Geometric topology

The geometric topology is a branch of mathematics that deals with manifolds and their embeddings. As deputy topics are mentioned here knot theory and braid groups. Over time, the term was increasingly used almost synonymous with low-dimensional topology, and this applies in particular to two -, three - and four-dimensional objects.

In the rapid development of topology after 1945 a distinction between the following areas has been met:

• Algebraic Topology, typified by the homotopy
• The geometric topology of the Poincaré conjecture as its largest, now dissolved, problem
• The differential topology made ​​up mostly with differentiable structures, with the Morse theory as a natural technique employed.

These areas are all based on the general or set-theoretic topology, which includes the study of general topological spaces. This subdivision will always artificial over the years.

History

As with the set-theoretic topology can not be clearly distinguished when this branch of mathematics was born. One of the first sets of the topology was the Jordan curve theorem. He says that the plane can be decomposed by a closed Jordan curve into two disjoint components, which is exactly one restricted. The sentence was formulated in 1887 by Camille Jordan, but his proof was flawed. The first correct proof has been furnished in 1905. The first classic result of geometric topology is the set of Schönflies. In 1910 this was proved by Arthur Moritz Schoenflies. Clearly it says that a closed Jordan curve can always be distorted into a circle. This statement can be understood as a generalization of the Jordan curve theorem. In 1908, the conjecture was put forward by Ernst Steinitz and Heinrich Tietze that every manifold having at least one triangulation and that two different triangulations have a common refinement. The second part of the statement is called the main conjecture of Steinitz. Tibor Radó showed in 1925 that the conjecture for surfaces is correct. For the three dimension, the conjecture was proved in 1952 by Edwin Moise. For dimensions greater than three, the main conjecture, however, does not apply. This was proved in 1961 by John Willard Milnor.

Some progress since the beginning of the 1960s meant that changed the geometric topology. The solution of the Poincaré conjecture in higher dimension by Stephen Smale in 1961 had the dimensions three and four seem to be the most difficult. And in fact, they required new methods, while the freedoms in the higher dimension meant that problems could be attributed to in surgery theory ( en. surgery ) available, calculating methods. The William Thurston formulated in the late 1970s geometrization presented a basic framework to disposal, which showed how strong geometry and topology are connected to each other in low dimensions. Thurston's proof of the geometrization of hook -manifolds used a wide range of tools from previously only weakly related to each other are areas of mathematics. Vaughan Jones ' discovery of the Jones polynomial in the early 1980s not only led the knot theory in new directions, but also gave the still unsettled relationship between low-dimensional topology and mathematical physics buoyancy.