Gelfand–Fuks cohomology

Gelfand–Fuks cohomology
Name	Gelfand–Fuks cohomology
Field	Algebraic topology; Differential topology; Lie algebra cohomology
Introduced	1960s
Founders	Israel Gelfand; Dmitry Fuks
Related	Lie algebra cohomology; Characteristic classes; Foliation theory

Gelfand–Fuks cohomology is a specialized cohomology theory for infinite-dimensional Lie algebras of smooth vector fields introduced in the 1960s by Israel Gelfand and Dmitry Fuks. It connects methods from Israel Gelfand's representation theory, Dmitry Fuks's topology, and constructions used in Mikhail Shubin's analysis to produce invariants relevant to Andrey Kolmogorov-type problems, Lev Pontryagin-style characteristic classes, and classification questions in Georges Reeb-influenced foliation theory. The theory plays a role parallel to finite-dimensional Chevalley–Eilenberg cohomology and interfaces with developments by Jean Leray, Raoul Bott, and Edward Witten.

Introduction

Gelfand–Fuks cohomology arose from attempts by Israel Gelfand and Dmitry Fuks to compute cohomological invariants for the Lie algebra of smooth vector fields on manifolds, inspired by prior work of Jean-Louis Koszul, Claude Chevalley, and Samuel Eilenberg. Early calculations were influenced by examples studied by Raoul Bott in the context of characteristic classes and by techniques from I. M. Gel'fand's school connecting functional analysis with algebraic topology. The subject attracted interest from researchers such as Vladimir Arnold, Benoît Mandelbrot, and Michael Atiyah because of applications to foliation theory, dynamical systems investigated by Stephen Smale, and anomalies considered by Gerard 't Hooft.

Definitions and Construction

The foundational construction parallels Chevalley–Eilenberg cohomology for finite-dimensional Lie algebras developed by Claude Chevalley and Samuel Eilenberg, but adapted to the infinite-dimensional setting considered by Israel Gelfand and Dmitry Fuks. One starts with the Lie algebra Vect(M) of smooth vector fields on a manifold M studied by André Haefliger and defines continuous cochains respecting topologies used by Laurent Schwartz and Joseph L. Doob. Gelfand–Fuks cohomology employs completions similar to those in Alexander Grothendieck's functional analytic frameworks and uses techniques related to Feigin–Fuchs modules investigated by Boris Feigin and Dmitry Fuchs. The resulting complex is analogous to constructions in the work of Jean-Pierre Serre and requires control of supports akin to methods from Edward Nelson and Ivo Dorfman.

Computation for Vector Fields and Foliations

Computations by Dmitry Fuks and collaborators build on spectral-sequence methods from Jean Leray and Henri Cartan. For the Lie algebra of formal vector fields the work relates to results of Boris Feigin and to the classification frameworks used by Phillip Griffiths and John Milnor. For foliations modeled on Haefliger structures studied by André Haefliger and Georges Reeb, Gelfand–Fuks cohomology yields invariants comparable to those in Raoul Bott and Isadore Singer index theory. Explicit calculations appear in contexts exploited by Michael Atiyah, Isadore Singer, William Thurston, and Dennis Sullivan for codimension-one and higher-codimension foliations, often invoking spectral sequences akin to those used by Jean-Michel Bismut.

Relationship to Characteristic Classes and Foliation Theory

Gelfand–Fuks cohomology provides a natural home for secondary characteristic classes introduced by Raoul Bott and André Haefliger, connecting to Pontryagin classes studied by Lev Pontryagin and Chern classes examined by Shiing-Shen Chern. These cohomology classes intervene in classification problems addressed by William Thurston and in rigidity results analogous to work by Grigory Margulis and Edward Witten. The theory links to the structural analysis of foliations advanced by Georges Reeb, André Haefliger, and Dennis Sullivan, and to index-theoretic perspectives developed by Atiyah–Singer collaborators such as Michael Atiyah and Isadore Singer.

Algebraic and Formal Variants

Algebraic variants developed by Boris Feigin, Dmitry Fuks, and others adapt Gelfand–Fuks ideas to formal geometry as in the work of Alexander Grothendieck and Maxim Kontsevich. Formal Gelfand–Fuks cohomology connects to deformation theory studied by Murray Gerstenhaber and Deligne-type formality investigated by Maxim Kontsevich and Pierre Deligne. In representation-theoretic settings parallels appear with Virasoro algebra cohomology explored by Victor Kac and Pavel Etingof, and with vertex algebra methods used by Edward Frenkel and Boris Feigin.

Applications and Examples

Applications include classification of foliations as pursued by William Thurston and André Haefliger, computations of anomaly terms in quantum field theory examined by Edward Witten and Gerard 't Hooft, and links to foliation index theory in the style of Michael Atiyah and Isadore Singer. Examples computed by Dmitry Fuks and successors illuminate rigidity phenomena investigated by Grigory Margulis and dynamical systems phenomena studied by Stephen Smale and Yakov Sinai. Further developments intersect with deformation quantization pioneered by Maxim Kontsevich, modular functor ideas influenced by Graeme Segal, and geometric representation theory advanced by Joseph Bernstein and George Lusztig.

Category:Algebraic topology Category:Differential topology Category:Lie algebra cohomology