cohomology group

cohomology group
Name	Cohomology group
Field	Mathematics
Introduced	20th century
Contributors	Henri Poincaré; Élie Cartan; Jean Leray; Samuel Eilenberg; Norman Steenrod

cohomology group A cohomology group is an algebraic invariant assigned to a mathematical object that captures global structure via homological duality, functoriality, and exact sequences. It arises in topology, algebraic geometry, and number theory through constructions that relate cycles, boundaries, sheaves, and derived functors, and it interacts with fundamental results such as the Lefschetz fixed-point theorem, Hodge theory, and the Riemann–Roch theorem.

Definition and Basic Concepts

Cohomology groups are defined using chain complexes, exact sequences, and functors associated to objects studied by Henri Poincaré, Élie Cartan, Samuel Eilenberg, Norman Steenrod, and Jean Leray; these constructions involve boundaries, coboundaries, and quotient modules that yield graded abelian groups or modules. In algebraic topology one constructs cochain complexes from singular simplices or cellular structures related to Poincaré duality, Alexander duality, Mayer–Vietoris sequence, and Eilenberg–MacLane space methods; in algebraic geometry one uses sheaf cohomology, Čech cohomology, and derived functors connected to Serre duality, Grothendieck, and Alexander Grothendieck techniques. The basic algebraic notion employs hom complexes, Ext and Tor functors, and long exact sequences as in the context of Hom functor and Ext functor constructions.

Examples and Computations

Classical computations include singular cohomology of spheres, tori, and projective spaces using cellular or simplicial methods connected to results by Henri Poincaré, Hermann Weyl, Lusternik–Schnirelmann theory, and James Clerk Maxwell (historical influence on vector analysis). Cohomology of complex projective space uses Chern classes from Chern–Weil theory and calculations that inform Riemann–Roch theorem applications associated with Bernhard Riemann and Georg Cantor-inspired set constructions. Étale cohomology, developed for arithmetic geometry problems linked to Alexander Grothendieck, provides computations for varieties over finite fields and is central to the proof of the Weil conjectures by Pierre Deligne; crystalline cohomology and de Rham cohomology relate to work by Jean-Pierre Serre and Alexander Grothendieck in p-adic and algebraic de Rham settings.

Cohomology Theories and Variants

Numerous cohomology theories exist, including singular cohomology, de Rham cohomology, Dolbeault cohomology, sheaf cohomology, étale cohomology, and K-theory cohomology, each tied to contributors such as Élie Cartan, Jean Leray, Georges de Rham, and Atiyah–Singer collaborators like Michael Atiyah and Isadore Singer. Other variants include Brown–Peterson cohomology, Morava K-theory, and extraordinary cohomology theories arising in stable homotopy theory pursued by researchers around J. Peter May, Haynes Miller, and Douglas Ravenel. Intersection cohomology, perverse sheaves, and mixed Hodge structures reflect advances by Mark Goresky, Robert MacPherson, and Pierre Deligne, with applications to singular spaces and representation-theoretic problems linked to George Lusztig and David Kazhdan.

Properties and Algebraic Structure

Cohomology groups form graded rings via cup product and support additional operations such as the Steenrod algebra operations discovered by Norman Steenrod and collaborators; they obey universal coefficient theorems, Künneth formulas, and spectral sequences including the Leray and Serre spectral sequences developed by Jean Leray and Jean-Pierre Serre. Duality theorems like Poincaré duality and Serre duality provide pairings between homology and cohomology groups, while operations from characteristic classes, Chern classes, and Pontryagin classes relate topological invariants to cohomological rings in work associated with Shiing-Shen Chern and Lev Pontryagin. Cohomology with coefficients, torsion phenomena, and Bockstein sequences connect to homological algebra traditions led by H. Cartan and Samuel Eilenberg.

Applications and Interpretations

Cohomology groups are applied in proofs of major results such as the Lefschetz fixed-point theorem used in dynamical systems studied by André Weil and Lefschetz-related work, in the proof of the Hodge conjecture-adjacent results, and in classification problems across algebraic geometry, differential topology, and mathematical physics influenced by Edward Witten and Michael Atiyah. In arithmetic geometry they underpin the proof of the Weil conjectures and modern approaches to the Langlands program pursued by researchers like Robert Langlands and Pierre Deligne; in gauge theory and low-dimensional topology they appear in Seiberg–Witten and Donaldson invariants connected to Simon Donaldson and Edward Witten.

Historical Development and Key Contributors

The conceptual origins trace to work by Henri Poincaré on algebraic topology, the formalization by Élie Cartan and Jean Leray of sheaf and spectral sequence methods, and the axiomatic development of cohomology theories by Samuel Eilenberg and Norman Steenrod. Grothendieck revolutionized sheaf cohomology and introduced étale cohomology for algebraic varieties over finite fields, leading to Deligne's proof of the Weil conjectures; subsequent generations including Michael Atiyah, Isadore Singer, Pierre Deligne, and Graeme Segal expanded connections with index theory, representation theory, and mathematical physics.

Category:Algebraic topology