [α] ↦ [d ∘ α],

and the natural map

H_{c}^{k}(V_{M}, B_{M}^{k}) → H_{c}^{k}(V_{M}, Ω_{M}^{k})

is surjective.

On the cohomology of vector fields on parallelizable manifolds 13

Proof. First we recall from Definition II.7 that d ∘ \overline{Ψ}_{k} = Ψ_{k}.

Next we observe that the exterior product of differential forms induces maps

Z_{M}^{p} × B_{M}^{q} → B_{M}^{p+q}, (α, β) ↦ α ∧ β,

and hence maps

Z_{M}^{p} × \overline{Ω}_{M}^{q} → \overline{Ω}_{M}^{p+q}, (α, [β]) ↦ [α ∧ β].

Since Ψ_{j} has values in B_{M}^{1} ∧ … ∧ B_{M}^{1} ⊆ B_{M}^{j}, each product Ψ_{j_{1}} ∧ … ∧ Ψ_{j_{r}} has values in B_{M}^{j_{1}+...+j_{r}}, so that

\overline{Ψ}_{j_{1}} ∧ Ψ_{j_{2}} ∧ … ∧ Ψ_{j_{r}}

is a well-defined cocycle with values in \overline{Ω}_{M}^{j_{1}+...+j_{r}}, satisfying

d ∘ (\overline{Ψ}_{j_{1}} ∧ Ψ_{j_{2}} ∧ … ∧ Ψ_{j_{r}}) = (d ∘ \overline{Ψ}_{j_{1}}) ∧ Ψ_{j_{2}} ∧ … ∧ Ψ_{j_{r}} = Ψ_{j_{1}} ∧ Ψ_{j_{2}} ∧ … ∧ Ψ_{j_{r}}. For j_{1} +…+j_{r} = k, this implies that the image of d contains all products [Ψ_{j_{1}} ∧Ψ_{j_{2}} ∧…∧Ψ_{j_{r}}] with ∑ j_{i} = k, and, in view of Theorem III.1, these products span H_{c}^{k}(V_{M}, Ω_{M}^{k}).

This proves the first part, and the second part of the assertion is an immediate consequence of the first one.

In view of

(4.3)  H_{c}^{q}(V_{M}, Ω_{M}^{p}) = 0 for q < p

(Corollary III.2(a)), the long exact cohomology sequence associated to the short exact sequence

0 → B_{M}^{p} → Ω_{M}^{p} → \overline{Ω}_{M}^{p} → 0

(which splits topologically by condition (F)), leads to isomorphisms

(4.4a)  H_{c}^{q}(V_{M}, \overline{Ω}_{M}^{p}) ≅ H_{c}^{q+1}(V_{M}, B_{M}^{p}) for q < p − 1.

For q = p − 1, Lemma IV.2 leads to an exact sequence

(4.4b)  0 = H_{c}^{p−1}(V_{M}, Ω_{M}^{p}) → H_{c}^{p−1}(V_{M}, \overline{Ω}_{M}^{p}) → H_{c}^{p}(V_{M}, B_{M}^{p}) → → H_{c}^{p}(V_{M}, Ω_{M}^{p}) → 0.

From [Ne06a, Prop. 6], we recall: Lemma IV.3. For each closed (p + q)-form ω ∈ Ω_{M}^{p+q}, the prescription

ω^{[p]}(X_{1}, …, X_{p}) := [i_{X_{p}} … i_{X_{1}}ω] ∈ \overline{Ω}_{M}^{q}

defines a continuous p-cocycle in Z_{c}^{p}(V_{M}, \overline{Ω}_{M}^{q}).

Calculating H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m})

In this subsection we address the problem to determine the second cohomology space H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m}) for a parallelizable manifold M of dimension N.

We start by observing that we have the following isomorphisms

H_{c}^{2}(V_{M}, \overline{Ω}^{m}) ≅ H_{c}^{2}(V_{M}, B_{M}^{m+1}) for N ≥ 3 by (4.2a)

≅ H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{m+1}) for m > 1 by (4.4a) (4.5)

≅ H_{c}^{1}(V_{M}, B_{M}^{m+2}) for N ≥ 2 by (4.2a)
≅ H_{c}^{0}(V_{M}, \overline{Ω}_{M}^{m+2}) for each m by (4.4a)
= H_{M}^{m+2} by Lemma IV.1.

We thus obtain Proposition IV.4. (a) For N ≥ 3 and m ≥ 2 the map

H_{M}^{m+2} → H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m}), [ω] ↦ [ω^{[2]}]

is an isomorphism.

(b) For N ≥ 2 and m ≥ 1 the map

H_{M}^{m+1} → H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{m}), [ω] ↦ [ω^{[1]}]

is an isomorphism. Proof. In view of (4.5), we only have to see that the isomorphisms are implemented by the maps ω ↦ ω^{[2]} , resp., ω ↦ ω^{[1]} .

(a) We start with a closed (m + 2)-form ω, representing an element of H_{M}^{m+2} , which we consider a V_{M} -fixed element in \overline{Ω}_{M}^{m+2} . A corresponding 1-cocycle with values in B_{M}^{m+2} is given by α(X) := L_{X}ω = d(i_{X}ω) = (d ∘ ω^{[1]})(X). This already shows that the corresponding \overline{Ω}_{M}^{m+1} -valued 1-cocycle is ω^{[1]} (cf. Lemma IV.3).

To distinguish it from the exterior differential d, we write d_{V(M)} for the Chevalley– Eilenberg differential on C^{•}(V_{M}, Ω_{M}^{•}). To see the corresponding 2-cocycle with values in B_{M}^{m+1} , we recall from [Ne06a, Lemma 5] that for ω^{[1]}(X) := i_{X}ω we have

(4.6)  (d_{V_{M}} ω^{[1]})(X, Y) = d(i_{X}i_{Y} ω) = −(d ∘ ω^{[2]})(Y, X).

Hence −ω^{[2]} ∈ Z_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m}) is the 2-cocycle corresponding to [ω] under the isomorphisms H_{M}^{m+2} ≅ H_{c}^{2}(V_{M}, \overline{Ω}^{m}) in (4.5).

(b) is proved as in (a).

Now we turn to the cases not covered by Proposition IV.4. The following proposition was the original motivation for the present paper:

Theorem IV.5. For N ≥ 2 we have

H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{1}) ≅ H_{M}^{3} ⊕ ℝ[\overline{Ψ}_{1} ∧ Ψ_{1}] ⊕ ℝ[\overline{Ψ}_{2}],

where H_{M}^{3} embeds via the map [ω] ↦ [ω^{[2]}].

Proof. First we assume that N ≥ 3. For m = 1, we first get from (4.2a) that H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{1}) ≅ H_{c}^{2}(V_{M}, B_{M}^{2}) and further from (4.4b) the exact sequence

0 → H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{2}) ↪ H_{c}^{2}(V_{M}, B_{M}^{2}) → H_{c}^{2}(V_{M}, Ω_{M}^{2}) = ℝ[Ψ_{1}^{2}] ⊕ ℝ[Ψ_{2}],

where the inclusion on the left maps [ω^{[1]}] to [ω^{[2]}] (cf. the proof of Proposition IV.4). Since we know from Proposition IV.4 that the map

H_{M}^{3} → H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{2}), [ω] ↦ [ω^{[1]}]

is an isomorphism, the assertion follows because the classes [Ψ_{1}^{2}] and [Ψ_{2}] in H_{c}^{2}(V_{M}, Ω_{M}^{2}) are contained in the image of H_{c}^{2}(V_{M}, B_{M}^{2}) (Lemma IV.2) and satisfy

[d ∘ (\overline{Ψ}_{1} ∧ Ψ_{1})] = [Ψ_{1}^{2}] and [d ∘ \overline{Ψ}_{2}] = [Ψ_{2}].

Now we consider the case N = 2. For m = 1 we get from (4.2a) and Proposition IV.4

H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{1}) ≅ H_{c}^{1}(V_{M}, B_{M}^{2}) ≅ H_{M}^{2} and H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{2}) ≅ H_{c}^{1}(V_{M}, B_{M}^{3}) = \{0\}.

Therefore the exact sequence B_{M}^{2} ↪ Ω_{M}^{2} → \overline{Ω}_{M}^{2} leads to an exact sequence

0 = H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{2}) → H_{c}^{2}(V_{M}, B_{M}^{2}) → H_{c}^{2}(V_{M}, Ω_{M}^{2}) → H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{2}) = 0,

showing that the map

H_{c}^{2}(V_{M}, B_{M}^{2}) → H_{c}^{2}(V_{M}, Ω_{M}^{2}) ≅ ℝ[Ψ_{1}^{2}] ⊕ ℝ[Ψ_{2}]

is an isomorphism.

On the other hand, (4.2b) yields an embedding

d∘: H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{1}) ↪ H_{c}^{2}(V_{M}, B_{M}^{2}) ≅ H_{c}^{2}(V_{M}, Ω_{M}^{2}) ≅ ℝ[Ψ_{1}^{2}] ⊕ ℝ[Ψ_{2}],

and Lemma IV.2 asserts that this embedding is surjective. This completes the proof.

Proposition IV.6. For N = 2 we have H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m}) = 0 for m ≥ 2 and

dim H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{1}) = 2.

Proof. For N = 2 we have Ω_{M}^{m} = 0 for m ≥ 3, so that \overline{Ω}_{M}^{m} vanishes in these cases. For m = 2 we get from (4.2b) an inclusion H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{2}) ↪ H_{c}^{2}(V_{M}, B_{M}^{3}) = 0. Therefore H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{m}) vanishes for m ≥ 2. The case m = 1 follows from Theorem IV.5.

So far we have covered all cases m ≥ 1 for N ≥ 2, and we are left with the case m = 0 or N = 1. First we recall the results for N = 1:

Example IV.7. (a) (cf. [FF01, Th. 30]) For N = 1 and M = 𝕊^{1} the algebra H^{•}(V_{𝕊^{1}}, F_{𝕊^{1}}) is a free graded commutative algebra generated by two elements α_{1}, α_{2} of degree 1 and one element β of degree 2. In particular

H_{c}^{1}(V_{𝕊^{1}}, F_{𝕊^{1}}) = ℝα_{1} ⊕ ℝα_{2} and H_{c}^{2}(V_{𝕊^{1}}, F_{𝕊^{1}}) = ℝα_{1}α_{2} ⊕ ℝβ.

Furthermore \overline{Ω}_{𝕊^{1}}^{1} = H_{𝕊^{1}}^{1} is one-dimensional and a trivial V_{𝕊^{1}} -module, so that

H^{2}(V_{𝕊^{1}}, \overline{Ω}_{𝕊^{1}}^{1}) = H^{2}(V_{𝕊^{1}}, ℝ) ≅ ℝ

(Theorem 29 in [FF01], p. 195).

(b) For M = ℝ^{N} the cohomology algebra H^{•}(V_{ℝ^{N}} , F_{ℝ^{N}}) ≅ H^{•}(𝔤𝔩_{N}(ℝ), ℝ) is the free exterior algebra generated by the classes [Φ_{k}], k = 1, …, N (cf. Definition II.7; [Tsu77], §4).

Now we turn to the case m = 0.

Theorem IV.8. For N ≥ 2 the map

H_{M}^{2} ⊕ H_{M}^{1} → H_{c}^{2}(V_{M}, F_{M}), ([α], [β]) ↦ [α + β ∧ \overline{Ψ}_{1}]

is a linear isomorphism.

Proof. In view of \overline{Ω}_{M}^{0} = F_{M}, (4.2a/b) provide an embedding

d_{∗}: H_{c}^{2}(V_{M}, F_{M}) ↪ H_{c}^{2}(V_{M}, B_{M}^{1}), [ω] ↦ [d ∘ ω]

which is an isomorphism for N > 2. We also get from Proposition IV.4 for N ≥ 2 an isomorphism:

H_{M}^{2} → H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{1}), [ω] ↦ [ω^{[1]}].

In view of Lemma III.3, we have

H_{c}^{2}(V_{M}, Ω_{M}^{1}) = H_{c}^{1}(V_{M}, F_{M}) · [Ψ_{1}] ≅ H_{M}^{1} ⊕ ℝ[\overline{Ψ}_{1} ∧ Ψ_{1}].

We now consider the exact sequence

\begin{matrix}H_{c}^{1}(V_{M}, B_{M}^{1}) → → H_{c}^{1}(V_{M}, Ω_{M}^{1})⟶^{0}H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{1}) ≅ H_{M}^{2} \\ ⟶^{−d_{∗}}H_{c}^{2}(V_{M}, B_{M}^{1})⟶^{f}H_{c}^{2}(V_{M}, Ω_{M}^{1})⟶^{g}H_{c}^{2}(V_{M}, \overline{Ω}_{M}^{1}), \end{matrix}

in which we know all terms except the middle one, which we are interested in. The onedimensional space H_{c}^{1}(V_{M}, Ω_{M}^{1}) is generated by [Ψ_{1}], which has values in B_{M}^{1} , so that its image in \overline{Ω}_{M}^{1} vanishes. This yields the 0-arrow in the upper row of the diagram (Lemma IV.2). To calculate the connecting map H_{c}^{1}(V_{M}, \overline{Ω}_{M}^{1}) ≅ H_{M}^{2} → H_{c}^{2}(V_{M}, B_{M}^{1}), we recall from (4.6) that we have for each closed 2-form ω the relation

δ([ω^{[1]}]) = [d_{V_{M}} ω^{[1]}] = −[d ∘ ω^{[2]}] = −d_{∗}[ω].

Let β be a closed 1-form on M, considered as a 1-cocycle V_{M} → F_{M}. Then

β ∧ \overline{Ψ}_{1} ∈ Z_{c}^{2}(V_{M}, F_{M})

satisfies

d ∘ (β ∧ \overline{Ψ}_{1}) = (d ∘ β) ∧ \overline{Ψ}_{1} + β ∧ d ∘ \overline{Ψ}_{1} = (d ∘ β) ∧ \overline{Ψ}_{1} + β ∧ Ψ_{1}.

The 1-cochain

γ: V_{M} → Ω_{M}^{1}, γ(X) := \overline{Ψ}_{1}(X) · β = (\overline{Ψ}_{1} · β)(X)

satisfies

d_{V(M)}γ = (d_{V(M)}\overline{Ψ}_{1}) ∧ β − \overline{Ψ}_{1} ∧ (d_{V(M)}β) = −\overline{Ψ}_{1} ∧ d_{V(M)}β.

Since the Cartan formula implies that

(d_{V(M)}β)(X) = L_{X}β = i_{X}(dβ) + d(i_{X}β) = d(β(X)),

we obtain d_{V(M)}γ = −\overline{Ψ}_{1} ∧ (d ∘ β), showing that (d ∘ β) ∧ \overline{Ψ}_{1} ∈ B_{c}^{2}(V_{M}, Ω_{M}^{1}). This implies that

d_{∗}[β · \overline{Ψ}_{1}] = [d ∘ (β ∧ \overline{Ψ}_{1})] = [β ∧ Ψ_{1}].

We also recall that

H_{c}^{2}(V_{M}, Ω_{M}^{1}) = (H_{M}^{1} ∧ [Ψ_{1}]) ⊕ ℝ(\overline{Ψ}_{1} ∧ Ψ_{1}) ≅ H_{M}^{1} ⊕ ℝ.

From the preceding considerations, we see that the subspace H_{M}^{1} ∧ [Ψ_{1}] lies in the range of f. Since g([\overline{Ψ}_{1} ∧ Ψ_{1}]) = [Ψ_{1}^{2}] is non-zero, we derive

im(f) = ker(g) = H_{M}^{1} ∧ [Ψ_{1}].

From the exactness in H_{c}^{1}(V_{M}, B_{M}^{1}) we now see that the natural map

H_{M}^{2} ⊕ H_{M}^{1} → H_{c}^{2}(V_{M}, B_{M}^{1}), ([α], [β]) ↦ [d ∘ α + β ∧ Ψ_{1}] = [d ∘ (α + β · \overline{Ψ})]

is a linear isomorphism. As the elements α+β·\overline{Ψ} form F_{M} -valued 2-cocycles, we finally conclude that the inclusion d_{∗} is also surjective for N = 2. This completes the proof.

Problem 1. Determine the spaces H^{2}(V_{M}, \overline{Ω}_{M}^{1}) for all connected smooth manifolds M, without assuming that M is parallelizable.

Appendix A. Lie algebra cocycles with values in associative algebras

Let A be a unital associative algebra and A_{L} := (A, [·, ·]) be the underlying Lie algebra. We consider A as an A_{L} -module with respect to the adjoint representation x.y := [x, y].

We write m_{A}: A × A → A for the product map and b_{A}: A × A → A for the commutator bracket and observe that both are A_{L} -invariant.

We take a closer look at the Lie algebra complex (C^{•}(A_{L}, A), d), where we use the multiplication on A, which is A_{L} -equivariant, to define a multiplication on C^{•}(A_{L}, A) by

(α ∧ β)(x_{1}, …, x_{p+q}) := \frac{1}{p!q!} ∑_{σ∈S_{p+q}} sgn(σ)m_{A}(α(x_{σ(1)}, …, x_{σ(p)}), β(x_{σ(p+1)}, …, x_{σ(p+q)}))

for α ∈ C^{p}(A_{L}, A) and β ∈ C^{q}(A_{L}, A) (cf. [Fu86, Sect. I.3.2]). With

Alt(γ)(x_{1}, …, x_{r}) := ∑_{σ∈S_{r}} sgn(σ)α(x_{σ(1)}, …, x_{σ(r)}),

this means that

α ∧ β = \frac{1}{p!q!} Alt(α · β) = \frac{1}{p!q!} Alt(m_{A} ∘ (α, β)).

An easy induction implies that n-fold products in C^{•}(A_{L}, A) are given by

α_{1} ∧ ⋯ ∧ α_{n} = \frac{1}{∏_{j=1}^{n} p_{j}!} Alt(α_{1} ⋯ α_{n}) for α_{i} ∈ C^{p_{i}}(A_{L}, A).

We thus obtain an associative differential graded algebra (C^{•}(A_{L}, A), d) (cf. [Ne04, App. F]; see also [Tsu81, p. 30]).

Note that id_{A} ∈ C^{1}(A_{L}, A) is a 1-cochain with
d_{A_{L}}(id_{A})(x, y) = x.y − y.x − [x, y] = [x, y] − [y, x] − [x, y] = [x, y],

so that d_{A_{L}}(id_{A}) = b_{A}. It follows in particular that b_{A} is a 2-coboundary. We therefore obtain a sequences of coboundaries

b_{A}^{n} = d_{A_{L}}(id_{A} ∧b_{A}^{n−1}) ∈ B^{2n}(A_{L}, A) ⊆ Z^{2n}(A_{L}, A).

Lemma A.1. If T: A → 𝔷 is a linear map vanishing on all commutators, then T ∘ b_{A}^{k} vanishes for each k ∈ ℕ, and if 𝔷 is considered as a trivial A_{L} -module, then

φ_{k} := T ∘ (id_{A} ∧b_{A}^{k−1}) = T ∘ (id_{A})^{2k−1} ∈ Z^{2k−1}(A_{L}, 𝔷),

is a cocycle given by

φ_{k}(x_{1}, …, x_{2k−1}) = ∑_{σ∈S_{2k−1}} sgn(σ)T (x_{σ(1)} ⋯ x_{σ(2k−1)}).

Proof. Arguing as in Remark I.2 of [Ne06b], we see that for each n ∈ ℕ_{0} , we have

b_{A}^{n} = \frac{1}{2^{n}} Alt(b_{A} ⋯ b_{A}) = Alt(m_{A} ⋯ m_{A}) = Alt(id_{A}^{2n}) = (id_{A})^{2n},

i.e.,

b_{A}^{n}(x_{1}, …, x_{2n}) = ∑_{σ∈S_{2n}} sgn(σ)x_{σ(1)} ⋯ x_{σ(2n)}.

Let τ be the cyclic permutation τ = (1 2 … 2n). Then sgn(τ) = −1, but, for each σ ∈ S_{2n}, we have

T (x_{στ(1)} ⋯ x_{στ(2n)}) = T (x_{σ(2)} ⋯ x_{σ(2n)}x_{σ(1)}) = T (x_{σ(1)}x_{σ(2)} ⋯ x_{σ(2n)}).

This leads to T ∘ b_{A}^{n} = −T_{A} ∘ b_{A}^{n}, which implies that T ∘ b_{A}^{n} vanishes.

Since b_{A} is a cocycle, we now obtain

dφ_{k} = d(T ∘ (id_{A} ∧b_{A}^{k−1})) = T ∘ (d(id_{A}) ∧ b_{A}^{k−1}) = T ∘ (b_{A}^{k}) = 0.

The following theorem describes an important application of the preceding construction, namely that for A = 𝔤𝔩_{n}(ℝ) we thus obtain a set of generators of the cohomology algebra ([Fu86, Th. 2.1.6]):

Theorem A.2. The cohomology algebra H^{•}(𝔤𝔩_{N}(ℝ), ℝ) is generated by the cohomology classes of the cocycles φ_{k} ∈ Z^{2k−1}(𝔤𝔩_{N}(ℝ), ℝ), k = 1, …, N, given by

φ_{k}(x_{1}, …, x_{2k−1}) = ∑_{σ∈S_{2k−1}} sgn(σ) Tr (x_{σ(1)} … x_{σ(2k−1)}) for x_{1}, …, x_{2k−1} ∈ 𝔤𝔩_{N}(ℝ).

Lie algebra cocycles with values in differential forms

Let (Ω^{•}, d) be a differential graded algebra whose multiplication is denoted α ∧ β, V a finite-dimensional vector space, and put A := Ω^{0} ⊗ End(V) and B := Ω^{•} ⊗ End(V). Then B also carries the structure of a differential graded algebra with differential d_{B} := d ⊗ id_{End(V)} which is not graded commutative. Note that A_{L} ≅ Ω^{0} ⊗ 𝔤𝔩(V) as Lie algebras.

Proposition A.3. For each k ∈ ℕ, we obtain cocycles ψ_{k} ∈ Z^{k}(A_{L}, Ω^{k}) and \overline{ψ}_{k} ∈ Z^{k}(A_{L}, Ω^{k−1}/dΩ^{k−2}) satisfying d ∘ \overline{ψ}_{k} = ψ_{k} by

ψ_{k}(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k}) := ∑_{σ∈S_{k}} Tr(x_{σ(1)} ⋯ x_{σ(k)})df_{1} ∧ … ∧ df_{k}

and

\overline{ψ}_{k}(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k}) := ∑_{σ∈S_{k}} Tr(x_{σ(1)} ⋯ x_{σ(k)})[f_{1} · df_{2} ∧ … ∧ df_{k}] ∈ Ω^{k−1}/dΩ^{k−2}.

Proof. We first consider the linear map

φ = d_{B}: A_{L} → Ω^{1} ⊗ End(V), f ⊗ x ↦ df ⊗ x

and note that

(d_{A_{L}}φ)(a, b) = a.db − b.da − d[a, b] = [a, db] − [b, da] − [da, b] − [a, db] = 0

implies that φ is a 1-cocycle. This implies that its ∧-powers φ^{k} ∈ Z^{k}(A_{L}, Ω^{k} ⊗ End(V)) are k -cocycles and hence that

ψ_{k} = Tr ∘φ^{k} ∈ Z^{k}(A_{L}, Ω^{k}),

because Tr: Ω^{k} ⊗ End(V) → Ω^{k} is a morphism of the A_{L} -module Ω^{k} ⊗ End(V) onto the trivial module Ω^{k}.

That \overline{ψ}_{k} is alternating follows from

[f_{1} ∧ df_{2} ∧ … ∧ df_{k}] + [f_{2} ∧ df_{1} ∧ df_{3} ∧ … ∧ df_{k}] = [d(f_{1}f_{2}) ∧ … ∧ df_{k}] = [d(f_{1}f_{2} ∧ df_{3} ∧ … df_{k})] = 0.

It remains to show that \overline{ψ}_{k} is a k -cocycle. To this end, we consider

id_{A} ∧φ^{k−1} ∈ C^{k}(A_{L}, Ω^{k−1} ⊗ End(V))

and observe that

d_{A_{L}}(id_{A} ∧φ^{k−1}) = d_{A_{L}}(id_{A}) ∧ φ^{k−1} = b_{A} ∧ φ^{k−1} = id_{A} ∧ id_{A} ∧φ^{k−1}.

If q: Ω^{k} → Ω^{k}/dΩ^{k−1}, β ↦ [β] is the quotient map, we have

(q ∘ Tr ∘(id_{A} ∧φ^{k−1}))(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k})

= ∑_{σ∈S_{k}} sgn(σ) Tr(x_{σ(1)} ⋯ x_{σ(k)})[f_{σ(1)}df_{σ(2)} ∧ … ∧ df_{σ(k)}]

= ∑_{σ∈S_{k}} Tr(x_{σ(1)} ⋯ x_{σ(k)})[f_{1}df_{2} ∧ … ∧ df_{k}] = \overline{ψ}_{k}(f_{1} ⊗ x_{1}, …, f_{k} ⊗ x_{k}).

To see that \overline{ψ}_{k} is a cocycle, it therefore suffices to show that

d\overline{ψ}_{k} = q ∘ Tr ∘(id_{A} ∧ id_{A} ∧φ^{k−1}) = 0.

Explicitly, we have

(q ∘ Tr ∘(id_{A}^{2} ∧φ^{k−1}))(f_{1} ⊗ x_{1}, …, f_{k+1} ⊗ x_{k+1})

= ∑_{σ∈S_{k+1}} sgn(σ) Tr(x_{σ(1)} ⋯ x_{σ(k+1)})[f_{σ(1)}f_{σ(2)} df_{σ(3)} ∧ … ∧ df_{σ(k+1)}].

If τ = (1 2 … k + 1) is the cyclic permutation, we obtain

sgn(στ) Tr(x_{στ(1)} ⋯ x_{στ(k+1)})[f_{στ(1)}f_{στ(2)} df_{στ(3)} ∧ … ∧ df_{στ(k+1)}]
= (−1)^{k} sgn(σ) Tr(x_{σ(1)} ⋯ x_{σ(k+1)})[f_{σ(2)}f_{σ(3)} df_{σ(4)} ∧ … ∧ df_{σ(k+1)} ∧ df_{1}]
= sgn(σ) Tr(x_{σ(1)} ⋯ x_{σ(k+1)})[f_{σ(2)}f_{σ(3)} df_{σ(1)} ∧ df_{σ(4)} ∧ … ∧ df_{σ(k+1)}].

Now the relation

[f_{1}f_{2}df_{3}∧d(⋯)]+[f_{2}f_{3}df_{1}∧d(⋯)]+[f_{3}f_{1}df_{2}∧d(⋯)]+ = [d(f_{1}f_{2}f_{3})∧d(⋯)] = [d(f_{1}f_{2}f_{3} ⋯)] = 0

implies that 3\overline{ψ}_{k} is a cocycle, so that \overline{ψ}_{k} is a cocycle.

Appendix B. Cohomology of formal vector fields

In this section, we will review results of Gelfand–Fuks on the cohomology of formal vector fields because this is needed to reduce Theorem III.1 to Tsujishita’s result.

Fix N ∈ ℕ and write F_{N} = ℝ[[x_{1}, …, x_{N}]] for the commutative algebra of formal power series in the variables x_{1}, … x_{N}, endowed with the projective limit topology, and let W_{N} be the Lie algebra of continuous derivations of F_{N}:

W_{N} = ⨁_{j=1}^{N} F_{N} \frac{∂}{∂x_{j}} .

Consider also the subalgebra L_{0} in W_{N} consisting of vector fields that vanish at the origin:

L_{0} = \{ ∑_{j=1}^{N} f_{j}(x)\frac{∂}{∂x_{j}} : f_{j}(0) = 0, j = 1, …, N\}, satisfying W_{N} ≅ L_{0} ⋊ ℝ^{N},

where the second factor corresponds to the abelian Lie subalgebra of constant (formal) vector fields.

For X ∈ W_{N}, we write
J(X) ∈ F_{N} ⊗ 𝔤𝔩_{N}(ℝ)

for the Jacobian of X, defined for X = ∑_{i} f_{i} \frac{∂}{∂x_{i}} by J(X)_{ij} = \frac{∂f_{i}}{∂x_{j}} . Then

−J: W_{N} → F_{N} ⊗ 𝔤𝔩_{N}(ℝ)

is a crossed homomorphism whose kernel is the subalgebra of constant vector fields. From the relation

−J([X, Y]) = −X.J(Y) + Y.J(X) + [−J(X), −J(Y)],

it follows that

−J_{0}: L_{0} → 𝔤𝔩_{N}(ℝ), X ↦ −J(X)(0)

is a surjective homomorphism of Lie algebras, restricting to an isomorphism on the subalgebra of linear vector fields.

Let Ω_{N}^{p} be the space of formal p-forms in N variables, considered as a free F_{N} -module and also as a W_{N} -module, with respect to the action defined by the Lie derivative.

Definition B.1. Using the crossed homomorphism θ := −J, we get with Lemma A.1, Proposition A.3 and Theorem I.7 the following cocycles

Φ_{k}^{W} ∈ Z_{c}^{2k−1}(W_{N}, F_{N}), Φ_{k}^{W} (X_{1}, …, X_{2k−1}) = ∑_{σ∈S_{2k−1}} sgn(σ) Tr (θ(X_{σ(1)}) … θ(X_{σ(2k−1)})) ,

Ψ_{k}^{W} ∈ Z_{c}^{k}(W_{N}, Ω_{N}^{k}), Ψ_{k}^{W} (X_{1}, …, X_{k}) = ∑_{σ∈S_{k}} sgn(σ) Tr (dθ(X_{σ(1)}) ∧ … ∧ dθ(X_{σ(k)})) ,

and \overline{Ψ}_{k}^{W} ∈ Z_{c}^{k}(W_{N}, Ω_{N}^{k−1}/dΩ_{N}^{k−2}), defined by

\overline{Ψ}_{k}^{W} (X_{1}, …, X_{k}) = ∑_{σ∈S_{k}} sgn(σ)[Tr (θ(X_{σ(1)})dθ(X_{σ(2)}) ∧ … ∧ dθ(X_{σ(k)}))].

The following theorem is a re-formulation of the results of Gelfand-Fuks ([GF70a]), describing the cohomology of W_{N} with values in the modules Ω_{N}^{p}. Let V = ℝ^{N}, considered as the canonical module of 𝔤𝔩_{N}(ℝ) and write

ev_{0}: Ω_{N}^{p} → Λ^{p}(V′), ω ↦ ω(0)

for the evaluation map. We consider V as a module of L_{0} by pulling back the module structure from 𝔤𝔩_{N}(ℝ), so that ev_{0} is a morphism of L_{0} -modules.

Theorem B.2.

(a) For each p, q ∈ ℕ_{0} , the map C_{c}^{p}(W_{N}, Ω_{N}^{q}) → C_{c}^{p}(L_{0}, Λ^{q}(V′)), ω ↦ ev_{0} ∘ω |_{L_{0}} induces an

isomorphism

H_{c}^{p}(W_{N}, Ω_{N}^{q}) → H_{c}^{p}(L_{0}, Λ^{q}(V′)).

(b) H_{c}^{•}(L_{0}, Λ^{•}(V′)) ≅ H_{c}^{•}(L_{0}, ℝ) ⊗ H_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{•}(V′)) as bigraded algebras.

(c) The inclusions 𝔤𝔩_{N}(ℝ) ↪ L_{0} ↪ W_{N} induce isomorphisms of graded algebras

H_{c}^{•}(W_{N}, ℝ) → H_{c}^{•}(L_{0}, ℝ) → H_{c}^{•}(𝔤𝔩_{N}(ℝ), ℝ).

In particular, H_{c}^{•}(W_{N}, ℝ) is an exterior algebra with generators of degree 2k − 1, k = 1, …, N.

(d) Let Ψ_{k}^{L} := ev_{0} ∘Ψ_{k}^{W} |_{L_{0}} ∈ Z_{c}^{k}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{k}(V′)). Then H_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{•}(V′)) is a quotient of the commutative algebra generated by the cohomology classes [Ψ_{k}^{L}], k = 1, …, N, by the ideal spanned by the elements of degree exceeding N.

Proof. We only explain how this can be derived from [Fu86].

(a) Write U(𝔤) for the enveloping algebra of a Lie algebra 𝔤. First we observe that the map

ζ: Ω_{N}^{p} → Hom_{L_{0}}(U(W_{N}), Λ^{p}(V′)) ≅ Hom(S(ℝ^{N}), Λ^{p}(V′)) ≅ F_{N} ⊗ Λ^{p}(V′),
ζ(α)(D) := (D.α)(0)

is an isomorphism of W_{N} -modules. Hence Ω_{N}^{p} is coinduced, as a W_{N} -module, from the L_{0} - module Λ^{p}(V′). Note that, since L_{0} is of finite codimensional in W_{N}, no problems arise from continuity requirements. Therefore (a) follows from a general result on coinduced modules ([Fu86, Th. 1.5.4]).

(b) The proof of part (b) is based on the Hochschild–Serre Spectral Sequence associated with the filtration on C^{•}(L_{0}, Λ^{•}(V′)) relative to the subalgebra 𝔤𝔩_{N}(ℝ) ⊂ L_{0} . By Theorem 1.5.1(ii) in [Fu86], H^{p}(L_{0}, Λ^{•}(V′)) is the E_{2}^{p,0} -term in the spectral sequence, while in the proof of Theorem 2.2.7′ in [Fu86], it is shown that E_{2}^{p,0} = E_{1}^{p,0} , and the term E_{1}^{p,0} is calculated explicitly.

(c), (d) follow from (a), combined with [Fu86, Thms. 2.2.7 and 2.2.7’].

From the formulas in [Fu86] it is not completely obvious that our formula for Ψ_{k}^{L} describes the same cocycle (up to a scalar factor). Fuks describes Ψ_{k}^{L} as an element of

Λ^{k}(V′) ⊗ Λ^{k}(S^{2}(V) ⊗ V′).

In our context, the cocycle Ψ_{k}^{L} ∈ Z_{c}^{k}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{k}(V′)) is given by the formula

Ψ_{k}^{L}(X_{1}, …, X_{k}) = (−1)^{k} ∑_{σ∈S_{k}} sgn(σ) Tr (dJ(X_{σ(1)})(0) ∧ … ∧ dJ(X_{σ(k)})(0)) ∈ Λ^{k}(V′).

For each X ∈ L_{0} , the constant term of the 1-form dJ(X) ∈ Ω_{N}^{1} ⊗ 𝔤𝔩_{N}(ℝ) corresponds to the quadratic term in X, which can be identified with an element of S^{2}(V′) ⊗ V ≅ Sym^{2}(V) ⊗ V .

For the basis element X_{i_{1},i_{2},ℓ} := x_{i_{1}}x_{i_{2}}∂_{ℓ}, we have, in terms of the matrix basis E_{ij} of 𝔤𝔩_{N}(ℝ):

J(X) = x_{i_{1}}E_{ℓi_{2}} + x_{i_{2}}E_{ℓi_{1}} and d(J(X)) = dx_{i_{1}}E_{ℓi_{2}} + dx_{i_{2}}E_{ℓi_{1}} = ∑_{τ∈S_{2}} dx_{i_{τ(1)}}E_{ℓi_{τ(2)}}.

This leads to

(−1)^{k}Ψ_{k}^{L}(X_{i_{11},i_{12},ℓ_{1}}, …, X_{i_{k1},i_{k2},ℓ_{k}})

= ∑_{σ∈S_{k}} ∑_{{τ_{i}∈S_{2} \atop i=1,...,k}} Tr (dx_{i_{σ(1),τ_{1}(1)}}E_{ℓ_{σ(1)}i_{σ(1)τ_{1}(2)}} ∧ ⋯ ∧ dx_{i_{σ(k),τ_{k}(1)}}E_{ℓ_{σ(k)}i_{σ(k)τ_{k}(2)}})

= ∑_{σ∈S_{k}} ∑_{{τ_{i}∈S_{2} \atop i=1,...,k}} δ_{i_{σ(1)τ_{1}(2)},ℓ_{σ(2)}}δ_{i_{σ(2)τ_{2}(2)},ℓ_{σ(3)}} ⋯ δ_{i_{σ(k)τ_{k}(2)},ℓ_{σ(1)}} · dx_{i_{σ(1),τ_{1}(1)}} ∧ ⋯ ∧ dx_{i_{σ(k),τ_{k}(1)}}.

From this formula it is not hard to verify that our Ψ_{k}^{L} are multiples of those in [Fu86].

Appendix C. Higher frame bundles and differential forms

In this appendix we shall prove our key Theorem III.1. The major part of the work consists in explaining why Theorem III.1 can be derived from Tsujishita’s [Tsu81]. For that we need the passage from the cohomology of formal vector fields to the cohomology of V_{M} and finally to link it to Theorem III.1.

Let M be an N -dimensional smooth manifold. For k ∈ ℕ_{0} ∪ {∞} we write J^{k}(M) for its k -frame bundle whose elements are k -jets [α] of local diffeomorphism α: (ℝ^{N}, 0) → M. Then J^{0}(M) = M and J^{1}(M) is the frame bundle of M whose elements are the isomorphisms ℝ^{N} → T_{m}(M), m ∈ M. Evaluating [α] in 0 leads to a natural map J^{k}(M) → M which exhibits J^{k}(M) as a fiber bundle over M. We write J_{m}^{k}(M) for the fiber over m ∈ M. For ℓ ≤ k we have projection maps

π_{ℓ,k}: J^{k}(M) → J^{ℓ}(M), [α] ↦ j_{0}^{ℓ}(α).

In particular, we have π_{0,k}([α]) = α(m) and π_{1,k}([α]) = T_{0}(α).

Let G^{k} denote the group of k -jets of local diffeomorphisms of ℝ^{N} fixing 0. As a set, G^{k} is the set of all polynomial maps φ: ℝ^{N} → ℝ^{N} of degree ≤ k without constant term for which dφ(0) is invertible. The multiplication is given by

φ_{1}φ_{2} := j_{0}^{k}(φ_{1} ∘ φ_{2}).

For ℓ ≤ k we have natural surjective homomorphisms

q_{ℓ,k}: G^{k} → G^{ℓ}, α ↦ j_{0}^{ℓ}(α),

cutting off all terms of order > ℓ. We put

G_{k} := ker(q_{k,∞}) ⊴ G^{∞}

and note that

(C.1)  G^{∞} ≅ G_{1} ⋊ GL_{N}(ℝ)

in a natural way.

For k < ∞, the group G^{k} is a finite-dimensional Lie group, G^{1} ≅ GL_{N}(ℝ), and

G^{k} ≅ G_{1}^{k} ⋊ GL_{N}(ℝ),

where G_{1}^{k} := ker(q_{1,k}) is a simply connected nilpotent Lie group. Identifying G^{∞} with the projective limit of all groups G^{k} ≅ G^{∞}/G_{k}, we obtain a topology for which it actually is a Lie group with Lie algebra L(G^{∞}) ≅ L_{0} and L_{k} := L(G_{k}) is a finite-codimensional ideal of L_{0} (cf. Appendix B). The normal subgroup G_{1} is pro-nilpotent, its exponential function exp_{G_{1}}: L_{1} → G_{1} is a diffeomorphism, and (C.1) is a semidirect product of Lie groups.

The group G^{k} acts on J^{k}(M) from the right by [α].φ := [α ∘ φ]. This action is transitive on the fibers and defines on J^{k}(M) the structure of a smooth G^{k} -principal bundle.

The group Diff(M) acts on each frame bundle J^{k}(M) by φ.[α] := [φ ∘ α], and the corresponding homomorphism of Lie algebras is given by

(C.2)  γ^{k}: V_{M} → V_{J^{k}(M)}, γ^{k}(X)_{[α]} = \frac{d}{dt} t=0 [Fl_{t}^{X} ∘ α].

Since the action of Diff(M) on J^{k}(M) commutes with the action of the structure group G^{k},

(C.3)  γ^{k}(V_{M}) ⊆ V_{J^{k}(M)}^{G^{k}}.

Lemma C.1. ([Tsu81, Lemma 4.2.2]) Let k ∈ ℕ ∪ {∞}, ∇ be an affine connection on M, and D ⊆ T M the open domain of the corresponding exponential function Exp^{∇}: D → M, which is an open neighborhood of the zero section. Then

s: J^{1}(M) → J^{k}(M), s(α) := j_{0}^{k}(Exp_{m}^{∇} ∘α)

is a smooth section.

Note that for each m ∈ M the intersection D_{m} := D ∩ T_{m}(M) is an open zero neighborhood and that Exp_{m}^{∇}: D_{m} → M is a smooth map with T_{0}(Exp_{m}^{∇}) = id_{T_{m}(M)} , hence a local diffeomorphism. Now the map J^{1}(M) × G_{1}^{k} → J^{k}(M), (α, g) ↦ s(α)g is a diffeomorphism and the map

F: J^{k}(M) → G_{1}^{k}, s(α)g ↦ g, α ∈ J^{1}(M),

is smooth and G_{1}^{k} -equivariant. For g ∈ GL_{N}(ℝ) we have s(α ∘ g) = s(α).g, which implies that

(C.4)  F (α.g) = g^{−1}F (α)g for all α ∈ J^{k}(M).

Hence F is equivariant with respect to the action of G^{k} ≅ G_{1}^{k} ⋊ GL_{N}(ℝ) on G_{1}^{k} from the right by g.(g_{1}, g_{0}) = g_{0}^{−1}gg_{1}g_{0} .

Since Tsujishita uses a realization of Lie algebra cohomology in terms of right invariant, resp., equivariant differential forms on the corresponding group, we briefly discuss the relevant identifications in the following remark.

Remark C.2. (a) Let G be a Lie group. A smooth G-module is a topological vector space V on which G acts smoothly by linear maps. We write

ρ_{V}: G × V → V, (g, v) ↦ ρ_{V} (g)(v) =: g.v

for the action map.

Further, let M be a smooth manifold on which G acts from the right by M × G → M, (m, g) ↦ m.g =: ρ_{g}^{M}(m). We call a p-form α ∈ Ω^{p}(M, V) equivariant if we have for each g ∈ G the relation

(ρ_{g}^{M})^{∗}α = ρ_{V} (g)^{−1} ∘ α.

We write Ω^{p}(M, V)^{G} for the space of G-equivariant p-forms on M. This is the space of G-fixed elements with respect to the action of G on Ω^{p}(M, V), given by g.α := ρ_{V} (g) ∘ (ρ_{g}^{M})^{∗}α.

(b) For the right action of G on M = G by left multiplication x.g := g^{−1}x we obtain the space Ω_{ℓ}^{p}(G, V) of left equivariant forms, i.e., forms satisfying

λ_{g}^{∗}ω = ρ_{V} (g) ∘ ω, g ∈ G.

In [ChE48] it is shown that the evaluation map

ev_{1}: (Ω_{ℓ}^{•}(G, V), d) → (C_{c}^{•}(𝔤, V), d_{𝔤}), ω ↦ ω_{1}

yields an isomorphism of cochain complexes (cf. [Ne04] for the unproblematic extension to infinitedimensional Lie groups).

(c) There is also a realization of the complex (C_{c}^{•}(𝔤, V), d_{𝔤}) by right equivariant differential forms on G: If η_{G}: G → G, g ↦ g^{−1} is the inversion map, then for each left equivariant p-form α ∈ Ω_{ℓ}^{p}(G, V) the p-form α̃ := η_{G}^{∗}α is right equivariant, i.e.,

ρ_{g}^{∗}α̃ = ρ_{V} (g)^{−1} ∘ α̃ for each g ∈ G.

We thus obtain an isomorphism of cochain complexes

η_{G}^{∗}: (Ω_{ℓ}^{•}(G, V), d) → (Ω_{r}^{•}(G, V), d).

Since T_{1}(η_{G}) = − id_{G}, we also obtain an isomorphism

ẽv_{1}: (Ω_{r}^{•}(G, V), d) → (C_{c}^{•}(𝔤, V), d_{𝔤}), ω ↦ (− id_{𝔤})^{∗}ω_{1}.

For each g ∈ G and ω ∈ Ω_{r}^{p}(G, V) the form λ_{g}^{∗}ω is also right equivariant and satisfies

(λ_{g^{−1}}^{∗}ω)_{1}(x_{1}, …, x_{p}) = ω_{g^{−1}}(g^{−1}.x_{1}, …, g^{−1}.x_{p}) = ρ_{V} (g).ω(Ad(g)^{−1}.x_{1}, …, Ad(g)^{−1}.x_{p}),

showing that ẽv_{1} intertwines the action of G by left translation on Ω_{r}^{p}(G, V) with the natural action of G on the cochain space C_{c}^{p}(𝔤, V).

(d) There is an alternative realization of the complex (C^{•}(𝔤, V), d_{𝔤}) as the space of left, resp., right invariant V -valued differential forms on G.

First we write the Chevalley–Eilenberg differential as

d_{𝔤} = d_{𝔤}^{0} + ρ_{V} ∧,

where d_{𝔤}^{0} is the differential corresponding to the trivial module structure on V and ρ_{V}: 𝔤 → 𝔤𝔩(V) is the homomorphism of Lie algebras, defining the 𝔤-module structure on V .

If κ_{G} ∈ Ω^{1}(G, 𝔤) denotes the left Maurer–Cartan form of G, we obtain κ_{V} := ρ_{V} ∘ κ_{G} ∈ Ω^{1}(G, 𝔤𝔩(V)) and a corresponding covariant differential on Ω^{•}(M, V):

d_{κ}ω = dω + κ_{V} ∧ ω.

Since κ is left invariant, this differential preserves the subspace of left invariant forms, and it is easy to see that evaluation in 1 intertwines it with d_{𝔤} on C_{c}^{•}(𝔤, V).

In a similar fashion, one obtains a realization by right invariant differential forms with the appropriate differential.

Let A be a finite-dimensional smooth G^{∞} -module, where we assume that the action of the non-connected Lie group GL_{N}(ℝ) on A is the one obtained by restricting the action of GL_{N}(ℂ) on the complexification A_{ℂ}, hence completely determined by the action of 𝔤𝔩_{N}(ℝ) on A. We then have natural identifications

(C.5)  C_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), A) ≅ C_{c}^{•}(L_{1}, A)^{𝔤𝔩_{N} (ℝ)} ≅ Ω_{r}^{•}(G_{1}, A)^{GL_{N} (ℝ)}

([Tsu81, Lemma 3.3.4], Remark C1b(c)). Although GL_{N}(ℝ) is not connected, this follows from a complexification argument since GL_{N}(ℂ) is connected.

Further, let ω ∈ C_{c}^{p}(L_{1}, A)^{GL_{N} (ℝ)} ≅ C_{c}^{p}(L_{1}, A)^{𝔤𝔩_{N} (ℝ)} and ω̃ ∈ Ω_{r}^{p}(G_{1}, A) the corresponding right equivariant A-valued p-form with ω̃_{1} = (−1)^{p}ω (cf. Remark C.2(c)). For g_{1} ∈ G_{1} we then have

ρ_{g_{1}}^{∗}F^{∗}ω̃ = (F ∘ ρ_{g_{1}})^{∗}ω̃ = (ρ_{g_{1}} ∘ F)^{∗}ω̃ = ρ_{A}(g_{1}) ∘ F^{∗}ω̃

and for g_{0} ∈ GL_{N}(ℝ) we obtain with (C.4)

ρ_{g_{0}}^{∗}F^{∗}ω̃ = (F ∘ ρ_{g_{0}})^{∗}ω̃ = (c_{g_{0}}^{−1} ∘ F)^{∗}ω̃ = F^{∗}((c_{g_{0}}^{−1})^{∗}ω̃) = ρ_{A}(g_{0}) ∘ F^{∗}ω̃,

hence the GL_{N}(ℝ)-equivariance of ω̃. We thus obtain for k = ∞ a morphism of chain complexes

(C.6)  F̃^{∗}: C_{c}^{•}(L_{1}, A)^{GL_{N} (ℝ)} → Ω^{•}(J^{∞}(M), A)^{G^{∞}}, ω ↦ F^{∗}ω̃.

The map γ_{A} defined below is the one used by Tsujishita in [Tsu81, p. 62].

Proposition C.3. Let A := J^{∞}(M) ×_{G^{∞}} A → M denote the bundle associated to J^{∞}(M) by the representation of G^{∞} on A and Γ(A) be its space of smooth sections on which V_{M} acts via γ^{∞}. We identify Γ(A) with C^{∞}(J^{∞}(M), A)^{G^{∞}}. Then the map

(C.7)  γ_{A}: = (γ^{∞})^{∗} ∘ F̃^{∗}: C_{c}^{•}(L_{1}, A)^{GL_{N} (ℝ)} ≅ C_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), A) → C_{c}^{•}(V_{M}, Γ(A))

is a morphism of chain complexes. Different choices of affine connections on M yield the same maps

H_{c}^{•}(L_{1}, A)^{GL_{N} (ℝ)} ≅ H_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), A) → H_{c}^{•}(V_{M}, Γ(A))

in cohomology. Proof. For the continuity of the so obtained cochains γ_{A}(ω) of V_{M}, we note that the continuous representation of G^{∞} on the finite-dimensional space A factors through some finitedimensional quotient group G^{k}. Hence the bundle A is also associated to J^{k}(M), Γ(A) can be realized as A-valued G^{k} -equivariant functions on J^{k}(M), and the action of V_{M} on this space via γ^{k}: V_{M} → V_{J^{k}(M)} is continuous because the finite dimensionality of J^{k}(M) implies that γ^{k} is a continuous morphism of topological Lie algebras.

To see that the choice of connection has no effect on the corresponding map in cohomology, let ∇′ be another affine connection on M and observe that ∇_{t} := t∇′ + (1 − t)∇ defines a smooth family of affine connections on M with ∇_{0} = ∇ and ∇_{1} = ∇′. It is easy to see that the corresponding functions s_{1}: J^{1}(M) → J^{∞}(M) depend smoothly on t, and so do the corresponding G_{1} -equivariant functions F_{t}: J^{∞}(M) → G_{1} . Hence, for each ω ∈ C_{c}^{p}(L_{1}, A)^{GL_{N} (ℝ)} the differential forms F_{1}^{∗}ω̃−F_{0}^{∗}ω̃ are equivariantly exact, i.e., the differential of a G^{∞} -equivariant A-valued (p − 1)-form (cf. [Ko74, p.143]). This implies the assertion.

Remark C.4. We also note that the 1-form δ(F) ∈ Ω^{1}(J^{∞}(M), L_{1}) satisfies the Maurer– Cartan equation, hence defines a crossed homomorphism

δ(F): V(J^{∞}(M)) → C^{∞}(J^{∞}(M), L_{1})

(Proposition I.8). Composing with the homomorphism γ^{∞}: V_{M} → V_{J^{∞}(M)} , we thus obtain a crossed homomorphism

δ(F) ∘ γ^{∞}: V_{M} → C^{∞}(J^{∞}(M), L_{1})^{G^{∞}}.

That δ(F) ∘ γ^{∞} maps into G^{∞} -equivariant functions is due to (C.3) and the G^{∞} -equivariance of F, resulting from (C.4). We further note that any element ω ∈ C_{c}^{p}(L_{1}, A)^{GL_{N} (ℝ)} defines by composition a V_{M} -equivariant p-cochain of the Lie algebra C^{∞}(J^{∞}(M), L_{1})^{G^{∞}} with values in C^{∞}(J^{∞}(M), A)^{G^{∞}} ≅ Γ(A). Now Theorem I.7 shows that pulling back with the crossed homomorphism δ(F) ∘ γ^{∞} yields Γ(A)-valued Lie algebra cochains and that this is compatible with the mutual differentials.

Example C.5. For A = Λ^{k}(V′), V = ℝ^{N}, and the canonical representation of G^{1} ≅ GL_{N}(ℝ) ≅ G^{∞}/G_{1} on this space, we obtain for the space of smooth sections Γ(A) ≅ Ω_{M}^{k}, so that we get a morphism of cochain complexes

γ_{A}: C_{c}^{•}(L_{1}, Λ^{k}(V′))^{GL_{N} (ℝ)} → C_{c}^{•}(V_{M}, Ω_{M}^{k}).

In this case the bundle A = Λ^{k}(T^{∗}(M)) is associated to the frame bundle J^{1}(M) which creates a simpler picture than working with the infinite-dimensional manifold J^{∞}(M).

Example C.6. If A = ℝ is the trivial module, we obtain in particular F_{M} -valued cocycles on V_{M} from any map

(γ^{k})^{∗}: (Ω_{J^{k}(M)}^{p})^{G^{k}} → C_{c}^{p}(V(M), F_{M}).

Here is a concrete example: For k = 1 we consider the 1-form ω ∈ Ω^{1}(J^{1}(M), ℝ)^{GL_{N} (ℝ)} defined as follows. From the homomorphism

χ: GL_{N}(ℝ) → ℝ_{+}^{×}, g ↦ | det(g)|

we obtain an associated bundle J^{1}(M) ×_{χ} ℝ_{+}^{×}, and since ℝ_{+}^{×} is contractible, this bundle has a global section (we could also take log ∘χ and obtain an affine bundle), which means that there is a smooth function

F: J^{1}(M) → ℝ_{+}^{×} with F (α.g) = F (α)| det(g)| for g ∈ GL_{N}(ℝ), α ∈ J^{1}(M). Then

δ(F) ∈ Ω^{1}(J^{1}(M), ℝ)^{GL_{N} (ℝ)}

is a GL_{N}(ℝ)-invariant 1-form.

If μ is a volume form on M, then we can construct F directly from μ by

F (v_{1}, …, v_{N}) := |μ_{m}(v_{1}, …, v_{N})|

for any basis (v_{1}, …, v_{N}) on T_{m}(M).

For a diffeomorphism φ ∈ Diff(M), we then have

\begin{aligned}(φ.F)(v_{1}, …, v_{N}) & = F (T (φ^{−1}).(v_{1}, …, v_{N})) = |μ(T (φ^{−1}).(v_{1}, …, v_{N}))| \\ & = |(φ^{−1})^{∗}μ(v_{1}, …, v_{N})|. \end{aligned}

Dividing by F (v_{1}, …, v_{N}), we obtain a smooth function

(φ.F)F^{−1} ∈ C^{∞}(M, ℝ_{+}^{×}),

and passing to the Lie derivative, we obtain for each vector field X on M a smooth function (L_{X}.F)F^{−1} ∈ C^{∞}(M, ℝ).

Now we turn to Tsujishita’s construction of the homomorphism

H_{c}^{•}(L^{1}, Λ^{k}(V′))^{GL(V)} → H_{c}^{•}(V_{M}, Ω_{M}^{k})

in term of the cocycles L_{X}∇ associated to affine connections (Lemma II.1).

Let ∇ be an affine connection on M. As in Lemma C.1, we obtain from ∇ a smooth section

s: J^{1}(M) → J^{2}(M), s(α) := j_{0}^{2}(Exp_{m}^{∇} ∘α) = j_{0}^{2}(Exp_{m}^{∇}) ∘ α.

From s(αg) = s(α)g for β ∈ GL(V) it follows that s is GL(V)-equivariant.

Let F: J^{2}(M) → G_{1}/G_{2} ≅ Sym^{2}(V, V) denote the unique smooth G^{1} -equivariant smooth function vanishing on s(J^{1}(M)). Identifying Sym^{2}(V, V) ≅ L_{1}/L_{2} with the corresponding subspace of V′ ⊗ 𝔤𝔩(V) ≅ (V′ ⊗ V′) ⊗ V corresponds to composition with the map ev_{0} dJ: L_{1} → V′ ⊗ 𝔤𝔩(V).

The map

δ(F) ∘ γ^{2}: V_{M} → C^{∞}(J^{2}(M), L_{1}/L_{2})^{G^{2}}

is a crossed homomorphism (Proposition I.8), and since L_{1}/L_{2} is abelian, it is a 1-cocycle. Composing with ev_{0} dJ: L_{1}/L_{2} → V′ ⊗ 𝔤𝔩(V), we thus get a 1-cocycle

ev_{0} dJδ(F) ∘ γ^{2}: V_{M} → C^{∞}(J^{2}(M), V′ ⊗ 𝔤𝔩(V))^{G^{2}}
≅ C^{∞}(J^{1}(M), V′ ⊗ 𝔤𝔩(V))^{GL(V)} ≅ Ω^{1}(M, End(T M)).

The following theorem is the link between the approaches in [Tsu81] and [Ko74].

Theorem C.7. If ∇ is torsion free, then

ev_{0} dJδ(F)(γ^{2}(X)) = L_{X}∇ ∈ Ω^{1}(M, End(T M)).

Proof. Since this is a local assertion, it suffices to assume that M is an open subset of V ≅ ℝ^{N}. Then we write

∇_{X}Y = dY (X) + Γ(X, Y)

for a symmetric (1, 2)-tensor Γ (recall that Γ is symmetric if and only if ∇ is torsion free). We then identify

J^{1}(M) = M × GL(V) and J^{2}(M) = M × G^{2} ≅ M × (Sym^{2}(V, V) ⋊ GL(V)).

Here we write elements of G^{2} as α = α_{1} + α_{2} with α_{1} ∈ GL(V) linear and α_{2} quadratic. Then the group structure of G^{2} is given by

α • β = α_{1}β_{1} + (α_{1}β_{2} + α_{2}β_{1}) = α_{1}β + α_{2}β_{1} = αβ_{1} + α_{1}β_{2}.

From this we see that

(C.10)  α^{−1} = α_{1}^{−1} − α_{1}^{−1}α_{2}α_{1}^{−1}.

To describes the section s: J^{1}(M) → J^{2}(M) explicitly, let γ_{v}(t) = Exp^{∇}(tv) be the geodesic with γ_{v}(0) = m and γ′_{v}(0) = v (which is defined for t sufficiently close to 0). Then the relation

γ′′_{v}(t) = −Γ(γ′_{v}(t), γ′_{v}(t))

leads to γ′′_{v}(0) = −Γ(v, v), so that

j_{0}^{2}(Exp_{m}^{∇})(v) = v + \frac{1}{2}γ′′(0) = v − \frac{1}{2}Γ(v, v).

We thus obtain

s(m, α_{1}) = (m, (1 − \frac{1}{2}Γ)α_{1}).

Accordingly, we find

F (m, α) = α_{1}^{−1}(α + \frac{1}{2}Γ ∘ α_{1}) = 1 + α_{1}^{−1}α_{2} + \frac{1}{2}α_{1}^{−1}Γ ∘ α_{1}

because

F (m, α(1 + β_{2})) = F (m, α + α_{1}β_{2}) = F (m, α) + β_{2} = F (m, α).β_{2}

and

F (m, αβ_{1}) = β_{1}^{−1}F (m, α)β_{1}.

From the relation

(β • α)_{1}^{−1}(β • α)_{2} = α_{1}^{−1}β_{1}^{−1}(β_{1}α_{2} + β_{2}α_{1}) = α_{1}^{−1}α_{2} + α_{1}^{−1}β_{1}^{−1}β_{2}α_{1},

we obtain the following formula for the differential of F in (v, β • α) ∈ T_{(m,α)}(M × G^{2}) with β ∈ T_{1}(G^{2}) ≅ L_{0}/L_{2} :

\begin{aligned}dF (m, α)(v, β • α) & = \frac{1}{2}α_{1}^{−1}dΓ(v) ∘ α_{1} + α_{1}^{−1}β_{2}α_{1} − \frac{1}{2}α_{1}^{−1}(β_{1}.Γ) ∘ α_{1} \\ & = α_{1}^{−1}(\frac{1}{2}dΓ(v) + β_{2} − \frac{1}{2}(β_{1}.Γ)) ∘ α_{1}. \end{aligned}

The natural lift of X ∈ V_{M} to J^{2}(M) is given by

\begin{aligned}γ^{2}(X)(m, α) & = j_{0}^{2}(X ∘ α) = (X(m), J(X)_{m} ∘ α + \frac{1}{2}(d^{2}X)_{m} ∘ α_{1}) \\ & = (X(m), (J(X)_{m} + \frac{1}{2}(d^{2}X)_{m}) • α) \end{aligned}

(J(X)_{m}v = (dX)_{m}(v) denoting the Jacobian of X), so that

γ^{2}(X)(s(m, α_{1})) = (X(m), J(X)_{m} ∘ α_{1} − \frac{1}{2}J(X)_{m}Γ ∘ α_{1} + \frac{1}{2}(d^{2}X)_{m} ∘ α_{1}).

This leads to

\begin{aligned}dF (γ^{2}(X))(s(m, α_{1})) & = δ(F)(γ^{2}(X))(s(m, α_{1})) \\ & = \frac{1}{2}α_{1}^{−1}(dΓ(X(m)) + (d^{2}X)_{m} − J(X)_{m}.Γ) ∘ α_{1}. \end{aligned}

This is a GL(V)-equivariant function J^{1}(M) → Sym^{2}(V, V), so that the corresponding (1, 2)- tensor field is given in the canonical coordinates by the smooth function M → Sym^{2}(V, V) given by

m ↦ dF (γ^{2}(X))(s(m, 1)) = \frac{1}{2}(dΓ(X(m)) + (d^{2}X)_{m} − J(X)_{m}Γ).

Suppose that β: V → V is a quadratic map and β̃ the corresponding symmetric bilinear map determined by β(v) = \frac{1}{2} β̃(v, v). Considering β as an element of L_{0}/L_{2} , we have

J(β)(v) = (dβ)_{v} = β̃(v, ·) ∈ 𝔤𝔩(V)

and thus

(dJ(β))_{0}(v)v = ev_{0}(dJ(β))(v)(v) = β̃(v, v) = 2β(v).

Applying ev_{0} ∘dJ to the smooth function above thus leads to

ev_{0} dJδ(F)(γ^{2}(X))(s(m, 1)) = dΓ(X(m)) + (d^{2}X)_{m} − J(X)_{m}Γ ∈ V′ ⊗ 𝔤𝔩(V).

In view of Remark II.2, it remains to show that

L_{X}Γ = dΓ(X) − J(X).Γ.

It suffices to verify this relation by evaluating on constant vector fields Y and Z ∈ V :

(L_{X}Γ)(Y, Z) = [X, Γ(Y, Z)] − Γ([X, Y], Z) − Γ(Y, [X, Z])
= (dΓ(X))(Y, Z) − dX(Γ(Y, Z)) + Γ(dX(Y), Z) + Γ(Y, dX(Z))
= (dΓ(X))(Y, Z) − (J(X).Γ)(Y, Z).

Now we can identify the image of the Λ^{k}(V′)-valued cocycle Ψ_{k}^{L} (Theorem B.2) under Tsujishita’s map γ_{Λ^{k}(V′)} , described in Proposition C.3, which maps it into an element of Z_{c}^{k}(V_{M}, Ω_{M}^{k}).

Theorem C.8. For each torsion free connection ∇ on the manifold M, we have

γ_{Λ^{k}(V′)}(Ψ_{k}^{L}) = Ψ_{k}.

Proof. We first observe that

Ψ_{k}^{L} = (ev_{0} dJ)^{∗}f_{k}, where f_{k}: (V′ ⊗ End(V))^{k} → Λ^{k}(V′)

is given by

\begin{aligned}((α_{1} ⊗ A_{1}), …, (α_{k} ⊗ A_{k})) & ↦ ∑_{σ∈S_{k}} sgn(σ) Tr(A_{σ(1)} ⋯ A_{σ(k)})α_{σ(1)} ∧ ⋯ ∧ α_{σ(k)} \\ & = ( ∑_{σ∈S_{k}} Tr(A_{σ(1)} ⋯ A_{σ(k)})) · α_{1} ∧ ⋯ ∧ α_{k}. \end{aligned}

In this sense, we have f_{k} = m_{k}⊗β_{k}, where m_{k}: (V′)^{⊗k} → Λ^{k}(V′) is the alternating multiplication map and β_{k}: End(V)^{k} → ℝ is the symmetric k -linear map defined by

β_{k}(A_{1}, …, A_{k}) = ∑_{σ∈S_{k}} Tr(A_{σ(1)} ⋯ A_{σ(k)}).

According to Proposition C.3, we have for X_{1}, …, X_{k} ∈ V_{M}:
γ_{Λ^{k}(V′)}(Ψ_{k}^{L})(X_{1}, …, X_{k}) = (F^{∗}Ψ̃_{k}^{L})(γ^{2}(X_{1}), …, γ^{2}(X_{k})),

where we identify

Ω_{M}^{k} ≅ C^{∞}(J^{∞}(M), Λ^{k}(V′))^{G^{∞}} ≅ C^{∞}(J^{2}(M), Λ^{k}(V′))^{G^{2}} ≅ C^{∞}(J^{1}(M), Λ^{k}(V′))^{GL(V)}.

This identification is based on the canonical map

J^{∞}(M) × Λ^{k}(V′) → Λ^{k}(T^{∗}(M)), ([α], ω) ↦ ω ∘ α_{1}^{−1}

which is constant on the diagonal G^{∞} -action and factors through the corresponding map on J^{1}(M), resp., J^{2}(M).

Since the cocycles Ψ_{k}^{L} factor through the quotient algebra L_{0}/L_{2} , the map γ_{Λ^{k}(V′)} can be constructed with J^{2}(M) instead of J^{∞}(M) and to pass from G^{2} -equivariant smooth functions on J^{2}(M) to GL(V)-equivariant smooth functions on J^{1}(M), we may simply restrict to s(J^{1}(M)), the 1-level set of the function F. We thus obtain with Theorem C.7 for any torsion free connection ∇:

γ_{Λ^{k}(V′)}(Ψ_{k}^{L})(X_{1}, …, X_{k}) = (F^{∗}Ψ̃_{k}^{L})(γ^{2}(X_{1}), …, γ^{2}(X_{k}))
= (−1)^{k}Ψ_{k}^{L}(δ(F)(γ^{2}(X_{1})), …, δ(F)(γ^{2}(X_{k})))
= (−1)^{k}f_{k}( ev_{0} dJδ(F)(γ^{2}(X_{1})), …, ev_{0} dJδ(F)(γ^{2}(X_{k})))
= (−1)^{k}f_{k}(L_{X_{1}}∇, …, L_{X_{k}}∇)
= Ψ_{k}(X_{1}, …, X_{k}),

where we identify L_{X}∇ with an element of Ω^{1}(M, End(T M)) ≅ C^{∞}(J^{1}(M), V′ ⊗ End(V))^{GL(V)} and use Ω_{M}^{k} ≅ C^{∞}(J^{1}(M), Λ^{k}(V′))^{GL(V)} .

Proof of Theorem III.1: We choose an affine torsion free connection ∇. From the formulation of Tsujishita’s Theorem in [Tsu81], Thm. 5.1.6, we get (with V = ℝ^{N}) an isomorphism of bigraded algebras:

H_{c}^{•}(V_{M}, Ω_{M}^{•}) ≅ H_{c}^{•}(V_{M}, F_{M}) ⊗ H_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{•}(V′)).

We further know from Theorem B.2, that H_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{•}(V′)) is generated by the classes of the cocycles Ψ_{k}^{L}. Hence it suffices to observe with Theorem C.8 that Tsujishita’s map

γ_{Λ^{k}(V′)}: C_{c}^{•}(L_{1}, Λ^{k}(V′))^{GL_{N} (ℝ)} ≅ C_{c}^{•}(L_{0}, 𝔤𝔩_{N}(ℝ), Λ^{k}(V′)) → C_{c}^{•}(V_{M}, Ω_{M}^{k})

(Proposition C.3) maps Ψ_{k}^{L} to Ψ_{k}.

Appendix D. Cohomology of vector fields with values in the trivial module

Here we shall review the results of Haefliger [Hae76].

We begin by recalling the definition of the Weil algebra Ŵ_{N}. Let E(u_{1}, …, u_{N}) be the exterior algebra in the generators u_{1}, …, u_{N}, and let ℝ[c_{1}, …, c_{N}], be the polynomial algebra in the generators c_{1}, …, c_{N}. Introduce gradings on these algebras by assigning degrees to the generators:

deg(u_{k}) = 2k − 1 and deg(c_{k}) = 2k.

Consider the quotient ℝ̂[c_{1}, …, c_{N}] of the polynomial algebra ℝ[c_{1}, …, c_{N}] by the ideal spanned by the elements of degrees exceeding 2N. The Weil algebra is the differential graded algebra

Ŵ_{N} = E(u_{1}, …, u_{N}) ⊗ ℝ̂[c_{1}, …, c_{N}]

with the differential defined by

d(u_{k}) = c_{k} and d(c_{k}) = 0.

Let H^{•}(Ŵ_{N}) be the cohomology of this differential graded algebra.

Gelfand–Fuks used the Weil algebra to describe the cohomology of the Lie algebra W_{N} of formal vector fields in dimension N:

Theorem D.1. ([GF70b]) H_{c}^{•}(W_{N}, ℝ) ≅ H^{•}(Ŵ_{N})

The explicit description of the cohomology of the Weil algebra was given by Vey:

Theorem D.2. ([Go74]) H^{•}(Ŵ_{N}) is spanned by 1 and the cocycles

(D.1)  (u_{i_{1}} ∧ … ∧ u_{i_{r}}) ⊗ (c_{j_{1}} … c_{j_{s}})

satisfying the following conditions: 1 ≤ i_{1} < … < i_{r} ≤ N; 1 ≤ j_{1} ≤ … ≤ j_{s} ≤ N; i_{1} ≤ j_{1} ; r > 0; j_{1} + … + j_{s} ≤ N; i_{1} + j_{1} + … + j_{s} > N.

Combining the preceding two theorems, we derive:

Corollary D.3.

(a) H^{q}(W_{N}, ℝ) = 0 for 1 ≤ q ≤ 2N and for q ≥ (N + 1)^{2} .

(b) dim H^{2N+1}(W_{N}, ℝ) = p(N + 1) − 1, where p is the partition function.

(c) dim H^{2N+2}(W_{N}, ℝ) = 0.

Proof. (a) The degree of the generator (D.1) is at least 2i_{1} − 1 + 2(j_{1} + … + j_{s}) > 2N − 1, and since it is odd, it is at least 2N + 1. On the other hand, it is at most

1 + 3 + … + (2N − 1) + 2(j_{1} + … + j_{s}) ≤ N^{2} + 2N < (N + 1)^{2}.

(b) If the degree is exactly q = 2N + 1, then the argument in (a) implies

2i_{1} − 1 + 2(j_{1} + … + j_{s}) = 2N + 1,

which leads to r = 1 and i_{1} + j_{1} + … + j_{s} = N + 1. Only the partition

N + 1 = (N + 1) + 0 + … + 0

does not contribute, and this proves (b).

(c) If the degree q of the generator in (D.1) is even and > N, then j_{1} + … + j_{s} ≤ N implies r ≥ 2 and hence

q = (2i_{1} − 1) + (2i_{2} − 1) + 2(j_{1} + … + j_{s}) > 2N − 1 + 3 = 2N + 2.

Let W̃(N) be the graded vector space

W̃(N) = ⨁_{q=2N}^{(N+1)^{2}−2} W̃(N)_{q},

where dim W̃(N)_{q} = dim H^{q+1}(Ŵ_{N}). We view W̃(N) as a super vector space, where the parity of W̃(N)_{q} is equal to the parity of q.

Let L_{N} be the free Lie superalgebra generated by W̃(N) (see e.g., [BMPZ92] for the description of the basis of a free Lie superalgebra). The ℤ-grading on W̃(N) extends to a ℤ-grading of L_{N}.

This Lie superalgebra was introduced by Haefliger [Hae76] in order to give the description of the cohomology of the Lie algebra of smooth vector fields on a manifold with coefficients in the trivial module.

Let Ṽ be a graded vector space obtained from L_{N} by a shift in the grading: Ṽ_{q} = (L_{N})_{q−1} . Since the grading of L_{N} starts at component 2N, the grading of Ṽ starts at component 2N +1.

Let Ω_{M}^{•} be the differential graded algebra of smooth differential forms on M, and consider the tensor product space Ω_{M}^{•} ⊗ Ṽ with the grading

deg(ω ⊗ v) = deg v − deg ω, ω ∈ Ω_{M}^{•}, v ∈ Ṽ.

Since Ω_{M}^{p} vanishes for p > N, the grading on the space Ω_{M}^{•} ⊗ Ṽ starts at degree N + 1. Next we consider the graded algebra S^{•}(Ω_{M}^{•} ⊗ Ṽ) of supersymmetric multilinear forms on the graded superspace Ω_{M}^{•} ⊗ Ṽ. We have

S^{•}(Ω_{M}^{•} ⊗ Ṽ) = ⊕_{p=0}^{∞} S^{p}(Ω_{M}^{•} ⊗ Ṽ),

where S^{p}(Ω_{M}^{•} ⊗ Ṽ)_{q} = 0 for q < p · (N + 1). Hence, apart from the component of degree 0, the grading on S^{•}(Ω_{M}^{•} ⊗ Ṽ) starts in degree N + 1.

Theorem D.4. ([Hae76], Theorem A) Let M be an N -dimensional smooth manifold. On the graded algebra S^{•}(Ω_{M}^{•} ⊗ Ṽ) there exists a differential of degree 1, which depends on a choice of the representatives of the Pontrjagin classes in Ω_{M}^{•}, and a homomorphism of differential graded algebras

S^{•}(Ω_{M}^{•} ⊗ Ṽ) → C_{c}^{•}(V_{M}, ℝ),

which induces an isomorphism in cohomology.

Corollary D.5. H_{c}^{s}(V_{M}, ℝ) = 0 for 1 ≤ s ≤ N.

If all Pontrjagin classes of M vanish and there is a splitting algebra homomorphism H_{M}^{•} ↪ Ω_{M}^{•} (which is, for example, the case when M is a torus or a compact Lie group), Haefliger gives a more explicit realization for the cohomology of V_{M}, which we described now.

Let H_{M}^{•} be the cohomology algebra of the manifold M. We view H_{M}^{•} as a commutative superalgebra and form the graded Lie superalgebra

L(M) = H^{•}(M, ℝ) ⊗ L_{N}

with the bracket of homogeneous elements given by

[ω ⊗ x, ω′ ⊗ x′] = (−1)^{deg x·deg ω′}ωω′ ⊗ [x, x′],

and the grading by

L(M)_{r} = ∑_{p−q=r} H_{M}^{p} ⊗ (L_{N})_{q}.

It follows in particular that L(M)_{q} = 0 for q < N.

Consider the cohomology H^{•}(L(M)) of the Lie superalgebra L(M) (see [Fu86], Sect. 1.6.3 for the definition of the cohomology for the Lie superalgebras, cf. Sect. 1.2 in [Hae76]). Since L(M) is graded, its cohomology inherits a grading H^{p}(L(M), ℝ) = ⊕_{q}H^{p}(L(M), ℝ)_{q} (see [Fu86], Sect. 1.3.7). Since the grading of L(M) starts at degree N, we get that

(D.2)  H^{p}(L(M), ℝ)_{q} = 0 for q < p · N.

Theorem D.6. ([Hae76], Theorem 3.4) Let M be a smooth manifold with a finite-dimensional cohomology H_{M}^{•} for which all Pontryagin classes vanish. Suppose, moreover, that there exists a homomorphism of differential graded algebras H_{M}^{•} → Ω_{M}^{•}, which induces an isomorphism in cohomology (this is the case if M is a compact Lie group). Then

H_{c}^{s}(V_{M}, ℝ) = ⊕_{p+q=s} H^{p}(L(M), ℝ)_{q}.

Corollary D.7. Under the assumptions of the preceding theorem, we have

(a) H_{c}^{s}(V_{M}, ℝ) = 0 for 1 ≤ s ≤ N.

(b) For N + 1 ≤ s ≤ 2N + 1, dim H_{c}^{s}(V_{M}, ℝ) = ∑_{k=0}^{N} dim H_{M}^{k} · dim W̃(N)_{s+k−1} .

(c) dim H_{c}^{N+1}(V_{M}, ℝ) = p(N +1)−1 and only H_{M}^{N} ≅ ℝ contributes in the decomposition above.

(d) If b_{1}(M) := dim H_{M}^{1} = dim H_{M}^{N−1} , then dim H_{c}^{N+2}(V_{M}, ℝ) = b_{1}(M) · (p(N + 1) − 1) and only H_{M}^{N−1} ≅ ℝ^{b_{1}(M)} contributes in the decomposition.

Proof. (a) If H^{p}(L(M), ℝ)_{q} is non-zero, then q ≥ pN, so that (D.2) implies s = p + q ≥ (N + 1)p ≥ N + 1 if s ≠ 0.

(b) We note that (D.2) implies H_{c}^{s}(V_{M}, ℝ) ≅ H^{1}(L(M), ℝ)_{s−1} for N + 1 ≤ s ≤ 2N + 1. However,

H^{1}(L(M), ℝ) = (L(M)/[L(M), L(M)])^{∗} = (H_{M}^{•} ⊗ W̃(N))^{∗}

and the claim (b) follows.

If M is an N -dimensional torus 𝕋^{N}, its cohomology H^{•}(𝕋^{N}) is an exterior algebra in N generators of degree 1. Combining the two previous corollaries, we get with b_{1}(𝕋^{N}) = N:

Corollary D.8. For the torus T = 𝕋^{N}, we have:

(a) H_{c}^{s}(V_{T} , ℝ) = 0 for 1 ≤ s ≤ N.

(b) dim H_{c}^{N+1}(V_{T} , ℝ) = p(N + 1) − 1.

(c) dim H_{c}^{N+2}(V_{T} , ℝ) = N · (p(N + 1) − 1).

Remark D.9. For M = 𝕊^{n}, the algebra H^{•}(V_{𝕊^{n}}, ℝ) is calculated by Cohen and Taylor in [CT78]; Theorems 1.3 and 3.2.

References

[AMR83] Abraham, R., J. E. Marsden, and T. Ratiu, “Manifolds, Tensor Analysis, and Applications,” Addison-Wesley, 1983.

[ABFP05] Allison, B., S. Berman, J. Faulkner, and A. Pianzola, Realizations of gradedsimple algebras as loop algebras. math.RA/0511723..

[BMPZ92] Bahturin, Y. A., Mikhalev, A. A., Petrogradsky, V. M., Zaicev, M. V., “Infinitedimensional Lie superalgebras,” Walter de Gruyter & Co., Berlin, 1992.

[Beg87] Beggs, E. J., The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford (2) 38 (1987), 131–154.

[BB99] Berman, S., and Y. Billig, Irreducible representations for toroidal Lie algebras, J. Algebra 221 (1999), 188–231..

[BR73] Bernshtein, I. N., and B. I. Rozenfel’d, Homogeneous spaces of infinite-dimen-sional Lie algebras and characteristic classes of foliations, Russ. Math. Surveys 28:4 (1973), 107–142.

[BN06] Benkart, G., and E. Neher, The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra 205:1 (2006), 117–145.

[Bi06] Billig, Y., A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not. 2006, Art. ID 68395, 46 pp..

[ChE48] Chevalley, C. and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Transactions of the Amer. Math. Soc. 63 (1948), 85–124.

[CT78] Cohen, F. R., and L. R. Taylor, Computations of Gelfand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, in “Geometric applications of homotopy theory I”, Proc. Conf. Evanston, Ill., 106–173; Lectures Notes Math. 657, Springer, 1978.

[dWL83] de Wilde, M., and P. B. A. Lecomte, Cohomology of the Lie algebra of smooth vector fields of a manifold, associated to the Lie derivative of smooth forms, J. Math. pures et appl. 62 (1983), 197–214.

[EM94] Eswara Rao, S., and Moody R.V., Vertex representations for n-toroidal Lie algebras and a generalization of the Virasoro algebra, Comm.Math.Phys. 159 (1994), 239-264..

[FF01] Feigin, B. L. and D. B. Fuchs, Cohomologies of Lie Groups and Lie Algebras, in “Lie Groups and Lie Algebras II”, A. L. Onishchik and E. B. Vinberg (Eds.), Encyclop. Math. Sci. 21, Springer-Verlag, 2001.

[FL80] Flato, M. and A. Lichnerowicz, Cohomologie des représentations définies par la dérivation de Lie et à valeurs dans les formes, de l’algèbre de Lie des champs de vecteurs d’une variété différentiable. Premiers espaces de cohomologie. Applications, C. R. Acad. Sci. Paris Sér. A-B 291:4 (1980), A331–A335.

[Fu86] Fuks, D. B. “Cohomology of Infinite-Dimensional Lie Algebras,” Consultants Bureau, New York, London, 1986.

[GF70a] Gelfand, I. M., Fuks, D. B., Cohomology of the Lie algebra of vector fields with nontrivial coefficients, Func. Anal. and its Appl. 4 (1970), 181–192.

[GF70b] —, Cohomology of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR 34 (1970), 322–337.

[Go74] Godbillon, C., Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki (1972/1973), Exp.No.421, Lecture Notes in Math. 383 (1974), 69–87.

[Hae76] Haefliger, A., Sur la cohomologie de l’algèbre de Lie des champs de vecteurs, Ann. Sci. Ec. Norm. Sup. 4^{e} série 9 (1976), 503–532.

[HS53]] Hochschild, G., and J.-P. Serre, Cohomology of Lie algebras, Annals of Math. 57:3 (1953), 591–603.

[Ka84] Kassel, C., Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J.Pure Applied Algebra 34 (1984), 265–275.

[Ko74] Koszul, J.-L., Homologie des complexes de formes différentielles d’ordre supérieur, in “Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I”, Ann. Sci. École Norm. Sup. (4) 7 (1974), 139–153.

[L99] T.A. Larsson, Lowest-energy representations of non-centrally extended diffeomorphism algebras. Commun.Math.Phys. 201, (1999) 461-470..

[Ma02] Maier, P., Central extensions of topological current algebras, in “Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups,” A. Strasburger et al Eds., Banach Center Publications 55, Warszawa 2002; 61–76.

[Ne04] Neeb, K.-H., Abelian extensions of infinite-dimensional Lie groups, Travaux mathématiques 15 (2004), 69–194.

[Ne06a] —, Lie algebra extensions and higher order cocycles, J. Geom. Sym. Phys. 5 (2006), 48–74.

[Ne06b] —, Non-abelian extensions of topological Lie algebras, Communications in Algebra 34 (2006), 991–1041.

[Neh04] Neher, E., Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26:3 (2004), 90–96.

[PS86] Pressley, A., and G. Segal, “Loop Groups,” Oxford University Press, Oxford, 1986.

[Ro71] Rosenfeld, B. I., Cohomology of certain infinite-dimensional Lie algebras, Funct. Anal. Appl. 13 (1971), 340–342.

[Tsu77] Tsujishita, T., On the continuous cohomology of the Lie algebra of vector fields, Proc. Jap. Math. Soc. 53:A (1977), 134–138.

[Tsu81] —, “Continuous cohomology of the Lie algebra of vector fields,” Memoirs of the Amer. Math. Soc. 34:253, 1981.

Yuly Billig School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 Canada billig@math.carleton.ca Karl-Hermann Neeb Technische Universität Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt Deutschland neeb@mathematik.tu-darmstadt.de