Differential topology

The differential topology is a branch of mathematics. There will be examines global geometric invariants that are not defined by a metric or a symplectic form. The studied invariants are usually invariants of topological spaces, which additionally carry a differentiable structure, ie differentiable manifolds. For example, the De Rham cohomology of a link between analytical properties and topological invariants of the manifold. Often means of analysis and the theory of differential equations are used to obtain information about the topology of the space. This happens for example in the Morse theory, or coming from the physics Yang-Mills theory.

The latter leads to so-called exotic R4s, ie four-dimensional Euclidean space, which, while homeomorphic but not diffeomorphic standard -R4. Such exotic rooms are available only from dimension four. Another prominent example is the Milnor's exotic 7- spheres. Its discovery in 1956 was the decisive turning point in the differential topology dar.

Pioneer of modern differential topology are Bernhard Riemann and Henri Poincaré. Important representatives in the 20th century, Hassler Whitney, John Willard Milnor and Simon Donaldson. Recent developments have connections to physics demonstrated, for especially the string theorists and Fields Medal carrier is Edward Witten.

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