Celebratio Mathematica

I. M. Gel’fand and D. B. Fuks: “Cohomology of Lie groups with real coefficients,” Dokl. Akad. Nauk SSSR 176 : 1 (1967), pp. 24–27. An English translation was published in Sov. Math., Dokl. 8 (1967). MR 226664 article

I. M. Gel’fand and D. B. Fuks: “Topological invariants of noncompact Lie groups connected with infinite dimensional representations,” Dokl. Akad. Nauk SSSR 177 : 4 (1967), pp. 763–766. An English translation was published in Sov. Math., Dokl. 8 (1967). MR 226665 article

I. M. Gel’fand and D. B. Fuks: “The topology of noncompact Lie groups,” Funkcional. Anal. i Priložen. 1 : 4 (October 1967), pp. 33–45. An English translation was published in Funct. Anal. Appl. 1:4 (1967). MR 226666 article

I. M. Gel’fand and D. B. Fuks: “Topology of noncompact Lie groups,” Funct. Anal. Appl. 1 : 4 (October 1967), pp. 285–295. English translation of Russian original published in Funkcional. Anal. i Priložen. 1:4 (1967). Zbl 0169.54702 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie groups with real coefficients,” Sov. Math., Dokl. 8 (1967), pp. 1031–1034. English translation of Russian original published in Dokl. Akad. Nauk SSSR 176:1 (1967). Zbl 0169.54801 article

I. M. Gel’fand and D. B. Fuks: “Topological invariants of noncompact Lie groups related to infinite- dimensional representations,” Sov. Math., Dokl. 8 (1967), pp. 1483–1486. English translation of Russian original published in Dokl. Akad. Nauk SSSR 177:4 (1967). Zbl 0169.54802 article

I. M. Gel’fand and D. B. Fuks: “Classifying spaces for principal bundles with Hausdorff bases,” Dokl. Akad. Nauk SSSR 181 : 3 (1968), pp. 515–518. An English translation was published in Sov. Math., Dokl. 9:3 (1968). MR 232391 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of vector fields on the circle,” Funkcional. Anal. i Priložen. 2 : 4 (1968), pp. 92–93. An English translation was published in Funct. Anal. Appl. 2:4 (1968). MR 245035 article

I. M. Gel’fand and D. B. Fuks: “The cohomologies of the Lie algebra of the vector fields in a circle,” Funct. Anal. Appl. 2 : 4 (October 1968), pp. 342–343. English translation of Russian original published in Funkts. Anal. Prilozh. 2:4 (1968). Zbl 0176.11501 article

I. M. Gel’fand and D. B. Fuks: “On classifying spaces for principal fiberings with Hausdorff bases,” Sov. Math., Dokl. 9 : 3 (1968), pp. 851–854. English translation of Russian original published in Dokl. Akad. Nauk SSSR 181:3 (1968). Zbl 0181.26602 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of vector fields on a manifold,” Funkcional. Anal. i Priložen. 3 : 2 (1969), pp. 87. An English translation was published in Funct. Anal. Appl. 3:2 (1969). MR 245036 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold,” Funkcional. Anal. i Priložen. 3 : 3 (1969), pp. 32–52. An English translation was published in Funct. Anal. Appl. 3:3 (1969). MR 256411 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebras of vector fields on a manifold,” Funct. Anal. Appl. 3 : 2 (April 1969), pp. 155. English translation of Russian original published in Funkts. Anal. Prilozh. 3:2 (1969). article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of tangential vector fields of a smooth manifold,” Funct. Anal. Appl. 3 : 3 (July 1969), pp. 194–210. English translation of Russian original published in Funkcional. Anal. i Priložen. 3:3 (1969). Zbl 0216.20301 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of formal vector fields,” Izv. Akad. Nauk SSSR Ser. Mat. 34 : 2 (1970), pp. 322–337. MR 266195 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of smooth vector fields,” Dokl. Akad. Nauk SSSR 190 : 6 (1970), pp. 1267–1270. An English translation was published in Sov. Math., Dokl. 11 (1970). MR 285023 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of the Lie algebra of tangent vector fields of a smooth manifold, II,” Funkcional. Anal. i Priložen. 4 : 2 (1970), pp. 23–31. An English translation was published in Funct. Anal. Appl. 4:2 (1970). MR 285024 article

I. M. Gel’fand and D. B. Fuks: “Upper bounds for the cohomology of infinite-dimensional Lie algebras,” Funkcional. Anal. i Priložen. 4 : 4 (1970), pp. 70–71. An English translation was published in Funct. Anal. Appl. 4:4 (1970). MR 287589 article

I. M. Gel’fand and D. B. Fuks: “Cycles that represent cohomology classes of the Lie algebra of formal vector fields,” Usp. Mat. Nauk 25 : 5(155) (1970), pp. 239–240. MR 293660 Zbl 0216.20401 article

I. M. Gel’fand and D. B. Fuks: “Cohomolgy of Lie algebras of vector fields with nontrivial coefficients,” Funkcional. Anal. i Priložen. 4 : 3 (1970), pp. 10–25. An English translation was published in Funct. Anal. Appl. 4:3 (1970). MR 298703 article

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of tangential vector fields, II,” Funct. Anal. Appl. 4 : 2 (April 1970), pp. 110–116. English translation of Russian original published in Funkts. Anal. Prilozh. 4:2 (1970). Zbl 0208.51401 article

This paper is a continuation of an earlier paper by us which we shall cite as [1969]. We recall that in [1969] we studied the cohomologies of the Lie algebra \( \mathfrak{U}(M) \) of smooth tangential vector fields of a smooth, compact orientable manifold \( M \) with coefficients in a trivial real representation. The main result of [1969] was a theorem about the finite dimensionality of these cohomologies (in every dimension). In the course of the proof we introduced in the standard complex \[ \mathscr{C}(M)= \{C^q(M),d^q\} \] of the Lie algebra \( \mathfrak{U}(M) \) a subcomplex \[ \mathscr{C}_1(M) = \{C_1^q(M),d^q\} ,\] designated “diagonal” by us, constructed from the spectral sequence \[ \mathscr{E} = \{E_r^{u,v}, \delta_r^{u,v}\} = E_r^{u,v} \to E^{u+r,v-r+1} ,\] which converged to the homologies of the diagonal complex, and an expression for its first (second) term was derived.

The present work consists of two parts. In the first a new interpretation of the second term of the spectral sequence \( \mathscr{E} \) is given which permits, in particular, proof of the triviality of some of its differentials. In the second part the relation between the passage to the limit of the spectral sequence \( \mathscr{E} \) (i.e., between the homologies of the diagonal complex) and the cohomologies of the algebra \( \mathfrak{U}(M) \) is discussed. The most complete information is obtained for the case when the spectral sequence \( \mathscr{E} \) is trivial (i.e., when \( E_2 = E_{\infty} \)). Making use of the results of both the parts we obtain, for some of the manifolds, in particular for toruses of any dimension and for all orientable two-dimensional manifolds, a description of the ring, adjoined (with respect to the filtrations introduced in 1.2 of [1969]) to the ring of cohomotogies of the Lie algebra of tangential vector fields. In the present work we shall follow the notation introduced in [1969] and will not repeat the definitions given there. For ease in reading we shall mention at the appropriate places the number of section of [1969] containing the necessary explanations.

The present work was preceded by a short note [1970].

I. M. Gel’fand and D. B. Fuks: “Cohomologies of Lie algebra of vector fields with nontrivial coefficients,” Funct. Anal. Appl. 4 : 3 (July 1970), pp. 181–192. English translation of Russian original published in Funkcional. Anal. i Priložen. 4:3 (1970). Zbl 0222.58001 article

In this paper we continue the investigation which we began in [1969, 1970a, 1970b] of the cohomologies of the Lie algebra of formal as well as smooth vector fields (on a smooth manifold). Specifically we study the cohomologies of the algebras with coefficients in various representations, mainly with coefficients in the spaces of exterior differential forms, formal or smooth, respectively. The results obtained in these papers, involving cohomologies with coefficients in spaces of forms of degree 0 (i.e., in the spaces of smooth functions or of formal power series), were also contained in the work of M. V. Losik [1970]. It is true that theorems about the algebra of formal vector fields were not singled out, but were essentially contained in them (see [Losik 1970, §2]). We shall not rely here on the results of M. V. Losik since we will prove them again. Our proof in the corresponding part is not, in principle, different from the proof of M. V. Losik, although it is considerably shorter. The investigation of cohomologies with coefficients in forms of degree greater than zero encounters a series of new difficulties which are overcome with the application of the results of [1970b].

I. M. Gel’fand and D. B. Fuks: “Upper bounds for cohomology of infinite-dimensional Lie algebras,” Funct. Anal. Appl. 4 : 4 (October 1970), pp. 323–324. English translation of Russian original published in Funkcional. Anal. i Priložen. 4:4 (1970). Zbl 0224.18013 article

In [1969] we studied cohomology with real coefficients of Lie algebras of smooth vector fields on smooth manifolds. Under certain conditions we proved the finite-dimensionality of this cohomology. One of the intermediate assertions was the assertion of the finite-dimensionality of the complete cohomology space of a Lie algebra of formal vector fields (see [1969, Proposition 6.2]; the treatment of this result via formal vector fields is obtained in §1 of [Gel’fand and Fuks 1970]). The purpose of the present note is the formalization and generalization of the method we used to prove this assertion. With the aid of similar methods, as remarked by B. I. Rosenfield, it may be shown that the cohomology space of the Lie algebra of formal tangent vector fields is finite-dimensional. At the same time, for algebras, for example, this method does not succeed in obtaining a proof of the same finite-dimensionality for Hamiltonian formal vector fields, although a certain reduction of the standard cochain complexes is obtained. We do not repeat here the method in question: it is not necessary for understanding what follows. The reader can observe for himself the direct connection between the material set forth below and the proof of Proposition 6.2 of [1969].

I. M. Gel’fand and D. B. Fuks: “On cohomologies of the Lie algebra of smooth vector fields,” Sov. Math., Dokl. 11 (1970), pp. 268–271. English translation of Russian original published in Dokl. Akad. Nauk SSSR 190:6 (1970). Zbl 0264.17005 article

I. M. Gel’fand and D. B. Fuks: “Cohomology of the Lie algebra of formal vector fields,” Math. USSR, Izv. 4 : 2 (April 1971), pp. 327–342. English translation of Russian original published in Izv. Akad. Nauk SSSR Ser. Mat. 34:2. Zbl 0216.20302 article

I. M. Gel’fand, D. A. Kazhdan, and D. B. Fuks: “The actions of infinite-dimensional Lie algebras,” Funkcional. Anal. i Priložen. 6 : 1 (1972), pp. 10–15. An English translation was published in Funct. Anal. Appl. 6:1 (1972). MR 301767 article

David A. Kazhdan	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funkcional. Anal. i Priložen. 6 : 3 (1972), pp. 25–29. An English translation was published in Funct. Anal. Appl. 6:3 (1973). MR 312531 article

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, D. A. Kazhdan, and D. B. Fuks: “The actions of infinite-dimensional Lie algebras,” Funct. Anal. Appl. 6 : 1 (January 1972), pp. 9–13. English translation of Russian original published in Funkcional. Anal. i Priložen. 6:1 (1972). Zbl 0267.18023 article

David A. Kazhdan	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations,” Funkcional. Anal. i Priložen. 7 : 4 (1973), pp. 29–37. An English translation was published in Funct. Anal. Appl. 7:4 (1974). MR 339195 article

I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funct. Anal. Appl. 6 : 3 (July 1973), pp. 193–196. Zbl 0259.57023 article

As noted in [Gel’fand and Fuks 1970a], computing the cohomology of the Lie algebra of Hamiltonian formal vector fields is an essentially more difficult problem than, say, computing the cohomology of the Lie algebra of all formed vector fields, which was done in [Gel’fand and Fuks 1970b]. The methods used in [1970b] allow one to find the homology of a certain direct summand of the cochain complex of the algebra of Hamiltonian fields without great difficulty, but they yield no information about the complementary summand. In order to test the hypothesis that this complementary summand is acyclic, we have made some computations on an electronic computer. As a result, the above hypothesis has been rejected: we have discovered new and nontrivial cohomology classes of the algebra of Hamiltonian formal vector fields in \( \mathbb{R}^2 \). The important difference between these classes and the cohomology classes of the algebra of all formal vector fields found in [1970b] is that the former cannot be represented by cocycles depending only on the 2-sets of their arguments (see [Gel’fand and Fuks 1970c]).

This partial result seems interesting to us for two reasons. The first is methodological: it turns out that the difficulties encountered in computing the cohomology of the algebra of Hamiltonian fields are fundamental in origin. The second is as follows. The construction of [Godbillon and Vey 1971] and [Bernshtein, and Rozenfel’d 1972] can be carried over to the Hamiltonian case, making it possible to construct, for each cohomology class of the algebra of Hamiltonian formal vector fields, a characteristic class of Hamiltonian fibers of corresponding codimension. In particular, the classes which we will point out here furnish characteristic classes of Hamiltonian fibers of codimension 2. It would be interesting to determine whether or not these characteristic classes are nontrivial.

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, B. L. Feĭgin, and D. B. Fuks: “Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations,” Funkcional. Anal. i Priložen. 8 : 2 (1974), pp. 13–29. An English translation was published in Funct. Anal. Appl. 8:2 (1974). MR 356082 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Usp. Mat. Nauk 29 : 1(175) (1974), pp. 195–223. An English translation was published in Russ. Math. Surv. 29:1 (1974). MR 386977 Zbl 0288.01022 article

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations, II,” Funkcional. Anal. i Priložen. 8 : 3 (1974), pp. 7–11. An English translation was published in Funct. Anal. Appl. 8:3 (1974). MR 418115 article

S. G. Gindikin, A. A. Kirillov, and D. B. Fuks: “The work of I. M. Gel’fand on functional analysis, algebra, and topology,” Russ. Math. Surv. 29 : 1 (February 1974), pp. 5–35. English translation of Russian original published in Usp. Mat. Nauk 29:1(175) (1974). Zbl 0294.01031 article

This survey is timed to coincide with the sixtieth birthday of I. M. Gel’fand. The authors have confined themselves to those branches of mathematics in which he has been engaged during the last decade, and in the various branches the chronology of the articles covered by the survey is different.

Gel’fand’s research in the theory of group representations, which has lasted for thirty years, falls into several cycles; the majority of his results are widely known and were dealt with in the survey [Vishik et al. 1964] on the occasion of his fiftieth birthday. For this reason we deal here only with the results of the last ten years.

His first articles on integral geometry appeared more than ten years ago, but this branch of mathematics is still in a formative phase. Because of this we include in the survey an outline of Gel’fand’s basic research in integral geometry, not excluding some that is comparatively old.

Topology is a new branch of his scientific activity; all the work in this field was carried out between 1968 and 1973.

Simon Grigorevich Gindikin	Related
Alexandre Aleksandrovich Kirillov	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations,” Funct. Anal. Appl. 7 : 4 (October 1974), pp. 278–284. English translation of Russian original published in Funkcional. Anal. i Priložen. 7:4 (1973). Zbl 0294.57016 article

The present study grew out of an attempt to comprehend the local geometric nature of some recently discovered characteristic classes of foliations. The analogy between smooth and piecewise-linear (PL) topologies suggests that such characteristic classes can be readily constructed for foliations on combinatorial manifolds reducible in every simplex to a family of parallel planes. Such foliations (which we call affine PL foliations) are defined in §1, subsection 2; they have a relatively uncomplicated classifying space (subsections 6 and 9) and are canonically smoothed (subsections 14 and 15). It turns out, however, that even the first of the characteristic classes, namely the Godbillon–Vey class, is identically equal to zero on these foliations (subsection 16). To render it nontrivial we extend the class of PL foliations by allowing the leaves in the simplexes to be planes parallel in the projective sense (“projective PL foliations;” see subsection 3). On these foliations the Godbillon–Vey class is no longer trivial (subsection 17).

Like the affine, PL foliations have a straightforward classifying space (subsections 7 and 10).

The present article is limited to foliations of codimension 1, but some of the results, including all those of §§1 and 2, are easily translated to arbitrary codimensions.

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomologies of the Lie algebra of formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations,” Funct. Anal. Appl. 8 : 2 (April 1974), pp. 99–112. English translation of Russian original published in Funkcional. Anal. i Priložen. 8:2 (1974). Zbl 0298.57011 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand and D. B. Fuks: “PL-foliations, II,” Funct. Anal. Appl. 8 : 3 (July 1974), pp. 197–200. English translation of Russian original published in Funkcional. Anal. i Priložen. 8:3 (1974). Zbl 0316.57010 article

In the first part of this work [1974] we defined the PL-analog of Haefliger structures (foliations with singularities) of codimension 1. We now investigate PL-foliations of any codimension. The fundamental results pertain to the corresponding classifying spaces, their homologies, and the relationship with the Haefliger classifying spaces \( B\Gamma^q \).

Our constructions refer to the case of oriented foliations. The theory can be readily extended, however, to nonoriented foliations and to foliations with additional transversal structures: nondivergent, simplectic, etc.

This part of the study can for the most part be read independently of [1974], although the latter contains the geometrical background of the formal constructions that follow.

D. B. Fuchs, A. M. Gabrielov, and I. M. Gel’fand: “The Gauss–Bonnet theorem and the Atiyah–Patodi–Singer functionals for the characteristic classes of foliations,” Topology 15 : 2 (1976), pp. 165–188. MR 431199 Zbl 0347.57009 article

We prove in this article some formulae of Gauss–Bonnet kind for the characteristic classes of foliations (see [Bernstein and Rosenfeld 1972; Bott and Haefliger 1972]). Namely, fixing a Riemannian metric on a manifold with a foliation allows one to determine explicitly for each of these characteristic classes a representing differential form (see, for example, [Bott 1972]). If a domain \( X \) with a piecewise smooth boundary transversal to the foliation is given in the manifold, then the integral of such a form over \( X \), corrected by adding the integrals over faces of different dimensions of certain forms depending on the foliation and the metric near \( \partial X \), depends only on the induced metric on \( \partial X \) (and, of course, on the foliation on \( X \)). By using these formulae one can, in particular, extend the definition of the characteristic classes of foliations to the piecewise smooth case.

Andrei M. Gabrielov	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomologies of infinite dimensional Lie algebras and Laplace operators,” Funkts. Anal. Prilozh. 12 : 4 (1978), pp. 1–5. An English translation was published in Funct. Anal. Appl. 12:4 (1979). MR 515625 Zbl 0396.17008 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, B. L. Feigin, and D. B. Fuks: “Cohomology of infinite-dimensional Lie algebras and Laplace operators,” Funct. Anal. Appl. 12 : 4 (1979), pp. 243–247. English translation of Russian original published in Funkts. Anal. Prilozh. 12:4 (1978). Zbl 0404.17008 article

Although appreciable progress has been made in the last 10 years in calculating the cohomology of infinite-dimensional Lie algebras, some of the problems of this circle appear to be unapproachable to this day; this relates, in the first place, to the algebras of Hamiltonian and divergence-free vector fields and to the algebras \( L_1(n) \subset W_n \). In the solutions of these problems there has been hardly any successful progress without the introduction of new methods.

Such methods could consist of the investigation of the Laplace operators induced by any metrics in the cochain complex. The impression is added that if the metric is introduced in a reasonable way, then the eigenvalues and eigenvectors of the Laplace operator, and hence also the cohomology, will turn out to be calculable. In the following papers we propose to investigate this possibility systematically. In this paper we will analyze one example in which the program indicated can be realized successfully all the way through. It is true that the question concerns a Lie algebra whose cohomology is known: the algebra \( L_1(1) \) of formal vector fields on the line having trivial 1-jet. In what follows we will denote this algebra by \( L_1 \) and whenever the question is of cohomology, we mean continuous cohomology with trivial coefficients.

The cohomology of the algebra \( L_1 \), as well as of the algebras \( L_k(1) \) with \( k > 1 \), was found in [Goncharova 1973]. Goncharova’s calculation is awkward and does not allow one to find the cohomology of the algebra \( L_1 \) without finding the cohomology of the other algebras \( L_k \)).

Our paper significantly, as it seems to us, clarifies Goncharova’s theorem and contains some new results. In particular, we give an explicit description of cocycles representing the cohomology classes of the algebra \( L_1 \) (we also apply this method of describing cocycles to the algebras \( L_k \)).

We recall that in the cohomology theory of infinite-dimensional Lie algebras, the cohomology of the algebra \( L_1 \) has special significance; the reason is explained in Goncharova’s paper. One can add that recently Bukhshtaber and Shokurov discovered a connection between these cohomologies and the Adams–Novikov spectral sequence in complex cobordism theory [Bukhshtaber and Shokurov 1978].

Our paper owes much to Goncharova’s paper: a whole series of our arguments is implicitly contained in it. It remains to indicate also the connection or in any case the analogy between what is presented below and the theory of Kats–Muda [1974].

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

B. L. Feigin and D. B. Fuks: “Cohomology of some nilpotent subalgebras of the Virasoro and Kac–Moody Lie algebras,” J. Geom. Phys. 5 : 2 (1988), pp. 209–235. Dedicated to I. M. Gelfand on his 75th birthday. MR 1029428 Zbl 0692.17008 article

Boris L. Feigin	Related
Israïl Moiseevich Gelfand	Related

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