Differential topology is the topology that studies differential manifolds and differentiable maps. With the progress of algebraic topology and differential geometry, it re-emerged in the 1930s. 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 the differential manifold, he also proposed the concept of fiber bundles, so that many geometric problems are related to homology (indicative class) and homotopy problems.
In 1953, Rene Thom's theory of collocation created a situation where differential topology and algebraic topology were advancing side by side. Many difficult differential topology problems were transformed into algebraic topology problems and solved, which also stimulated algebraic topology. Further development. In 1956, Milno discovered that in addition to the usual differential structure on the seven-dimensional sphere, there was also an unusual differential structure. Subsequently, the manifolds that cannot be assigned any differential structure were constructed by humans. These all show that the three categories of topological manifolds, differential manifolds, and piecewise linear manifolds in between have huge Difference, differential topology has since been recognized as an independent branch of topology. In 1960, Smail proved the Poincaré conjecture for differential manifolds with more than five dimensions. J.W. Milno et al. developed a basic method to deal with differential manifolds ─ ─ 剜讓擜, so that the classification of manifolds with more than five dimensions has gradually become algebraic.
The prominent areas are the relationship between the above three categories of manifolds and the classification of three-dimensional and four-dimensional manifolds. The major achievements in the early 1980s included the proof of the four-dimensional Poincaré conjecture and the discovery of the unusual differential structure in four-dimensional Euclidean space. This kind of research is generally called geometric topology in order to emphasize its geometric color, which is different from the algebraic homotopy theory.