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110 relations: Acta Mathematica, Affine bundle, Algebraic topology, Annals of Mathematics, Associated bundle, Atlas (topology), Čech cohomology, Basis (linear algebra), British English, Bundle (mathematics), Bundle map, Cartan's theorem, Category (mathematics), Category theory, Characteristic class, Charles Ehresmann, Chern class, Circle, Circle bundle, Circle group, Cohomology, Commutative diagram, Compact space, Comptes rendus de l'Académie des Sciences, Connected space, Connection (mathematics), Continuous function, Cotangent bundle, Covering space, CW complex, Cylinder, Dependent type, Differentiable manifold, Differential geometry, Differential topology, Discrete space, Equivariant bundle, Equivariant map, Euler class, Exact sequence, Fibered manifold, Fibration, Frame bundle, Gauge theory, General linear group, Gennadi Sardanashvily, Graduate Studies in Mathematics, Group (mathematics), Group action, Group representation, ..., Gysin homomorphism, Hairy ball theorem, Hassler Whitney, Heinz Hopf, Herbert Seifert, Homeomorphism, Homotopy, Homotopy lifting property, Hopf fibration, Hypersphere, I-bundle, Image (mathematics), Jacques Feldbau, Jean-Pierre Serre, Klein bottle, Lie group, Local homeomorphism, Manifold, Map (mathematics), Mapping torus, Mathematics, Möbius strip, Metric tensor, Natural bundle, Neighbourhood (mathematics), Norman Steenrod, Obstruction theory, Open and closed maps, Open set, Optical fiber cable, Principal bundle, Principal homogeneous space, Proceedings of the National Academy of Sciences of the United States of America, Product topology, Project Euclid, Projective bundle, Proper map, Pullback bundle, Quasi-fibration, Quotient space (topology), Riemannian manifold, Section (fiber bundle), Sheaf (mathematics), Smoothness, Space (mathematics), Special unitary group, Sphere bundle, Submersion (mathematics), Surjective function, Tangent bundle, Topological group, Topological space, Topology, Torsor (algebraic geometry), Torus, Unit tangent bundle, Universal bundle, Vector bundle, Vector space, 3-manifold. Expand index (60 more) »
Acta Mathematica
Acta Mathematica is a peer-reviewed open-access scientific journal covering research in all fields of mathematics.
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Affine bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.
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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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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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Associated bundle
In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G. For a fibre bundle F with structure group G, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ.
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Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas.
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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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Basis (linear algebra)
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
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British English
British English is the standard dialect of English language as spoken and written in the United Kingdom.
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Bundle (mathematics)
In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure.
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Bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles.
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Cartan's theorem
In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan.
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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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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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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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Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a French mathematician who worked in differential topology and category theory.
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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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Circle
A circle is a simple closed shape.
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Circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle \scriptstyle \mathbf^1.
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Circle group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
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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.
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Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
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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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Comptes rendus de l'Académie des Sciences
Comptes rendus de l'Académie des Sciences (English: Proceedings of the Academy of sciences), or simply Comptes rendus, is a French scientific journal which has been published since 1666.
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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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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.
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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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Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.
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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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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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Cylinder
A cylinder (from Greek κύλινδρος – kulindros, "roller, tumbler"), has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes.
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Dependent type
In computer science and logic, a dependent type is a type whose definition depends on a value.
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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 geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
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Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
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Equivariant bundle
In differential geometry, given a compact Lie group G, an equivariant bundle is a fiber bundle such that the total space and the base spaces are both ''G''-spaces and the projection map \pi between them is equivariant: \pi \circ g.
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Equivariant map
In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.
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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 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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Fibered manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion i.e. a surjective differentiable mapping such that at each point the tangent mapping is surjective, or, equivalently, its rank equals dim.
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Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle.
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Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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Gennadi Sardanashvily
Gennadi Sardanashvily (Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 - September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University.
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Graduate Studies in Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
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Gysin homomorphism
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle.
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Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres.
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Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician.
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Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
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Herbert Seifert
Herbert Karl Johannes Seifert (27 May 1907, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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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 lifting property
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E. For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces.
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Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.
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Hypersphere
In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center.
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I-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold.
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
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Jacques Feldbau
Jacques Feldbau was a French mathematician, born on 22 October 1914 in Strasbourg, of an Alsatian Jewish traditionalist family.
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Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
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Klein bottle
In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
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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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Mapping torus
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism: The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold, Mf will be a manifold of dimension one higher, and it is said to "fiber over the circle".
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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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Möbius strip
The Möbius strip or Möbius band, also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary.
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Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
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Natural bundle
In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle F^s(M) for some s \geq 1.
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Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
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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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Obstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
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Open and closed maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets.
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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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Optical fiber cable
An optical fiber cable, also known as a fiber optic cable, is an assembly similar to an electrical cable, but containing one or more optical fibers that are used to carry light.
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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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Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.
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Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America (PNAS) is the official scientific journal of the National Academy of Sciences, published since 1915.
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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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Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent and society publishers.
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Projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
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Proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.
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Pullback bundle
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space.
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Quasi-fibration
In algebraic topology, a branch of mathematics, a quasi-fibration, introduced by Albrecht Dold and René Thom, is a continuous map of topological spaces f\colon X \to Y such that the fibers f^(y) are homotopy equivalent to the homotopy fiber of f via the canonical map.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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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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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
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Sphere bundle
In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension n. Similarly, in a disk bundle, the fibers are disks D^n.
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Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
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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 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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Torsor (algebraic geometry)
In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme (or even algebraic space) with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along "some" covering map Y \to X is the trivial torsor Y \times G \to Y (G acts only on the second factor).
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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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Unit tangent bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M) or simply UTM, is the unit sphere bundle for the tangent bundle T(M).
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Universal bundle
In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group, is a specific bundle over a classifying space, such that every bundle with the given structure group over is a pullback by means of a continuous map.
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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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3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space.
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References
[1] https://en.wikipedia.org/wiki/Fiber_bundle
