Geometric topology
The geometric topology is a branch of mathematics that deals with manifolds and their embeddings busy. The knot theory and braid groups should be mentioned here as representative topics . Over time, the term became more and more used almost synonymously 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 was made between the following areas:
- the algebraic topology , embodied by the homotopy theory
- the geometrical topology with the Poincaré conjecture as its biggest, now solved, problem
- the differential topology, which is largely concerned with differentiable structures , with Morse theory as a natural technique.
These areas are all based on general or set theoretical topology , which includes the study of general topological spaces. This division seems more and more artificial over the years.
history
As with set theoretic topology, it cannot be clearly defined when this sub-area of mathematics came into being. One of the first sentences in topology was the Jordanian curve theorem . It says that the plane can be broken down into two disjoint components by means of a closed Jordan curve , of which exactly one is bounded . The sentence was formulated by Camille Jordan in 1887 , but his proof was flawed. The first correct proof was given in 1905. The first classic result of geometric topology is Schönflies' theorem . In 1910 this was proven by Arthur Moritz Schoenflies . It clearly states that a closed Jordan curve can always be distorted into a circle . This statement can be understood as a generalization of the Jordanian curve theorem. In 1908, Ernst Steinitz and Heinrich Tietze suggested that every manifold has at least one triangulation and that two different triangulations have a common refinement. The second part of the statement is called the Steinitz Main Conjecture . Tibor Radó was able to show in 1925 that the conjecture for areas is correct. For dimension three, Edwin Moise was able to prove the conjecture in 1952 . However, the main assumption does not apply to dimensions greater than three. This was proven by John Willard Milnor in 1961 .
A number of advances since the early 1960s have resulted in the geometric topology changing. The solution of the Poincaré conjecture in higher dimensions by Stephen Smale in 1961 made dimensions three and four appear to be the most difficult. And indeed they required new methods, while the freedoms meant in the higher dimension that questions on the surgery theory (s. Surgery available, calculating methods were reduced). The geometrization conjecture formulated by William Thurston in the late 1970s provided a framework that showed how strongly geometry and topology are connected in low dimensions. Thurston's proof of the geometrization of hook manifolds used a wide range of tools from previously weakly related branches of mathematics. Vaughan Jones ' discovery of the Jones polynomial in the early 1980s not only took knot theory in new directions, but also gave impetus to the still unexplained relationships between low-dimensional topology and mathematical physics .
Web links
Individual evidence
- ^ Edwin E. Moise: Geometric topology in dimensions 2 and 3 . Springer-Verlag, New York 1977, ISBN 978-0-387-90220-3 , Preface.