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