Differential topology
The differential topology is a branch of mathematics . It examines global geometric invariants that are not defined by a metric or a symplectic shape . The examined invariants are mostly invariants of topological spaces , which additionally have a differentiable structure , i.e. differentiable manifolds . For example, De Rham cohomology makes a connection between analytical properties and topological invariants of the manifold. Means of analysis and the theory of differential equations are often used to obtain information about the topology of space. This happens, for example, in the Morse theory or the Yang-Mills theory, which comes from physics .
The latter leads to so-called exotic R 4 s, i.e. H. four-dimensional Euclidean spaces that are homeomorphic , but not diffeomorphic to the standard R 4 . Such exotic spaces only appear from dimension four. Another well-known example is Milnor's exotic 7- spheres . Its discovery in 1956 marked a decisive turning point in differential topology.
Bernhard Riemann and Henri Poincaré are pioneers of modern differential topology . Important representatives in the 20th century are Hassler Whitney , John Willard Milnor and Simon Donaldson . Recent developments have demonstrated the physics compounds for which especially the string theorists and Fields Medal -carrier Edward Witten is.
literature
- Morris W. Hirsch : Differential Topology. Corrected 5th printing. Springer-Verlag, New York, NY et al. 1994, ISBN 3-540-90148-5 ( Graduate Texts in Mathematics 33).
- Theodor Bröcker , Klaus Jänich : Introduction to the differential topology . Springer-Verlag, Heidelberger Taschenbücher 1973, reprinted 1990, ISBN 3540064613 .