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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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, a sequence is an ordered list of objects (or events An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. 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. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is 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.

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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The twentieth century of the Common Era began on In Algebraic topology, a simplicial k - chain is a formal linear combination of k - simplices. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules 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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. In Mathematics, specifically in Algebraic topology, the cup product is a method of adjoining two Cocycles of degree p and q to form In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

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 It can be broadly In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. In Physics, a sign convention is a choice of the signs (plus or minus of a set of quantities in a case where the choice of sign is arbitrary

Contents

History

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic 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. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and Cohomology

There were various precursors to cohomology. In the mid-1920s, J.W. Alexander and Solomon Lefschetz founded the intersection theory of cycles on manifolds. The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of Solomon Lefschetz ( 3 September 1884 – 5 October 1972) was an American Mathematician who did fundamental work on A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Algebraic geometry, general position in a notion of genericity for a set of points or other geometric objects This enables us to define a multiplication of homology classes

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. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line

In 1931, Georges de Rham related homology and exterior differential forms, proving De Rham's theorem. Year 1931 ( MCMXXXI) was a Common year starting on Thursday (link will display full 1931 calendar of the Gregorian calendar. Georges de Rham ( 10 September 1903 &ndash 9 October 1990) was a Swiss Mathematician, known for his contributions to In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable This result is now understood to be more naturally interpreted in terms of cohomology.

In 1934, Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups. Year 1934 ( MCMXXXIV) was a Common year starting on Monday (link will display full 1934 calendar of the Gregorian calendar. Lev Semenovich Pontryagin ( Russian Лев Семёнович Понтрягин ( 3 September 1908 &ndash 3 May 1988) was a In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in terms of group characters. In Mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and Cohomology In Mathematics, Alexander duality refers to a Duality theory presaged by a result of 1915 by J In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a character is (most commonly a special kind of function from a group to a field (such as the Complex numbers)

At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. Year 1935 ( MCMXXXV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Moscow (Москва́ romanised: Moskvá, IPA: see also other names) is the Capital and the largest city of Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet

In 1936 Norman Steenrod published a paper constructing Čech cohomology by dualizing Čech homology. Year 1936 ( MCMXXXVI) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian calendar. Norman Earl Steenrod ( April 22, 1910 – October 14, 1971) was a preeminent Mathematical Topologist who most widely known for his contributions 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. Year 1938 ( MCMXXXVIII) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar. Hassler Whitney ( 23 March 1907 &ndash 10 May 1989) was an American Mathematician. Eduard Čech (ˈʧɛx ( June 29, 1893 – March 15, 1960) was a Czech Mathematician born in Stračov, Bohemia In Mathematics, specifically in Algebraic topology, the cup product is a method of adjoining two Cocycles of degree p and q to form In Algebraic topology the cap product is a method of adjoining a chain of degree p with a Cochain of degree q, such that q Their theory was still limited to finite cell complexes.

In 1944, Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology. Year 1944 ( MCMXLIV) was a Leap year starting on Saturday (link will display full calendar of the Gregorian calendar. Samuel Eilenberg ( September 30, 1913 — January 30, 1998) was a Polish and American Mathematician of In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space

In 1945, Eilenberg and Steenrod stated the axioms defining a homology or cohomology theory. Year 1945 ( MCMXLV) was a Common year starting on Monday (link will display the full calendar In Mathematics, specifically in Algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of Topological spaces In their 1952 book, Foundations of Algebraic Topology, they proved that the existing homology and cohomology theories did indeed satisfy their axioms. Year 1952 ( MCMLII) was a Leap year starting on Tuesday (link will display full calendar of the Gregorian calendar. [1]

In 1948 Edwin Spanier, building on work of Alexander and Kolmogorov, developed Alexander-Spanier cohomology. Year 1948 ( MCMXLVIII) was a Leap year starting on Thursday (link will display the 1948 calendar of the Gregorian calendar. Edwin Henry Spanier ( August 8, 1921, Washington DC &ndash October 11, 1996, Scottsdale Arizona) was an American In Mathematics, particularly in Algebraic topology Alexander-Spanier cohomology is a Cohomology theory arising from Differential forms with

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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in In Topology, a CW complex is a type of Topological space introduced by J An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, specifically in Algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of Topological spaces

Some cohomology theories in this sense are:

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. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable 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 In Mathematics, K-theory is a tool used in several disciplines In Mathematics, an n+1 cobordism is a Triple (WMN where W is an (n+1-dimensional Manifold, whose There are others, coming from stable homotopy theory. In Mathematics, stable homotopy theory is that part of Homotopy theory (and thus Algebraic topology) concerned with all structure and phenomena that remain

Other cohomology theories

Theories in a broader sense of cohomology include:

References

  1. ^ Spanier, E. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper In Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to Lie algebra cohomology is a Cohomology theory for Lie algebras. In Mathematics, Hochschild homology is a Homology theory for Associative algebras over rings. In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having 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 groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological In mathematics crystalline cohomology is a Weil cohomology theory for schemes invented by Grothendieck (in his letter to Tate (Grothendieck 1966 and his lecture In Mathematics, the flat topology is a Grothendieck topology used in Algebraic geometry. Motivic cohomology is a cohomological theory in Mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s In Topology, a branch of Mathematics, intersection homology is a generalization of homology theory for singular spaces, discovered by Mark Goresky In Mathematics, specifically in Symplectic topology and Algebraic geometry, a quantum cohomology ring is an extension of the ordinary Cohomology H. (2000) "Book reviews: Foundations of Algebraic Topology" Bulletin of the American Mathematical Society 37(1): pp. 114-115

See also

This is a list of some of the ordinary and generalized (or extraordinary homology and cohomology theories in Algebraic topology that are defined on the categories of CW

Dictionary

cohomology

-noun

  1. (mathematics) A theory associating a system of quotient groups to each topological space.
  2. (mathematics) A system of quotient groups associated to a topological space.
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