Betti cohomology
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| Betti cohomology | |
|---|---|
| Name | Betti cohomology |
| Field | Algebraic topology |
| Introduced | 20th century |
| Prominent figures | Enrico Betti, Henri Poincaré, Élie Cartan, Alexander Grothendieck, Jean-Pierre Serre |
Betti cohomology is a cohomological tool that assigns graded vector spaces to complex algebraic varieties and topological spaces, capturing topological invariants related to cycles and differential forms. Developed through contributions from Enrico Betti, Henri Poincaré, Élie Cartan, Alexander Grothendieck, and Jean-Pierre Serre, it connects classical invariants such as Betti numbers, singular homology, and de Rham cohomology with modern formulations in algebraic geometry and topology. Betti cohomology plays a central role in statements and conjectures involving the Hodge conjecture, Weil conjectures, and comparison isomorphisms across cohomology theories.
Definition and basic properties
Betti cohomology is defined by taking the underlying complex analytic space of a complex algebraic variety and computing its singular cohomology with coefficients in a chosen ring or field, typically Z, Q, or C, producing graded abelian groups H^n(X, R). Historically motivated by work of Enrico Betti and formalized by Henri Poincaré and Samuel Eilenberg, the theory satisfies the Eilenberg–Steenrod axioms for ordinary cohomology such as homotopy invariance and the long exact sequence of a pair, and it is equipped with a graded-commutative cup product coming from the diagonal map and the cross product in cohomology ring constructions prominent in Leray–Serre spectral sequence contexts. For a smooth projective complex variety X, Betti cohomology groups carry additional structures via the Hodge decomposition and polarizations studied by Carl Ludwig Siegel and David Mumford.
Relation to singular cohomology and Betti numbers
Betti cohomology ordinarily coincides with singular cohomology of the complex manifold underlying a complex algebraic variety, and thus the ranks of these groups are the Betti numbers b_n(X) that appear in topological classification theorems by Poincaré duality and calculation techniques using the Mayer–Vietoris sequence, the Künneth theorem, and cellular decompositions leading back to methods used by G. H. Hardy and contemporaries. In many situations, comparison isomorphisms relate Betti cohomology with de Rham cohomology via the de Rham theorem and with étale cohomology via comparison theorems developed in the work culminating with contributions from Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and Alexander Beilinson. Betti numbers serve as topological invariants in classification problems addressed by René Thom, John Milnor, Michael Atiyah, and Isadore Singer.
Comparison with other cohomology theories
Betti cohomology interfaces with other cohomology theories such as de Rham cohomology, ℓ-adic cohomology, and crystalline cohomology in comparison theorems central to work by Alexander Grothendieck and Pierre Deligne, and these comparisons underpin major results including the proof of the Weil conjectures by Pierre Deligne and the development of the Hodge theory pioneered by W. V. D. Hodge. Unlike ℓ-adic cohomology used in arithmetic contexts by André Weil and Jean-Pierre Serre, Betti cohomology naturally reflects complex-analytic topology as in studies by H. Cartan and Kiyoshi Oka, and contrasts with syntomic cohomology and motivic cohomology in approaches to regulators and special values of L-functions considered by Vladimir Voevodsky and Don Zagier.
Functoriality and cup product structure
Betti cohomology is functorial for morphisms of complex algebraic varieties and continuous maps between complex analytic spaces, yielding pullback maps f^*: H^*(Y, R) → H^*(X, R) and pushforward (Gysin) maps in proper situations studied by Grothendieck and Friedrich Hirzebruch. The cup product endows H^*(X, R) with a graded-commutative algebra structure used in intersection theory in the style of William Fulton and in index theorems of Atiyah–Singer fame. Cup products interact with Poincaré duality and cap products in manifold contexts treated by Marston Morse and Hassler Whitney, and they are compatible with spectral sequences such as the Leray spectral sequence and the Atiyah–Hirzebruch spectral sequence exploited by Michael Atiyah and Friedrich Hirzebruch.
Computations and examples
Explicit computations of Betti cohomology appear in classical examples: for projective spaces one recovers results classically known to Bernhard Riemann and Felix Klein; for smooth projective curves the dimensions relate to the genus in the work of Riemann and Oskar Zariski; for complex tori and abelian varieties computations link to the theory of Jacobians and Néron models studied by André Weil and David Mumford; and for hypersurfaces computations connect with the Lefschetz hyperplane theorem due to Solomon Lefschetz and Hodge numbers explored by W. V. D. Hodge. Tools for computation include cellular homology used by J. H. C. Whitehead, Morse theory from Marston Morse, and spectral sequences applied in research by Jean Leray and Jean-Pierre Serre.
Applications in algebraic geometry and topology
Betti cohomology underpins many applications: in algebraic geometry it contributes to the formulation of the Hodge conjecture and the study of moduli spaces by David Mumford and Pierre Deligne, informs period mappings in Torelli theorems investigated by Friedrich Schottky and Igor Dolgachev, and aids in defining invariants used in enumerative geometry explored by Maxim Kontsevich. In topology Betti cohomology classifies manifolds up to homotopy type in the tradition of Henri Poincaré and René Thom, supports classification results by John Milnor, and interacts with index theory and fixed-point theorems developed by Michael Atiyah, Isadore Singer, and Lefschetz. Contemporary research connects Betti cohomology with motivic ideas from Vladimir Voevodsky and arithmetic geometry programs influenced by Alexander Grothendieck and Pierre Deligne.