Topology was originally called situation analysis, a term put forward by German mathematician Leibniz 1679. /kloc-In the mid-9th century, German mathematician Riemann emphasized that the study of functions and integrals requires the study of situational analysis. From then on, a systematic study of modern topology began. Topology does not discuss the concept of congruence between two graphs, but the concept of topological equivalence. For example, circles, squares and triangles have different shapes and sizes, but they are all equivalent graphs under topological transformation; Football and rugby are also equivalent-from a topological point of view, their topological structure is exactly the same.
The surface of swimming ring and the surface of football have different topological properties, for example, there is a "hole" in the middle of swimming ring. In topology, the space represented by football is called sphere, the space represented by swimming ring is called torus, and sphere and torus are "different" spaces. Topology was originally called situation analysis, a term put forward by German mathematician Leibniz 1679. Euler solved the seven-bridge problem in 1736 and published the polyhedron formula in 1750. In 1833, Gauss defined the encircling number of two closed curves in space by line integral in electrodynamics. The word topology was put forward by J.B. Listing (1847) and originated from the Greek word τ? πο? And λ? γο? ("location" and "research"). This is the embryonic stage of topology.
185 1 year, German mathematician Riemann put forward the geometric concept of Riemann surface in the study of complex variable functions, and stressed that in order to study functions and integrals, it is necessary to study situation analysis. Riemann solved the homeomorphism classification problem of orientable closed surfaces by himself.
The founder of combinatorial topology is French mathematician Poincare. He introduced topological problems in his work of analysis and mechanics, especially in the study of complex functions and single values of curves determined by differential equations. His main interests are manifold. During 1895 ~ 1904, he established the basic method of studying manifolds by subdivision. He introduced many invariants: basic group, homology, Betty number and torsion coefficient, discussed the topological classification of three-dimensional manifolds and put forward the famous Poincare conjecture.
Another source of topology is the rigor of analysis. The strict definition of real numbers pushed Cantor to systematically study point sets in Euclidean space from 1873, and obtained many topological concepts, such as aggregation point (limit point), open set, closed set, density, connectivity and so on. Under the influence of set theory, the concept of functional (that is, function) appeared in the analysis, and the function set was taken as a geometric object, and its limit was discussed. This eventually leads to the concept of abstract space. It was Fréchet who first studied abstract space. He introduced the concept of metric space in 1906. In the Outline of Set Theory (19 14), F Hausdorf defined a relatively general topological space with an open neighborhood, which marked the emergence of the general topology that used axiomatic methods to study continuity. Subsequently, the Polish School and the Soviet School systematically studied the basic properties of topological space (separability, compactness, connectivity, etc.). ). Post-complement (consistency space, paracompactness, etc. ) and the arrangement of the Bourbaki School Since the mid-1930s, general topology has matured and become the same basis for mathematical research after the Second World War.
For example, the study of point set in Euclidean space has always been an important part of topology, which has developed to the intersection of general topology and algebraic topology, and can also be regarded as a part of geometric topology. Since 1950s, the work of American school, represented by R.H. Bing, has deepened the understanding of manifolds and raised the question of whether two given mappings are homotopy, which has played a role in proving the four-dimensional Poincare conjecture. The study of dimension and continuum caused by peano curve is also regarded as a branch of general topology. During the period of19/kloc-0 ~1912, L. E.J Brouwer proposed the method of approximating continuous mapping with simple mapping. Many important geometric phenomena are used to prove that different embryos are different in Euclidean space of different dimensions. The mapping degree between manifolds of the same dimension is introduced to study homotopy classification, and the fixed point theory is established. He made combinatorial topology reach its due level in terms of precise concepts and rigorous argumentation. Then, J.W. Alexander proved the topological invariance of Betty number and torsion coefficient in 19 15.
With the rise of abstract algebra, around 1925, A.E. Nott proposed to establish combinatorial topology on the basis of group theory. Under her influence, H. hopf defined the homology group in 1928. Since then, combinatorial topology has gradually evolved into algebraic topology that uses abstract algebra to study topological problems. For example, dimensions and Euler numbers, S. Allen Berg and N. E. Stranrod summed up the theory of homology at that time in an axiomatic way in 1945, and later wrote "Fundamentals of Algebraic Topology" (1952), which greatly promoted the spread, application and further development of algebraic topology. They summarized the basic spirit of algebraic topology as: transforming topological problems into algebraic problems and solving them through calculation. Until today, the invariants provided by homology theory are still the most easily calculated and commonly used invariants in topology. Homotopy ethics studies the homotopy classification of space and mapping. Wolonid Hurwicz introduced the N-dimensional homotopy group of topological space from 1935 to 1936. Its elements are homotopy classes of mapping from N-dimensional sphere to this space, and the one-dimensional homotopy group is the basic group. Homotopy group provides another transition from topology to algebra, and its geometric meaning is more obvious than homology group, but it is extremely difficult to calculate. The calculation of homotopy groups, especially spherical homotopy groups, promotes the development of topology and produces colorful theories and methods. 1950, the French mathematician Searle used the spectral sequence algebra tool developed by J Leray to study the homology theory of fiber bundles and made a breakthrough in the calculation of homotopy groups.
Since the late 1950s, under the influence of algebraic geometry and differential topology, K- theory and several other generalized homology theories have emerged. It is a transition from topology to algebra. Although the geometric meaning is different, the algebraic properties are very similar to homology or cohomology, which is a powerful weapon of algebraic topology. Theoretically, it is obvious that homology theory (ordinary and generalized) is essentially a part of homotopy theory. Differential topology is a topology that studies differential manifolds and differentiable mappings. With the development of algebraic topology and differential geometry, it reappeared in 1930' s. H. Whitney gave a general definition of differential manifold in 1935, and proved that it can always be embedded in high-dimensional Euclidean space. In order to study the vector field on differential manifold, he also put forward the concept of fiber bundle, which linked many geometric problems with homology (indicator class) and homotopy problems.
1953, R.Thom's edge matching theory created a situation in which differential topology and algebraic topology kept pace. Many difficult differential topology problems were solved by transforming them into algebraic topology problems, which also stimulated the further development of algebraic topology. 1956, Milnor discovered that in addition to the usual differential structure, there are unusual differential structures on the seven-dimensional sphere. Subsequently, manifolds that can't be given any differential structure are reconstructed, which shows that there are great differences between topological manifolds, differential manifolds and piecewise linear manifolds between them, and differential topology has been recognized as an independent branch of topology since then. 1960, Smale proved the Poincare conjecture of differential manifolds with more than five dimensions. J.W. Milnor and others have developed the basic method of dealing with differential manifolds ── planing technology, which makes the classification of manifolds with more than five dimensions tend to algebra gradually.
In recent years, manifold research has made a lot of progress, such as the discipline of geometry and geometric methods. Prominent places such as the relationship between the above three types of manifolds and the classification of three-dimensional and four-dimensional manifolds. Major achievements in the early 1980s include the proof of four-dimensional Poincare conjecture and the discovery of unusual differential structures in four-dimensional Euclidean space. This kind of research is usually called geometric topology to emphasize its geometric color, which is different from the same ethics with strong algebraic flavor.