Cohomology theories

Cohomology theories
Name	Cohomology theories
Field	Algebraic topology; Algebraic geometry; Homological algebra
Introduced	20th century
Notable contributors	Henri Poincaré; Élie Cartan; Jean Leray; Solomon Lefschetz; André Weil; Alexander Grothendieck

Cohomology theories are collections of contravariant functors from categories of mathematical objects to graded abelian groups or modules that assign algebraic invariants capturing global structure. They provide dual or complementary invariants to homology, enable classification of geometric and arithmetic phenomena, and serve as organizing principles connecting topology, geometry, and number theory.

Introduction

Cohomology theories arose to formalize and extend insights from Henri Poincaré, Élie Cartan, Solomon Lefschetz, and Jean Leray and now appear in diverse settings such as algebraic topology, algebraic geometry, differential geometry, number theory, and mathematical physics. Typical cohomology assigns graded groups H^n(X;A) to a space X and coefficient object A, producing long exact sequences, cup products, and spectral sequences exemplified by constructions in the work of André Weil, Alexander Grothendieck, and Jean-Pierre Serre.

Historical development

Early roots trace to invariants introduced by Henri Poincaré and dualities studied by Élie Cartan and Solomon Lefschetz, with major milestones including Jean Leray's sheaf-theoretic methods during the period surrounding World War II and the expansion of homological algebra promoted by Hermann Weyl, Samuel Eilenberg, and Norman Steenrod. The categorical and axiomatic reframing owes much to Samuel Eilenberg and Norman Steenrod's axioms, while revolutionary advances in the 1950s–1960s emerged via Jean-Pierre Serre's work on sheaf cohomology, André Weil's conjectures, and Alexander Grothendieck's development of étale cohomology and derived categories, later refined by Pierre Deligne and Grothendieck's collaborators at the Institut des Hautes Études Scientifiques. Subsequent decades saw interactions with Michael Atiyah, Isadore Singer, Raoul Bott, William Browder, and the influence of institutions such as Institute for Advanced Study and Princeton University.

Axiomatic frameworks

Axiomatic approaches formalize properties that characterize cohomology theories. The classical axioms of Samuel Eilenberg and Norman Steenrod for singular cohomology prescribe homotopy invariance, excision, dimension, and exactness; notable extensions include Brown representability theorems associated with Edwards Brown and stable homotopy frameworks developed by J. F. Adams and G. W. Whitehead. In algebraic geometry, the Grothendieck school—centered at École Normale Supérieure and University of Paris—introduced axioms for Weil cohomology theories used in the formulation of the Weil conjectures and realized by theories such as Betti, de Rham, crystalline, and étale cohomology established by Michael Artin and Jean-Pierre Serre.

Major examples and variants

Prominent topological examples include singular cohomology, de Rham cohomology developed by Georges de Rham, Čech cohomology associated with Eduard Čech, and generalized cohomology theories like K-theory advanced by Michael Atiyah and Friedrich Hirzebruch, and complex cobordism studied by René Thom and Douglas Ravenel. In algebraic geometry, essential variants are étale cohomology (Grothendieck, Michael Artin), crystalline cohomology (Berthelot), rigid cohomology (Berthelot, Pierre Berthelot), and de Rham cohomology in the algebraic setting as treated by Alexander Grothendieck. Arithmetic and motivic frameworks involve ℓ-adic cohomology used by Pierre Deligne in proofs of the Weil conjectures and the emerging theory of motives advocated by Grothendieck and developed by Yves André, Alexander Beilinson, and Vladimir Voevodsky.

Tools and computational methods

Core computational tools include spectral sequences such as the Leray spectral sequence (Jean Leray), the Serre spectral sequence (Jean-Pierre Serre), the Adams spectral sequence (J. F. Adams), and Grothendieck's spectral sequences arising from composition of derived functors. Derived categories and derived functor formalism introduced by Grothendieck and Jean-Louis Verdier underpin computations alongside sheaf-theoretic techniques from Henri Cartan and Jean Leray. Homotopical and model category methods advanced by Daniel Quillen and André Joyal facilitate calculations in stable homotopy and motivic contexts, while computational algebra systems inspired by work at CNRS and Institut des Hautes Études Scientifiques implement algorithms for cohomology of algebraic varieties and schemes.

Applications and connections

Cohomology theories play central roles across mathematics and mathematical physics: they underlie index theorems such as the Atiyah–Singer index theorem (Michael Atiyah, Isadore Singer), inform classification results in topology via Bott periodicity (Raoul Bott), and enable modern approaches to number theory through ℓ-adic cohomology in the work of Pierre Deligne and Andrew Wiles. Connections to representation theory appear in the study of character varieties and modularity by Jean-Pierre Serre and Pierre Deligne, while interactions with mathematical physics occur in mirror symmetry dialogues involving Maxim Kontsevich, Edward Witten, and the Princeton University and Harvard University research communities.

Advanced topics and generalizations

Advanced developments include the theory of mixed Hodge structures pioneered by Pierre Deligne, the theory of perverse sheaves introduced by Alexandre Beilinson, Joseph Bernstein, and Pierre Deligne, and motivic cohomology formulated by Vladimir Voevodsky and Alexander Beilinson. Higher-categorical and ∞-categorical generalizations influenced by Jacob Lurie and André Joyal yield spectral algebraic geometry frameworks tied to stable homotopy theory and derived algebraic geometry, intersecting work at institutions like Massachusetts Institute of Technology and University of California, Berkeley. Contemporary research engages with the Langlands program through geometric methods championed by Robert Langlands, Ngô Bảo Châu, and Edward Frenkel, and explores categorical and homotopical refinements connected to Mikhail Kapranov and Maxim Kontsevich.

Category:Algebraic topology