Fibration, the Glossary
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.[1]
Table of Contents
50 relations: Algebraic topology, Annals of Mathematics, Approximate fibration, Cambridge University Press, Cofibration, Commutative diagram, Compact-open topology, Connected space, Continuous function, Contractible space, Covering space, CW complex, Euler characteristic, Exact sequence, Fiber bundle, Field (mathematics), Functor, Fundamental group, Fundamental groupoid, Homology (mathematics), Homomorphism, Homotopy, Homotopy fiber, Homotopy group, Homotopy lifting property, Hopf fibration, Hurewicz theorem, Isomorphism, Loop space, McGraw Hill Education, N-sphere, Obstruction theory, Paracompact space, Path (topology), Path space fibration, Postnikov system, Princeton University Press, Projection (mathematics), Pullback bundle, Puppe sequence, Ring (mathematics), Serre spectral sequence, Spectral sequence, Split exact sequence, Subspace topology, Suspension (topology), Topological space, Unit interval, University of Chicago Press, Weak equivalence (homotopy theory).
Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
See Fibration and Algebraic topology
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Fibration and Annals of Mathematics
Approximate fibration
In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property holds only approximately. Fibration and approximate fibration are algebraic topology.
See Fibration and Approximate fibration
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge.
See Fibration and Cambridge University Press
Cofibration
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces is a cofibration if it has the homotopy extension property with respect to all topological spaces S. That is, i is a cofibration if for each topological space S, and for any continuous maps f, f': A\to S and g:X\to S with g\circ i.
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.
See Fibration and Commutative diagram
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.
See Fibration and Compact-open topology
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 non-empty open subsets.
See Fibration and Connected space
Continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.
See Fibration and Continuous function
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.
See Fibration and Contractible space
Covering space
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. Fibration and covering space are algebraic topology.
See Fibration and Covering space
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. Fibration and cW complex are algebraic topology and topological spaces.
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. Fibration and Euler characteristic are algebraic topology.
See Fibration and Euler characteristic
Exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
See Fibration and Exact sequence
Fiber bundle
In mathematics, and particularly topology, a fiber bundle (''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure.
See Fibration and Fiber bundle
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.
See Fibration and Field (mathematics)
Functor
In mathematics, specifically category theory, a functor is a mapping between categories.
Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. Fibration and fundamental group are algebraic topology.
See Fibration and Fundamental group
Fundamental groupoid
In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. Fibration and fundamental groupoid are algebraic topology.
See Fibration and Fundamental groupoid
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.
See Fibration and Homology (mathematics)
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
See Fibration and Homomorphism
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
Homotopy fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag (See Chapter 11 for construction.) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. Fibration and homotopy fiber are algebraic topology.
See Fibration and Homotopy fiber
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.
See Fibration and Homotopy group
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. Fibration and homotopy lifting property are algebraic topology.
See Fibration and Homotopy lifting property
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. Fibration and Hopf fibration are algebraic topology.
See Fibration and Hopf fibration
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism.
See Fibration and Hurewicz theorem
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.
Loop space
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Fibration and loop space are topological spaces.
McGraw Hill Education
McGraw Hill is an American publishing company for educational content, software, and services for pre-K through postgraduate education.
See Fibration and McGraw Hill Education
N-sphere
In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.
Obstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
See Fibration and Obstruction theory
Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
See Fibration and Paracompact space
Path (topology)
In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis.
See Fibration and Path (topology)
Path space fibration
In algebraic topology, the path space fibration over a based space (X, *) is a fibration of the form where. Fibration and path space fibration are algebraic topology.
See Fibration and Path space fibration
Postnikov system
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X.
See Fibration and Postnikov system
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University.
See Fibration and Princeton University Press
Projection (mathematics)
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure).
See Fibration and Projection (mathematics)
Pullback bundle
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space.
See Fibration and Pullback bundle
Puppe sequence
In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe.
See Fibration and Puppe sequence
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.
See Fibration and Ring (mathematics)
Serre spectral sequence
In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. Fibration and Serre spectral sequence are algebraic topology.
See Fibration and Serre spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations.
See Fibration and Spectral sequence
Split exact sequence
In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
See Fibration and Split exact sequence
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).
See Fibration and Subspace topology
Suspension (topology)
In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points.
See Fibration and Suspension (topology)
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. Fibration and topological space are topological spaces.
See Fibration and Topological space
Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
See Fibration and Unit interval
University of Chicago Press
The University of Chicago Press is the university press of the University of Chicago, a private research university in Chicago, Illinois.
See Fibration and University of Chicago Press
Weak equivalence (homotopy theory)
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape".
See Fibration and Weak equivalence (homotopy theory)
References
[1] https://en.wikipedia.org/wiki/Fibration
Also known as Fibration sequence, Fibrations, Hurewicz Fibration, Serre fibration, Weak fibration.