Gel'fand and Fuks' “Globalizing” of cohomology of formal vector fields

I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do realize that it is a very specific question, so my hopes of getting a lot of attention are slim, but I'm not sure how to simplify it any further. If nothing else, maybe this can be a reference to others for interesting Gel'fand-Fuks results.

Let us denote by $$W_n$$ the Lie algebra of formal vector fields of $$\mathbb{R}^n$$ at the origin, i.e. if $$\{\partial_1,\dots,\partial_n\}$$ is a basis of $$\mathbb{R}^n$$, then an element of $$W_n$$ is a linear combination $$\sum_{i=1}^n f_i \partial_i$$ where the $$f_i \in \mathbb{R}[[x_1,\dots,x_n]]$$ are formal power series in variables $$x_1,\dots, x_n$$. The Lie algebra structure is defined via the obvious commutator, where $$\partial_i x_j = \delta_{ij}$$.

Gel'fand and Fuks show (e.g. in [1, Chapter 2.2] or [2, Section 4.6] that the Lie algebra cohomology of $$W_n$$ is the limit of a multiplicative (Hochschild-Serre) spectral sequence $$E^{p,q}_2$$ which is the tensor product of $$E^{\bullet,0}_2 = H^\bullet(\mathfrak{gl}(n,\mathbb{R})), \quad E^{0,\bullet}_2 = \mathbb{K}([\Psi_1,\dots,\Psi_n]/I),$$ where $$\deg \Psi_k = 2k$$ and the ideal $$I$$ is generated by all products of the $$\Psi_k$$ with degree greater $$2n$$. Recall that $$H^\bullet(\mathfrak{gl}(n,\mathbb{R})) = \Lambda^\bullet(\phi_1,\phi_3,\dots,\phi_n)$$ with $$\deg \phi_k = k$$. The generators $$\phi_k$$ are then transgressive in this spectral sequence, meaning that $$d_{2i}(\phi_i) = \Psi_i$$, where $$d_{2i}$$ is the differential of the $$2i$$-th page, and together with the multiplicativity of the spectral sequence, this allows one to fully calculate $$H^\bullet(W_n)$$. For example one can find that there survives a nontrivial cohomology class in $$H^7(W_3)$$ induced by a cochain $$c_7$$ arising from $$\phi_3 \otimes \Psi_2 \in E^{3,4}_2$$ (this object will be important in the following). If there is any interest, I can make a little picture showing how that object survives in the spectral sequence.

Furthermore, Gel'fand and Fuks globalize these cochains in a way to be able to talk about Lie algebra cohomology of the Lie algebra of vector fields $$\Gamma(TM)$$, where $$M$$ is some $$n$$-dimensional, compact manifold; see [3, 1.3 - 1.4]. Namely, they claim that, given a nontrivial class in de Rham cohomology $$[\omega] \in H^{n-p}(M)$$ and a class $$[c] \in H^q(W_n)$$, one can construct a noncanonical isomorphism into Lie algebra cohomology* of $$\Gamma(TM)$$ given the following information: Given an open cover $$\{U_1,\dots, U_N\}$$ of $$M$$, local coordinates $$x_1,\dots,x_n$$ for all these open sets $$U_i$$ and a subordinate partition of unity $$\chi_1,\dots,\chi_N$$. Then, given vector fields $$X_1,\dots,X_{q-p} \in \Gamma(TM)$$, construct the forms $$\alpha_{U_i,X_1,\dots,X_{q-p}} := c(\widehat{X_1}(U_i),\dots,\widehat{X_{q-p}}(U_i),\partial_{i_1},\dots,\partial_{i_p}) \cdot dx^{i_1}\wedge \dots \wedge dx^{i_p} \in \Gamma(T^*U_i).$$ Here, $$\widehat{X_i}(U)$$ denotes taking the formal vector field of $$X_i$$ in the coordinates $$x_i$$ on $$U$$. Now, we get a cochain $$\zeta([\omega],[c]) \in C^{q-p}(\Gamma(TM))$$ by setting $$\zeta([\omega],[c])(X_1,\dots,X_{q-p}) := \int_M \omega \wedge \left( \sum_i \chi_i \alpha_{U_i,X_1,\dots,X_{q-p}} \right).$$

But here, note that all the forms $$\alpha_{U_i,X_1,\dots,X_{q-p}}$$ and hence also $$\zeta([\omega],[c])$$ would vanish if $$c$$ were to vanish whenever it had $$p$$ basis vector fields $$\partial_i$$ inserted into it.

Now, recall the form $$c_7$$ on $$W_3$$ from earlier, arising from $$\phi_3 \in H^3(\mathfrak{gl}(n,\mathbb{R}))$$ and $$\Psi_2 \in E_2^{0,4}$$. Tracking how the spectral sequence $$E^{p,q}_2$$ came about, we find that $$\phi_3$$ is induced by a cocycle on the subalgebra $$\mathfrak{gl}(n,\mathbb{R}) \subset W_n$$ induced by formal vector fields of linear order, so only nonvanishing on elements of the form $$x^i \partial_j$$. The element $$\Psi_2$$ on the other hand is nonvanishing** only on permutations of pairs $$(\partial_i, \partial_j, x_r x_s \partial_k, x_l x_m \partial_u)$$. In total, this means that $$c_7$$ (arising as the wedge product of $$\Psi_2$$ and $$\phi_3$$) will vanish on any term of the form $$(\partial_1,\partial_2,\partial_3, \cdot, \cdot, \cdot, \cdot)$$.

Assuming I have not made a mistake (which I find heavily doubtful), my conclusion would be that the isomorphism which Gelfand and Fuks propose breaks a bit, for example when $$M = S^3$$, because then, by my previous argumentation, the map $$\zeta$$ would for example vanish on $$ \otimes [c_7]$$. This clearly runs counter to the claim that their map is an isomorphism.

I am unfortunately not able to follow their proof for why their proposed map is an isomorphism, as it is heavily based on another, very technical paper . I would just like to understand if there is any obvious mistake I am making. Specifically, the points I am most unsure about is whether the map $$c_7$$ that I described really exists as a nontrivial cocycle, and whether it really vanishes if one inserts three constant order vector fields, but I think I have laid out my reasoning enough for why I think this is the case.

$$^*$$ actually, this is an isomorphism not into Lie algebra cohomology, but into a spectral sequence of the closely related diagonal cohomology, but I did not want to get into this, the question is long enough as is. In any case, this map is supposed to be injective.

$$^{**}$$ this is because on $$E^{0,\bullet}_2$$, we consider cochains on relative Lie algebra cohomology of $$W_n$$ relative to the subalgebra $$\mathfrak{gl}(n,\mathbb{R})$$ of linear formal vector fields. One can show that here, cochains are only nonzero on pairs of elements which are invariant under the $$\mathfrak{gl}(n,\mathbb{R})$$-action, which first forces an equal amount of $$x$$'s and $$\partial$$'s, and then together with the symmetry conditions forces the cochain to vanish on all pairs which do not only include constant terms and quadratic terms.

 Fuks, D. B., Cohomology of infinite dimensional Lie algebras, Moskva: ”Nauka”. 272 p. R. 2.40 (1984). ZBL0592.17011.

 Gel’fand, I. M.; Fuks, D. B., Cohomology of the Lie algebra of formal vector fields, Math. USSR, Izv. 4(1970), 327-342 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 34, 322-337 (1970). ZBL0216.20302.

 Gel’fand, I. M.; Fuks, D. B., Cohomologies of Lie algebra of tangential vector fields. II, Funct. Anal. Appl. 4, 110-116 (1970); translation from Funkts. Anal. Prilozh. 4, No. 2, 23-31 (1970). ZBL0208.51401.

 Gel’fand, I. M.; Fuks, D. B., Cohomologies of the Lie algebra of tangential vector fields of a smooth manifold, Funct. Anal. Appl. 3, 194-210 (1969); translation from Funkts. Anal. Prilozh. 3, No. 3, 32-52 (1969). ZBL0216.20301.