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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 $[1] \otimes [c_7]$. 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 [4]. 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.

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

[2] 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.

[3] 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.

[4] 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.

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