UDC 513.83

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR I. M. GELFAND, D. B. FUKS

ON THE COHOMOLOGIES OF THE LIE ALGEBRA OF SMOOTH VECTOR FIELDS

In our recent paper \((^1)\) we began the study of the cohomologies of the topological Lie algebra of smooth* vector fields on a connected compact orientable smooth manifold. The present note contains several new facts about these cohomologies. The results stated in paragraphs 1—5 are essentially contained in the papers \((^{1,2})\); the results of paragraphs 6—11 are new. The central result is a complete description of the cohomologies of the Lie algebra on tori of arbitrary dimension, and also on two-dimensional manifolds (paras. 5, 8, 10, 11).

1. Let us recall that the cohomologies of a topological Lie algebra** \(\mathfrak g\) with coefficients in \(\mathbf R\) are defined as the cohomologies of the standard complex \(\{C^q,d^q\}\), where \(C^q\) is the space of continuous skew-symmetric \(q\)-linear real-valued functionals on \(\mathfrak g\), and the homomorphism \(d^q:C^q\to C^{q+1}\) is defined by the formula

\[ (d^qP)(\xi_1,\ldots,\xi_{q+1}) = \sum_{1\le s<t\le q+1} (-1)^{s+t-1} P\bigl([\xi_s,\xi_t],\xi_1,\ldots,\widehat{\xi_s},\ldots,\widehat{\xi_t},\ldots,\xi_{q+1}\bigr) \]

(here \(\xi_1,\ldots,\xi_{q+1}\in\mathfrak g,\ P\in C^q\)). The elements of the space \(C^q\) are called \(q\)-dimensional chains of the algebra \(\mathfrak g\). The complex \(\{C^q,d^q\}\) is endowed with a canonical multiplicative structure, turning the space

\[ H^*(\mathfrak g;\mathbf R)=\sum_{q\ge 0} H^q(\mathfrak g;\mathbf R) \]

of its cohomologies into a graded (associative) algebra.

We shall denote the standard complex of the topological Lie algebra \(W_n\) of formal vector fields at the origin of the coordinate space \(\mathbf R^n\) (see \((^2)\), para. 0.1) by \(\{C^q(n),d^q(n)\}\), and the cohomologies of this algebra (with coefficients in \(\mathbf R\)) by

\[ \mathfrak H^*(n)=\sum_{q\ge 0}\mathfrak H^q(n). \]

The topological Lie algebra of smooth vector fields on a smooth manifold \(M\) will be denoted by \(\mathfrak A(M)\); the standard complex corresponding to this algebra will be denoted by \(\{C^q(M),d^q(M)\}\), or, more briefly, by \(\mathfrak C(M)\); and the cohomologies of the algebra \(\mathfrak A(M)\) (with coefficients in \(\mathbf R\)) by

\[ \mathfrak H^*(M)=\sum \mathfrak H^q(M). \]

1. Let \(X_n\) denote the full inverse image of the \(2n\)-dimensional skeleton of the base of the universal \(U(n)\)-bundle \((EU(n),p,BU(n))\) under the map \(p\) (we have in mind the usual cellular decomposition of the space \(BU(n)\)—see, for example, \((^3)\), p. 89). Obviously, \(X_n\) is a \(2n\)-connected \(n(n+2)\)-dimensional cellular complex; in particular, \(X_1\) is the three-dimensional sphere.

For every \(q\) there is an equality \(\mathfrak H^q(n)=H^q(X_n;\mathbf R)\). Multiplication in the ring \(\mathfrak H^*(n)\) is trivial, i.e., the product of any two elements of positive dimension is equal to zero.

This theorem constitutes the main content of our paper \((^2)\).

* Smoothness is everywhere understood as belonging to the class \(C^\infty\).

** In saying “algebra,” “space,” etc., we everywhere take the field \(\mathbf R\) of real numbers as the ground field.

1. The group \(\mathrm{Diff}^n\) of diffeomorphisms of the space \(\mathbb{R}^n\) preserving the origin acts in the obvious way on the algebra \(W_n\), and hence also in the spaces \(C^*(n)\), \(\mathfrak{H}^*(n)\) of its cochains and cohomologies.

The action of the group \(\mathrm{Diff}^n\) in \(\mathfrak{H}^*(n)\) is trivial. For any element of the space \(\mathfrak{H}^{n(n+2)}(n)\) there exists in \(C^{n(n+2)}(n)\) a representing cocycle invariant with respect to the group \(\mathrm{Diff}^n\). For \(q<n(n+2)\), an element of \(\mathfrak{H}^q(n)\), generally speaking, cannot be represented by a \(\mathrm{Diff}^n\)-invariant cocycle, but can always be represented by a cocycle invariant with respect to the group \(GL(n,\mathbb{R})\subset \mathrm{Diff}^n\), composed of linear transformations.

This proposition is easily derived from \((^2)\), although it is not explicitly formulated there.

1. Let \(M\) be a connected compact orientable smooth \(n\)-dimensional manifold. In the complex \(\mathscr{C}(M)\) we define a filtration, assigning to a cochain \(P\in C^q(M)\) filtration \(\le k\) if, for any smooth vector fields \(\xi_1,\ldots,\xi_q\in\mathfrak{A}(M)\) with \(P(\xi_1,\ldots,\xi_q)\ne0\), there exist points \(x_1,\ldots,x_k\in M\) such that \(\xi_s(x_{j_s})\ne0\) for some \(j_s\) for each \(s\). The subspace of the space \(C^q(M)\) consisting of elements of filtration \(\le k\) is denoted by \(C_k^q(M)\). Obviously,
   \[ 0=C_0^q(M)\subset\cdots\subset C_q^q(M)=\cdots=C^q(M). \]

This filtration is compatible with the differential \(d^q(M)\), so that for each \(k\) a subcomplex
\[ \mathscr{C}_k(M)=\{C_k^q(M),\,d^q(M)\} \]
of the complex \(\mathscr{C}(M)\) is defined. The complex \(\mathscr{C}_0(M)\) is trivial; the complex \(\mathscr{C}_1(M)\) is called diagonal.

The filtration is also compatible with multiplication in the complex \(\mathscr{C}(M)\). For details see \((^1)\), §§ 1.2—1.5.

1. In § 2 of the paper \((^1)\), in the complex \(\mathscr{C}_k(M)\) (for each \(k\)) a new filtration is introduced. With the aid of this filtration a spectral sequence
   \[ \{\,{}^{(k)}E_r^{u,v},\ {}^{(k)}d_r^{u,v}:{}^{(k)}E_r^{u,v}\to{}^{(k)}E_r^{u+r,v-r+1}\,\} \]
   is defined, converging to the cohomologies of the quotient complex \(\mathscr{C}_k(M)/\mathscr{C}_{k-1}(M)\). The second term of this spectral sequence admits the following description. Denote by \(M^k\) the product of \(k\) copies of the manifold \(M\), and by \(M_*^k\) the part of this product consisting of points \((x_1,\ldots,x_k)\in M^k\) among whose coordinates \(x_1,\ldots,x_k\) there are coincident ones. Put
   \[ H(k,p,q)=H_p\left(M^k,M_*^k;\ \bigoplus_{\substack{m_1+\cdots+m_k=q\\ m_1>0,\ldots,m_k>0}} \left(\mathfrak{H}^{m_1}(n)\otimes\cdots\otimes\mathfrak{H}^{m_k}(n)\right)\right)= \]
   \[ =H^{kn-p}(M^k\setminus M_*^k;\mathbb{R})\otimes H^q\left(X_n\#\cdots\#X_n,*;\mathbb{R}\right) \]
   where in the last expression there are \(k\) factors \(X_n\) under the brace.

(\(\#\) denotes the tensor product of spaces, i.e. the direct product with the coordinate cross collapsed to a point; \(*\) is the marked point.) In the space \(H(k,p,q)\) acts the group \(S(k)\) of permutations of \(k\) elements (the group acts simultaneously in \(M^k\) and in \(X_n\#\cdots\#X_n\)). The space \({}^{(k)}E_2^{u,v}\) is isomorphic to the space of \(S(k)\)-invariant elements of the space \(H(k,-u,v)\); in particular, \({}^{(k)}E_2^{u,v}\) can be nontrivial only for
\[ -kn\le u\le0,\qquad 2n<v\le n(n+2). \]

For \(k=1\) the spectral sequence
\[ \{\,{}^{(k)}E_r^{u,v},\ {}^{(k)}d_r^{u,v}\,\} \]
converges to the cohomologies of the diagonal complex and is denoted simply by
\[ \{\,E_r^{u,v},\,d_r^{u,v}\,\}. \]
From the proposition formulated above it follows that, for \(v>0\), the equality
\[ E_2^{u,v}=H_{-u}(M,\mathbb{R})\otimes\mathfrak{H}^v(n) =H^{u+n}(M;\mathbb{R})\otimes H^v(X_n;\mathbb{R}) \]
holds.

Proofs of the assertions set forth in this item are contained in §§ 3—8 of the paper \((^1)\) and in § 6 of the paper \((^2)\).

1. Multiplication
   \[ C_k^p(M)\otimes C_l^q(M)\to C_{k+l}^{p+q}(M) \]
   induces, for each \(r\), multiplication
   \[ {}^{(k)}E_r^{u_1,v_1}\otimes{}^{(l)}E_r^{u_2,v_2} \to{}^{(k+l)}E_r^{u_1+u_2,v_1+v_2}, \]
   connected with the differential by the formula
   \[ {}^{(k+l)}d_r^{u_1+u_2,v_1+v_2}(\eta_1\eta_2) = ({}^{(k)}d_r^{u_1,v_1}\eta_1)\eta_2 + (-1)^{u_1+v_1}\eta_1({}^{(l)}d_r^{u_2,v_2}\eta_2). \]

In particular, there arises a multiplication
\[ E_r^{u_1,v_1}\otimes\cdots\otimes E_r^{u_k,v_k}\to{}^{(k)}E_r^{u_1+\cdots+u_k,\;v_1+\cdots+v_k}, \]
where the product of the elements
\[ \Phi_i\otimes\alpha_i\in H^{\,n-r_i}(M;\mathbf R)\otimes\mathfrak h^{\,q_i+r_i}(n) =E_2^{-r_i,\;q_i+r_i} \]
is equal to the element of the space
\[ {}^{(k)}E_2^{-r_1-\cdots-r_k,\;q_1+\cdots+q_k+r_1+\cdots+r_k}, \]
obtained from
\[ \pi(\Phi_1\otimes\cdots\otimes\Phi_k)\otimes\alpha_1\otimes\cdots\otimes\alpha_k, \]
where
\[ \pi:H^*(M^k,\mathbf R)\to H^*(M^k\setminus M_*^k,\mathbf R) \]
is the homomorphism induced by the inclusion \(M^k\setminus M_*^k\to M^k\), symmetrization, and the assignment of the corresponding sign. The homomorphism \(\pi\), as is easy to show, is an epimorphism, and therefore the multiplication
\[ E_2^{u_1,v_1}\otimes\cdots\otimes E_2^{u_k,v_k} \to{}^{(k)}E_2^{u_1+\cdots+u_k,\;v_1+\cdots+v_k} \]
is epimorphic.

1. If the spectral sequence \(\{E_r^{u,v},d_r^{u,v}\}\) is trivial, i.e. \(d_r^{u,v}=0\) for \(r\ge 2\), then:

(1) All the spectral sequences \(\{{}^{(k)}E_r^{u,v},{}^{(k)}d_r^{u,v}\}\) are trivial, and hence the cohomology of the complex \(\mathfrak C_k(M)/\mathfrak C_{k-1}(M)\) coincides with \({}^{(k)}E_2\).

(2) Every cohomology class of the complex \(\mathfrak C_k(M)/\mathfrak C_{k-1}(M)\) is represented by a product of \(k\) cochains from \(\mathfrak C_1(M)\).

(3) The product of \(k\) cochains from \(\mathfrak C_1(M)\) is cohomologous to zero in \(\mathfrak C(M)\) if and only if it is cohomologous to zero in \(\mathfrak C_k(M)/\mathfrak C_{k-1}(M)\).

(4) The space \(\mathfrak H^*(M)\) is the direct sum of the cohomology spaces of the complexes \(\mathfrak C_k(M)/\mathfrak C_{k-1}(M)\), \(k=1,2,\ldots\).

(5) The ring \(\mathfrak H^*(M)\) is generated (multiplicatively) by the cohomology classes of the complex \(\mathfrak C_1(M)\); in particular, it has a finite number of generators.

This proposition is easily derived from what was said in item 6 (in the proof of assertion (3) one uses the triviality of the multiplication in the ring \(\mathfrak H^*(n)\)—see item 2).

1. In the case when the spectral sequence of the diagonal complex is trivial, Proposition 7 gives a complete description of the ring \(\mathfrak H^*(M)\): in this case there is an isomorphism
   \[ \mathfrak H^*(M)=\sum_{k,u,v}{}^{(k)}E_2^{u,v}, \]
   and the multiplication
   \[ \mathfrak H^*(M)\otimes\mathfrak H^*(M)\to\mathfrak H^*(M) \]
   is generated by the multiplications
   \[ {}^{(k)}E_2\otimes{}^{(l)}E_2\to{}^{(k+l)}E_2. \]
   Thus, the computation of the ring \(\mathfrak H^*(M)\) is reduced, in the indicated case, to the computation of the cohomology rings of the spaces \(M^k\setminus M_*^k\) and \(X_n\), which is already carried out by standard topological methods.

2. The element
   \[ \Phi\otimes a\in H^{n-r}(M;\mathbf R)\otimes\mathfrak h^{q+r}(n)=E_2^{-r,\;q+r} \]
   can be represented by a cochain of the complex \(\mathfrak C_1(M)\) in the following way. Fix on the manifold \(M\) a finite covering by charts \(\{U_\nu;x_1^{(\nu)},\ldots,x_n^{(\nu)}\}\) (here \(x_1^{(\nu)},\ldots,x_n^{(\nu)}\) are local coordinates in \(U_\nu\)) and a partition of unity \(\gamma_\nu\) subordinate to this covering. Also fix a closed differential form \(\varphi\) in the cohomology class \(\Phi\), and a cocycle \(a\in C^{q+r}(n)\) representing the cohomology class \(\alpha\). Let \(\xi_1,\ldots,\xi_q\in\mathfrak A(M)\). For each point \(x\in U_\nu\) the vector fields \(\xi_1,\ldots,\xi_q\) determine, in view of the existence of a coordinate system in \(U_\nu\), elements \(\xi_1(x),\ldots,\xi_q(x)\in W_n\). Put
   \[ \psi_\nu= \sum_{1\le i_1<\cdots<i_r\le n} a\bigl(\xi_1(x),\ldots,\xi_q(x),e_{i_1},\ldots,e_{i_r}\bigr) \,dx_{i_1}^{(\nu)}\wedge\cdots\wedge dx_{i_r}^{(\nu)}, \]
   where \(e_1,\ldots,e_n\) are the basis vector fields in \(\mathbf R^n\), and define a differential form \(\psi\) of degree \(r\) on \(M\) as \(\sum_\nu\gamma_\nu\psi_\nu\). Assigning to the elements \(\xi_1,\ldots,\xi_q\in\mathfrak A(M)\) the number
   \[ \int_M \psi\wedge\varphi, \]
   we obtain a continuous skew-symmetric \(q\)-linear real functional \(\lambda\) on \(\mathfrak A(M)\), i.e. an element of \(C^q(M)\). It is verified directly that: (a) the cochain \(\lambda\) belongs to the diagonal complex; b) the element it represents from \(E_0\) belongs to the kernel of the differentials \(d_0\) and \(d_1\) and determines in \(E_2\) the element

\(\Phi\otimes a\) (independently of the choice of local coordinates, partition of unity, the form \(\varphi\), and the cochain \(a\)); (c) to the cochain \([d^q(M)]\lambda\) there corresponds the number, assigned to vector fields \(\xi_1,\ldots,\xi_{q+1}\in\mathfrak A(M)\),

\[ \int_M \widetilde\psi\wedge\varphi,\qquad \text{where}\quad \widetilde\psi=\sum_\nu \gamma_\nu\,d\psi'_\nu, \]

\[ \psi'_\nu= \sum_{1\le i_1<\cdots<i_{r-1}\le n} a\bigl(\xi_1(x),\ldots,\xi_{q+1}(x),e_{i_1},\ldots,e_{i_{r-1}}\bigr)\, dx^{(\nu)}_{i_1}\wedge\cdots\wedge dx^{(\nu)}_{i_{r-1}} . \]

10. It is clear that if the cochain \(\lambda\) constructed in item 8 is a cocycle of the complex \(\mathcal C(M)\), then the element \(\Phi\otimes a\in E_2^{-r,q+r}\) belongs to the kernel of all differentials. Here are two cases when \(\lambda\) is a cocycle for trivial reasons: (a) the element \(a\in C^{q+r}(n)\) (see item 8) is invariant with respect to the group \(\operatorname{Diff}_n\); (b) the element \(a\) is invariant with respect to the group \(GD(n,\mathbf R)\), and the transition functions from one set of local coordinates \(x_1^\nu,\ldots,x_n^\nu\) to another are all linear. In both cases the forms \(\psi'_\nu\) (as well as the forms \(\psi_\nu\)) agree on overlaps of charts, i.e. they serve as restrictions of a global form \(\psi'\), and therefore

\[ \widetilde\psi=\left(\sum_\nu\gamma_\nu\right)d\psi'=d\psi',\qquad \{[d^q(M)]\lambda\}(\xi_1,\ldots,\xi_{q+1})=\int_M d\psi'\wedge\varphi=0. \]

Let us give two consequences of this observation.

(1) If \(M\) is a torus (the product of \(n\) circles), then the spectral sequence \(\{E_r^{u,v},d_r^{u,v}\}\) is trivial.

This follows from what was said above (case (b)) and item 3.

(2) If \(n=2\), then the spectral sequence \(\{E_r^{u,v},d_r^{u,v}\}\) is trivial.

Indeed, for \(n=2\) the space \(E_2^{u,v}\) can be nontrivial only for \(u=-2,-1,0\) and \(v=5,7,8\). By dimensional considerations, only the differential

\[ d^{-2,8}:\ E_2^{-2,8}\to E^{0,7} \]

can be nontrivial, and it is trivial by what was said above (case (a)) and item 3.

11. The final solution of the problem of computing the ring \(\mathfrak H^*(M)\) for the case when \(M\) is a torus of arbitrary dimension or a two-dimensional manifold is obtained by a simple comparison of the results of items 5, 8, and 10. For example, the ring \(\mathfrak H^*(S^2)\) is generated by the generators \(a_1,a_2,a_3,a_4,a_5;\ b_1,b_2,b_3,b_4,b_5\) of dimensions \(3,3,5,6,6;\ 5,5,7,8,8\), respectively, subject, besides the usual anticommutativity relations, to the relations

\[ b_i b_j=0\quad (i,j\ \text{arbitrary});\qquad a_i b_j+a_j b_i=0\quad (i\ne j);\qquad a_4b_4=a_5b_5=0. \]

Moscow State University
named after M. V. Lomonosov

Received
19 XI 1969

CITED LITERATURE

1. I. M. Gelfand, D. B. Fuks, “Functional analysis and its applications,” 3, no. 3, 32 (1969).
2. I. M. Gelfand, D. B. Fuks, Izv. Akad. Nauk SSSR, Ser. Mat., 34, no. 2 (1970).
3. Chen Shan-shen, Complex Manifolds, IL, 1961.