This group gathers theorems about manifolds and the shapes of low-dimensional spaces, the part of topology concerned with surfaces, curves, knots, and smooth structure. It covers the Jordan curve theorem and the Schoenflies theorem on closed curves in the plane, the classification of surfaces, the Poincare conjecture and Thurston's geometrization theorem on three-dimensional spaces, Freedman's and Donaldson's theorems on four-dimensional manifolds, the Poincare-Hopf theorem on vector fields, and Smale's theorem on turning a sphere inside out.
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Geometric and Differential Topology
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Poincare Conjecture (Wikipedia)
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In the mathematical field of geometric topology, the Poincaré conjecture (UK: , US: , French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds the 4-ball in four-dimensional space).
Jordan Curve Theorem (Wikipedia)
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In topology, a discipline within mathematics, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every simple closed curve in the plane (known as a Jordan curve) divides the plane into two regions: the interior, bounded by the curve, and an unbounded exterior, containing all of the nearby and faraway exterior points.
Schoenflies Theorem (Wikipedia)
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In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies.
Classification of surfaces (Wikipedia)
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In topology, a surface is a two-dimensional manifold.
Poincare-Hopf theorem, Wikipedia
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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 in differential topology that relates zeros of continuous vector field on compact differential manifold to Euler characteristic of that manifold.
Wikipedia: Freedman classification
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In topology in mathematics, Freedman's classification (or Freedman's theorem) is a central result about four-dimensional topological manifolds (short 4-manifolds).
Geometrization conjecture, Wikipedia
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In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it.
Donaldson's theorem, Wikipedia
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In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable.
Sphere eversion, Wikipedia
Has Member: Smale's Theorem (Sphere Eversion), Lead paragraphQuote, Has Member: Smale's Theorem (Sphere Eversion), Lead paragraph
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out").
Whitney embedding theorem (Wikipedia)
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particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney
Whitney immersion theorem (Wikipedia)
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In differential topology, the Whitney immersion theorem (named after Hassler Whitney)
H-cobordism (Wikipedia)
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In geometric topology and differential topology, an (n + 1)-dimensional cobordism
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