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Cohomology

Index Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. [1]

186 relations: Abelian category, Abelian group, Abstract algebra, Alexander duality, Alexander Grothendieck, Alexander–Spanier cohomology, Algebraic geometry, Algebraic K-theory, Algebraic topology, André–Quillen cohomology, Andrey Kolmogorov, Associative algebra, Étale cohomology, Čech cohomology, Bilinear form, Bilinear map, Birkhäuser, Brown–Peterson cohomology, BRST quantization, Cambridge University Press, Cap product, Category (mathematics), Cellular homology, Chain (algebraic topology), Chain complex, Character (mathematics), Characteristic class, Chern class, Closed set, Cobordism, Codimension, Coherent sheaf cohomology, Cohomology ring, Cohomotopy group, Commentarii Mathematici Helvetici, Commutative ring, Compact space, Complex analysis, Complex cobordism, Complex projective space, Complex-oriented cohomology theory, Connected space, Constant sheaf, Continuous function, Contractible space, Covering space, Crystalline cohomology, Cup product, CW complex, Cyclic homology, ..., Daniel Quillen, De Rham cohomology, Deligne cohomology, Derived category, Derived functor, Diagonal morphism, Differentiable manifold, Differential form, Dimension, Direct sum, Dual space, Eduard Čech, Edwin Spanier, Eilenberg–MacLane space, Eilenberg–Steenrod axioms, Elliptic cohomology, Equivalence class, Equivariant cohomology, Euler class, Exact functor, Exact sequence, Excision theorem, Ext functor, Exterior algebra, Field (mathematics), Finitely generated module, Flat topology, Floer homology, Formal group law, Frank Adams, Free abelian group, Free module, Function (mathematics), Functor, Fundamental class, Fundamental group, Galois cohomology, General position, Genus (mathematics), Geometry, George W. Whitehead, Georges de Rham, Graded ring, Graded-commutative ring, Group cohomology, Hassler Whitney, Henri Poincaré, Highly structured ring spectrum, Hochschild homology, Hodge structure, Holomorphic function, Homological algebra, Homology (mathematics), Homotopy, Homotopy category of chain complexes, Homotopy group, Hyperhomology, Hyperplane, Ideal (ring theory), Integer, Intersection homology, Intersection theory, James Waddell Alexander II, Jean Leray, K-theory, Künneth theorem, Khovanov homology, Lev Pontryagin, Lie algebra cohomology, Local cohomology, Manifold, Map (mathematics), Mathematics, Mayer–Vietoris sequence, Module (mathematics), Morava K-theory, Morphism of algebraic varieties, Moscow, Motivic cohomology, N-sphere, Natural transformation, Noetherian ring, Nonabelian cohomology, Normal bundle, Norman Steenrod, Open set, Orientability, Phantom map, Poincaré duality, Polynomial ring, Pontryagin class, Pontryagin duality, Princeton University Press, Principal bundle, Product topology, Pullback (cohomology), Pullback (differential geometry), Quantum cohomology, Quotient ring, Real number, Real projective space, Relative homology, René Thom, Ring (mathematics), Ring homomorphism, Ring spectrum, Samuel Eilenberg, Section (fiber bundle), Sheaf (mathematics), Sheaf cohomology, Simplicial complex, Simplicial homology, Singular homology, Solomon Lefschetz, Spectrum (topology), Springer Science+Business Media, Stable homotopy theory, Steenrod algebra, Stiefel–Whitney class, Subspace topology, Tensor product of algebras, Topological group, Topological pair, Topological space, Topology, Tor functor, Torsion subgroup, Torus, Transversality (mathematics), Triangulated category, Universal coefficient theorem, University of Chicago Press, Vector bundle, Vector space, Weak equivalence (homotopy theory), Weil cohomology theory. Expand index (136 more) »

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

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Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Alexander duality

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.

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Alexander Grothendieck

Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry.

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Alexander–Spanier cohomology

In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Algebraic K-theory

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.

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Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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André–Quillen cohomology

In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex.

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Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov (a, 25 April 1903 – 20 October 1987) was a 20th-century Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

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Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

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Étale cohomology

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 space, introduced by Grothendieck in order to prove the Weil conjectures.

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Čech cohomology

In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space.

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Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

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Bilinear map

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments.

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Birkhäuser

Birkhäuser is a former Swiss publisher founded in 1879 by Emil Birkhäuser.

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Brown–Peterson cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by, depending on a choice of prime p. It is described in detail by.

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BRST quantization

In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Cap product

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 ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

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Cellular homology

In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes.

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Chain (algebraic topology)

In algebraic topology, a -chain is a formal linear combination of the -cells in a cell complex.

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Chain complex

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.

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Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers).

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Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.

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Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

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Cobordism

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold.

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Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

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Coherent sheaf cohomology

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties.

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Cohomology ring

In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication.

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Cohomotopy group

In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions.

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Commentarii Mathematici Helvetici

The Commentarii Mathematici Helvetici is a peer-reviewed scientific journal in mathematics.

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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

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Complex cobordism

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds.

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Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers.

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Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map E^2(\mathbb\mathbf^\infty) \to E^2(\mathbb\mathbf^1) is surjective.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Constant sheaf

In mathematics, the constant sheaf on a topological space X associated to a set A is a sheaf of sets on X whose stalks are all equal to A. It is denoted by or AX.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.

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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

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Crystalline cohomology

In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by and developed by.

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Cup product

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. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.

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CW complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

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Cyclic homology

In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds.

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Daniel Quillen

Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician.

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De Rham cohomology

In mathematics, de Rham cohomology (after Georges de Rham) 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.

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Deligne cohomology

In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold.

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Derived category

In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.

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Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones.

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Diagonal morphism

In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism satisfying where \pi_k is the canonical projection morphism to the k-th component.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

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Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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Eduard Čech

Eduard Čech (29 June 1893 – 15 March 1960) was a Czech mathematician born in Stračov (then Bohemia, Austria-Hungary, now Czech Republic).

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Edwin Spanier

Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology.

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Eilenberg–MacLane space

In mathematics, and algebraic topology in particular, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name.

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Eilenberg–Steenrod axioms

In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common.

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Elliptic cohomology

In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology.

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Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.

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Equivariant cohomology

In mathematics, equivariant cohomology is a cohomology theory from algebraic topology which applies to topological spaces with a group action.

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Euler class

In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles.

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Exact functor

In homological algebra, an exact functor is a functor that preserves exact sequences.

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Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

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Excision theorem

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology—given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X,A) and (X \ U,A \ U) are isomorphic.

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Ext functor

In mathematics, the Ext functors of homological algebra are derived functors of Hom functors.

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Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

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Flat topology

In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry.

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Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.

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Formal group law

In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group.

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Frank Adams

John Frank Adams FRS (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory.

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Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.

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Free module

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Functor

In mathematics, a functor is a map between categories.

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Fundamental class

In mathematics, the fundamental class is a homology class associated to an oriented manifold M of dimension n, which corresponds to the generator of the homology group H_n(M;\mathbf)\cong\mathbf.

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Fundamental group

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

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Galois cohomology

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.

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General position

In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects.

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Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

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Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

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George W. Whitehead

George William Whitehead, Jr. (August 2, 1918 – April 12, 2004) was an American professor of mathematics at the Massachusetts Institute of Technology, a member of the United States National Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences.

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Georges de Rham

Georges de Rham (10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.

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Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_.

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Graded-commutative ring

In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements x, y satisfy where |x|, |y| denote the degrees of x, y. A commutative (non-graded) ring, with trivial grading, is a basic example.

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Group cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

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Hassler Whitney

Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.

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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Highly structured ring spectrum

In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory.

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Hochschild homology

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings.

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Hodge structure

In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold.

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

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Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.

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Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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Homotopy category of chain complexes

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences.

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Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.

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Hyperhomology

In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes.

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Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Intersection homology

In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.

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Intersection theory

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.

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James Waddell Alexander II

James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others.

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Jean Leray

Jean Leray (7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.

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K-theory

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.

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Künneth theorem

In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product.

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Khovanov homology

In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain complex.

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Lev Pontryagin

Lev Semyonovich Pontryagin (Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.

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Lie algebra cohomology

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras.

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Local cohomology

In algebraic geometry, local cohomology is an analog of relative cohomology.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Mayer–Vietoris sequence

In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.

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Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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Morava K-theory

In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.

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Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.

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Moscow

Moscow (a) is the capital and most populous city of Russia, with 13.2 million residents within the city limits and 17.1 million within the urban area.

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Motivic cohomology

Motivic cohomology is an invariant of algebraic varieties and of more general schemes.

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N-sphere

In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension.

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Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

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Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.

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Nonabelian cohomology

In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological space.

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Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).

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Norman Steenrod

Norman Earl Steenrod (April 22, 1910October 14, 1971) was a mathematician most widely known for his contributions to the field of algebraic topology.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

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Phantom map

In algebraic topology, a phantom map is a map between spectra such that the induced map between homology theories is zero.

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Poincaré duality

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.

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Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

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Pontryagin class

In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes.

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Pontryagin duality

In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.

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Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

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Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group.

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Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.

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Pullback (cohomology)

In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map.

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Pullback (differential geometry)

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*.

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Quantum cohomology

In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold.

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Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Real projective space

In mathematics, real projective space, or RPn or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in Rn+1.

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Relative homology

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces.

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René Thom

René Frédéric Thom (2 September 1923 – 25 October 2002) was a French mathematician.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

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Ring spectrum

In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map and a unit map where S is the sphere spectrum.

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Samuel Eilenberg

Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-born American mathematician who co-founded category theory with Saunders Mac Lane.

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Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

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Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

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Sheaf cohomology

In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.

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Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).

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Simplicial homology

In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex.

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Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X).

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Solomon Lefschetz

Solomon Lefschetz (Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.

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Spectrum (topology)

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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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 after sufficiently many applications of the suspension functor.

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Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.

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Stiefel–Whitney class

In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle.

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Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

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Tensor product of algebras

In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra.

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Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

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Topological pair

In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A\hookrightarrow X. Sometimes i is assumed to be a cofibration.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Tor functor

In homological algebra, the Tor functors are the derived functors of the tensor product of modules over a ring.

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Torsion subgroup

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A).

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Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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Transversality (mathematics)

In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position.

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Triangulated category

In mathematics, a triangulated category is a category together with the additional structure of a "translation functor" and a class of "distinguished triangles".

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Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories.

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University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

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Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Weak equivalence (homotopy theory)

In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same "shape".

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Weil cohomology theory

In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups.

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Betti cohomology, Cochain (algebraic topology), Cohomological, Cohomologies, Cohomology class, Cohomology classes, Cohomology group, Cohomology groups, Cohomology theories, Cohomology theory, Differential cohomology, Extraordinary cohomology theories, Extraordinary cohomology theory, Extraordinary homology theory, Generalized cohomology, Generalized cohomology theories, Generalized cohomology theory, Generalized homology theory, Higher cohomology, Integral cohomology group, Multiplicative cohomology theory, Singular cohomology.

References

[1] https://en.wikipedia.org/wiki/Cohomology

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