Differential topology is the field of mathematics dealing with the topological and smooth properties of smooth manifolds. It studies coarse structural features, such as the number of holes in a manifold, its homotopy type and the composition of its diffeomorphism group, rather than the geometric measurements of size and distance that belong to differential geometry, and it has substantial ties to algebraic topology because many of its properties can be expressed in algebraic terms.
Facts
Central QuestionDifferential topology asks how smooth manifolds can be classified up to a smooth, invertible change of coordinates, a diffeomorphism, and which coarse features, such as the number of holes or the homotopy type, survive that classification. 1 Key DebateWhether every smooth 4-manifold homeomorphic to the 4-sphere is also diffeomorphic to it, the smooth four-dimensional Poincare conjecture, remains one of the field's central open problems; the analogous statement is known to be true in dimensions 1 through 3 and known to be false in dimension 7. 1 Classification
Pure or Applied Connections
Includes
Source Donaldson's theorem, Wikipedia
Source Hassler Whitney (Wikipedia)
Source Wikipedia: Hopf fibration
Source Local Diffeomorphism (Wikipedia)
Source Odd number theorem - Wikipedia
Source Poincare-Hopf theorem, Wikipedia
Source Sphere eversion, Wikipedia
Sources
1. Differential Topology (Wikipedia)
Wikimedia FoundationLead section
The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism.
Body section on smooth structures in dimension 4
One of the central open problems in differential topology is the four-dimensional smooth Poincaré conjecture, which asks if every smooth 4-manifold that is homeomorphic to the 4-sphere, is also diffeomorphic to it.
View the Source Odd number theorem - Wikipedia
Includes: Odd Number Theorem, Lead sentenceQuote, Includes: Odd Number Theorem, Lead sentence
The odd number theorem is a theorem in differential topology about strong gravitational lensing, which states that the number of m
Local Diffeomorphism (Wikipedia)
Includes: Local Diffeomorphism, Lead sentenceQuote, Includes: Local Diffeomorphism, Lead sentence
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that
Poincare-Hopf theorem, Wikipedia
Includes: Poincare-Hopf Theorem, Lead sentenceQuote, Includes: Poincare-Hopf Theorem, Lead sentence
or Hopf index theorem) is an important theorem in differential topology that relates zeros of continuous vector field on compact d
Wikipedia: Hopf fibration
Includes: Hopf Fibration, Lead sentenceQuote, Includes: Hopf Fibration, Lead sentence
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in fo
Sphere eversion, Wikipedia
Includes: Smale's Theorem (Sphere Eversion), Lead sentenceQuote, Includes: Smale's Theorem (Sphere Eversion), Lead sentence
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (th
Donaldson's theorem, Wikipedia
Includes: Donaldson's Theorem, Lead sentenceQuote, Includes: Donaldson's Theorem, Lead sentence
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection for
Hassler Whitney (Wikipedia)
Includes: Hassler Whitney, Lead paragraph [in-branch]Quote, Includes: Hassler Whitney, Lead paragraph [in-branch]
manifolds, embeddings, immersions
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