User:LMias/sandbox

Definition
For a smooth, finite-dimensional manifold $$M$$, the space of smooth vector fields $$\mathfrak{X}(M)$$ carries a Fréchet space structure with the Whitney $$C^\infty$$ topology, and a (jointly) continuous Lie bracket $$[ \cdot, \cdot] : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$$. The Gelfand-Fuks cohomology of $$M$$ is the cohomology of the complex of $$C^\bullet_\text{GF}(M)$$ of continuous Chevalley-Eilenberg cochains (with coefficients in $$ \mathbb{K} = \mathbb{R}$$ or $$\mathbb{C}$$) with respect to the Chevalley-Eilenberg differential.

The restriction to continuous cochains allows the use of analytic techniques to gain control over multilinear cochains, like the Schwartz Kernel theorem, which is extensively used within most results about Gelfand-Fuks cohomology.

Calculation apparatus
In contrast to classical finite-dimensional Lie algebras or even the infinite-dimensional Kac-Moody algebras, the Lie algebra of vector fields naturally intertwines heavily with geometry and analytic concepts. As such,

The technique originally employed by Gelfand and Fuks to gain information about Gelfand-Fuks cohomology can be summarized as follows:

f \frac{\partial g}{\partial x_i} \cdot \partial_j - g \frac{\partial f}{\partial x_j} \cdot \partial_i, \quad f,g \in \mathbb{K} x_1,\dots,x_n. $$ This is isomorphic to the (topological) Lie algebra of infinity-jets of vector fields at a fixed point in $$\mathbb{R}^n$$, and is in this sense the infinitesimal analogue of Gelfand-Fuks cohomology. This cohomology can be calculated exactly by means of a Hochschild-Serre spectral sequence with respect to the subalgebra of $$\left\{\sum_{i,j = 1}^n a_{ij} x_i \partial_j : a_{ij} \in \mathbb{K} \right\} \cong gl_n(\mathbb{K})$$, all of whose pages and differentials are explicitly understood.
 * 1) The infinitesimal case: For a fixed dimension $$n$$, begin by calculating the continuous Lie algebra cohomology of the Lie algebra of formal vector fields $$W_n := \mathbb{K} x_1,\dots,x_n \otimes \mathbb{K}^n$$ with Lie bracket $$ [f \partial_i, g \partial_j] :=
 * 1) The local case: Continue with the cohomology for $$M = \mathbb{R}^n$$, which essentially represents the local case for a topological trivial open set $$U \subset M $$. Since Euclidean space is star-shaped, we can consider the scalings $$T_t : \mathbb{R}^n \to \mathbb{R}^n, x \mapsto t \cdot x$$ for $$t>0$$ and their pushforwards/pullbacks on vector fields and Gelfand-Fuks cochains. In the limit $t \to 0$, this relates Gelfand-Fuks cochains for $$M = \mathbb{R}^n$$ to formal Gelfand-Fuks cochains on the infinity-jets at zero, and one is able to show $$H^\bullet_\text{GF}(\mathbb{R}^n) \cong H^\bullet(W_n).$$
 * 2) The global case: For general manifolds, one employs so-called $$k$$-diagonal cohomology, ... definition ... Using the Schwartz kernel theorem, one can show that $$k$$-diagonal cochains can be identified with antisymmetrizations of distributions of sections of the k-fold external tensor power of the tangent bundle. Gelfand and Fuks then introduce a filtration over the differential order of these cochains to construct spectral sequences which allow a description of k-diagonal cohomology. More precisely, they construct .... blah blah second page and whatever

For the third step, an alternative approach is given by Miaskiwskyi, using the sheaf structure of the Lie algebra of vector fields to exhibit various cosheaf structures on the $$k$$-diagonal cochains. Intertwining the cosheaf-theoretic Cech differential with the differential of the Gelfand-Fuks complex gives rise to a double complex whose associated spectral sequences are closely related to those of Gelfand and Fuks. The result is a spectral sequence converging to $$k$$-diagonal cohomology whose second page is the direct sum of the second pages of Gelfand's and Fuks' spectral sequences up to order k.

There are spectral sequences converging to the quotients of the k-diagonal subcomplexes, or alternatively, the direct sum of these spectral sequences converges to the cohomology of the $$k$$-diagonal subcomplex.

Properties
E^{0,\bullet}_2 = \Lambda^\bullet [ \phi_1, \phi_3,\dots, \phi_{2n -1}] $$, $$ E^{\bullet,0}_2 = \mathbb{K}[\Psi_2, \Psi_4\dots,\Psi_{2n}] / \langle \Psi_{i_1} \dots \Psi_{i_k} : i_1 + \dots + i_k > 2n \rangle$$, $$ E^{p,q}_2 = E^{p,0}_2 \otimes E^{0,q}_2$$, and all differentials of the spectral sequence are fully specified on the generators by $$ d_{i + 1}\phi_i = \Psi_{i+1} $$ for all $$i \in \{1,3,\dots,2n-1\}.$$
 * $$H^k_\text{GF}(M)$$ is finite-dimensional for all $$k \geq 0$$ and all smooth manifolds $$M$$.
 * For all $$n \in \mathbb{N}$$, we have $$H^k_\text{GF}(\mathbb{R}^n) = 0$$ in degrees $$1 \leq k \leq 2n$$ and $$k > n(n + 2)$$. There exists a multiplicative spectral sequence $$\{E^{p,q}_k\}_{p,q,k \geq 0}$$ with $$
 * For M = R^n the cohomology is scaling-invariant (???)
 * $$H^\bullet_\text{GF}(\mathbb{R}^n) = H^\bullet_\Delta(\mathbb{R}^n)$$.
 * For M = S^1 the cohomology is freely generated from one-dimensional generators in degree 2 and 3, which take a certain shape. The cocycle in degree 2 is what gives rise to the Virasoro algebra as a central extension of $$\mathfrak{X}(S^1)$$.