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In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. Euclid, detail from The School of Athens by Raphael. ...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...
In mathematics, the Oseledec theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. ...
In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...
In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...
In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications cohomology, a contravariant theory, is more natural than homology. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping f : X → Y composition with f gives rise to a function F o f on X. Cohomology groups also have natural products, making calculation easier. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
(19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ...
In geometry, a simplex (plural: simplices) or n-simplex is an n-dimensional analogue of a triangle. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
Contravariant is a mathematical term with a precise definition in tensor analysis. ...
Partial plot of a function f. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ...
With hindsight, general homology theory should probably have been given an inclusive meaning covering both homology and cohomology: the direction of the arrows in a chain complex is not much more than a sign convention. In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ...
In mathematics, in the field of homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...
In some physics textbooks and articles, certain quantities are defined with the opposite sign from that which is used in other publications. ...
History
Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...
In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ...
There were various precursors to cohomology. In the mid-1920s, J.W. Alexander and Lefschetz founded the intersection theory of cycles on manifolds. On an n-dimensional manifold M, a p-cycle and a q-cycle with nonempty intersection will, if in general position, have intersection a (p+q−n)-cycle. This enables us to define a multiplication of homology classes The 1920s were a decade sometimes referred to as the Jazz Age or the Roaring Twenties, usually applied to America. ...
J. W. Alexander James Waddell Alexander II (September 19, 1888 – September 23, 1971) was an important topologist of the pre-WWII era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ...
Solomon Lefschetz (3 September 1884-5 October 1972) was a US mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. ...
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
2-dimensional renderings (ie. ...
In geometry, general position for a set of points, or other configuration, means the general case situation, as opposed to some more special or coincidental cases that are possible. ...
- Hp(M) × Hq(M) → Hp+q-n(M).
Alexander had by 1930 defined a first cochain notion, based on a p-cochain on a space X having relevance to the small neighborhoods of the diagonal in Xp+1. In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ...
In 1931, De Rham related homology and exterior differential forms, proving De Rham's theorem. This result is now understood to be more naturally interpreted in terms of cohomology. 1931 (MCMXXXI) was a common year starting on Thursday (link is to a full 1931 calendar). ...
Georges de Rham (10 September 1903-9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. ...
A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...
In 1934, Pontrjagin proved the Pontrjagin duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in terms of group characters. 1934 (MCMXXXIV) was a common year starting on Monday (link will take you to calendar). ...
Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908- 3 May 1988) was a Soviet/Russian mathematician. ...
In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...
In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ...
In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In a 1935 conference in Moscow, Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. 1935 (MCMXXXV) was a common year starting on Tuesday (link will take you to calendar). ...
Moscow (Russian: МоÑкваÌ, Moskva, IPA: (help· info)) is the capital of Russia and the countrys principal political, economic, financial, educational and transportation center, located on the river Moskva. ...
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ...
In 1936 Steenrod published a paper constructing Čech cohomology by dualizing Čech homology. 1936 (MCMXXXVI) was a leap year starting on Wednesday (link will take you to calendar). ...
Norman Earl Steenrod (April 22, 1910–October 14, 1971) was a leading mathematician, working in the field of topology. ...
Čech cohomology is a particular type of cohomology in mathematics. ...
From 1936 to 1938, Hassler Whitney and Eduard Čech developed the cup product (making cohomology into a graded ring) and cap product, and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes. 1938 (MCMXXXVIII) was a common year starting on Saturday (link will take you to calendar). ...
Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ...
Eduard Čech (June 29, 1893 - March 15, 1960) was a mathematician born in Stracov, Bohemia (then Austria-Hungary now Czech Republic). ...
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. ...
In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In 1944, Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology. 1944 (MCMXLIV) was a leap year starting on Saturday (the link is to a full 1944 calendar). ...
Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ...
In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...
In 1945, Eilenberg and Steenrod stated the axioms defining a homology or cohomology theory. In their 1952 book, Foundations of Algebraic Topology, they proved that the existing homology and cohomology theories did indeed satisfy their axioms. 1945 (MCMXLV) was a common year starting on Monday (the link is to a full 1945 calendar). ...
In mathematics, specifically in algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of topological spaces have in common. ...
1952 (MCMLII) was a Leap year starting on Tuesday (link will take you to calendar). ...
In 1948 Spanier, building on work of Alexander and Kolmogorov, developed Alexander-Spanier cohomology. 1948 (MCMXLVIII) was a leap year starting on Thursday (the link is to a full 1948 calendar). ...
Spanier, Spanjer (means Spanish) refers to: Alexander-Spanier cohomology theory, named after Edwin Henry Spanier Edwin Henry Spanier (1921-1996) Graham Spanier Muggsy Spanier Wolfgang Spanier, see German article This is a disambiguation page — a list of articles associated with the same title. ...
In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ...
Cohomology theories
Eilenberg-Steenrod theories A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...
In mathematics, a subcategory S of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in...
In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ...
In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In abstract algebra, a homomorphism is a structure-preserving map. ...
In mathematics, specifically in algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of topological spaces have in common. ...
Some cohomology theories in this sense are:
In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...
Čech cohomology is a particular type of cohomology in mathematics. ...
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...
Extraordinary cohomology theories When one axiom (dimension axiom) is relaxed, one obtains the idea of extraordinary cohomology theory; this allows theories based on K-theory and cobordism theory. There are others, coming from stable homotopy theory. In mathematics, K-theory is, firstly, an extraordinary cohomology theory which consists of topological K-theory. ...
In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ...
In mathematics, stable homotopy theory is a branch of algebraic topology. ...
Other cohomology theories Theories in a broader sense of cohomology include: In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ...
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ...
Lie algebra cohomology is a cohomology theory for Lie algebras. ...
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. ...
In mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck. ...
In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...
Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by Grothendieck during the 1960s. ...
In mathematics, intersection cohomology is a theory from algebraic topology, initially developed by Goresky and MacPherson, to apply to spaces with singularities. ...
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