# Category: Differential topology

Lie algebroid
In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of
Regular homotopy
In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. S
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a
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). It is a fiber bundle over M
Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two ve
H-cobordism
In geometric topology and differential topology, an (n + 1)-dimensional cobordism W between n-dimensional manifolds M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion m
Lie algebra bundle
In mathematics, a weak Lie algebra bundle is a vector bundle over a base space X together with a morphism which induces a Lie algebra structure on each fibre . A Lie algebra bundle is a vector bundle
Nonholonomic system
A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differe
L² cohomology
In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrab
Double (manifold)
In the subject of manifold theory in mathematics, if is a manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where fo
Jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sect
Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly sup
Yamabe invariant
In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under
Milnor's sphere
In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of -connected manifolds of dimension (since -connected -
Connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, f : M → N is an immersion if is an i
Akbulut cork
In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compa
Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topol
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topolo
Symplectic space
No description available.
Seifert conjecture
In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if s
Sphere
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the
Double vector bundle
In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent of a vector bundle and the double tangent bundle .
Minimax eversion
In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models. It is a variational method, and consists of special homotopies (they are shortest paths with respe
Plumbing (mathematics)
In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of . It was first described by John Milnor a
Critical value
Critical value may refer to: * In differential topology, a critical value of a differentiable function ƒ : M → N between differentiable manifolds is the image (value of) ƒ(x) in N of a critical point
Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle , the vertical bundle and horizontal b
Cerf theory
In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions on a smooth manifold , their generic singularities
Canonical coordinates
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates a
Clutching construction
In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
Vector flow
In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, R
Associated bundle
In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to ,
Pseudogroup
In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating h
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. It formalizes the id
Glossary of differential geometry and topology
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: * Glossary of general topology * Glossary of algebraic to
Kervaire manifold
In mathematics, specifically in differential topology, a Kervaire manifold is a piecewise-linear manifold of dimension constructed by Michel Kervaire by plumbing together the tangent bundles of two -s
Inverse bundle
In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let be a fibre bundle. A bundle is called the inverse bundle of if their Whitney sum is a
Sphere eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). Remarkably, it is possible to smoot
Spin geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fun
Thom's first isotopy lemma
In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map between smooth manifolds and a closed Whitney stratified subset, if is proper and is a submer
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object is said to be embedde
Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space,
Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "cloc
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 m
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of on
Differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely r
Perfect obstruction theory
In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of: 1. * a perfect two-term complex in the derived category of quasi-coherent étale sheaves on X, a
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying lin
Symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of sy
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often refe
Whitney topologies
In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mapp
Mazur manifold
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold (with boundary) which is not diffeomorphic to the standard 4-ball. The
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).
Kervaire invariant
In mathematics, the Kervaire invariant is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 i
Tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the cont
Gluing axiom
In mathematics, the gluing axiom is introduced to define what a sheaf on a topological space must satisfy, given that it is a presheaf, which is by definition a contravariant functor to a category whi
Two-dimensional Yang–Mills theory
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously de
Donaldson theory
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (19
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in, which generalizes the cup product. The Massey product was created by William S. Massey, an American a
Whitney umbrella
In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface plac
Parallelizable manifold
In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a
Covariant classical field theory
In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields
Timeline of bordism
This is a timeline of bordism, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds. Jean Dieudonné wrote that cobordism returns to the attemp
Obstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic cla
Conley index theory
In dynamical systems theory, Conley index theory, named after Charles Conley, analyzes topological structure of invariant sets of diffeomorphisms and of smooth flows. It is a far-reaching generalizati
Implicit function
In mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is An implicit fun
Serre–Swan theorem
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules a
Symplectization
In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Kervaire semi-characteristic
In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire, is an invariant of closed manifolds M of dimension taking values in , given by . Michael Atiyah and Isadore Singer show
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types
Cotangent space
In differential geometry, the cotangent space is a vector space associated with a point on a smooth (or differentiable) manifold ; one can define a cotangent space for every point on a smooth manifold
Lie derivative
In differential geometry, the Lie derivative (/liː/ LEE), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-fo
In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field. The primary motivation is as a te
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Secondary vector bundle structure
In mathematics, particularly differential topology, the secondary vector bundle structurerefers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smo
Stunted projective space
In mathematics, a stunted projective space is a construction on a projective space of importance in homotopy theory, introduced by James. Part of a conventional projective space is collapsed down to a
Atiyah conjecture
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
Thom's second isotopy lemma
In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces i
Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic
Degree of a continuous mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the r
Sharp map
In differential geometry, the sharp map is the mapping that converts 1-forms into corresponding vectors, given a non-degenerate (0,2)-tensor.
Disc theorem
In the area of mathematics known as differential topology, the disc theorem of states that two embeddings of a closed k-disc into a connected n-manifold are ambient isotopic provided that if k = n the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvatur
Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows wi
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of i
Polyvector field
A Polyvector field within Mathematics topology is concerned with the properties of a geometric object. A multivector field, polyvector field of degree k , or k-vector field, on a manifold , is a gener
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces o
Pontryagin class
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple o
Vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division alg
Neat submanifold
In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. To define this more precisely, first let be a manifold with bo
Generalized Stokes theorem
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement ab
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that i
Whitney conditions
In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A stratification of a topologica
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 e
Congruence (manifolds)
In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences are an important concept in general relativity
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. It may be described also as
Introduction to gauge theory
A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a
Integrability conditions for differential systems
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differe
Partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval [0,1] such that for every point : * there is a neighbourhood of where all but a
Donaldson's theorem
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalis
Band sum
In geometric topology, a band sum of two n-dimensional knots K1 and K2 along an (n + 1)-dimensional 1-handle h called a band is an n-dimensional knot K such that: * There is an (n + 1)-dimensional 1-
Exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a s
Topological degree theory
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely c