Differential topology is a branch of mathematics that focuses on the study of differentiable manifolds and the differentiable maps between them. A differentiable manifold is a space that, locally, resembles Euclidean space and allows for the notion of differentiability. This field explores how the shapes of these manifolds can be understood and categorized through the properties of differentiable functions. Differential topology is closely related to differential geometry; however, while differential geometry is concerned with metrics and curvature, differential topology primarily deals with properties that are invariant under smooth deformations, known as diffeomorphisms.
One of the central concepts in differential topology is the idea of a smooth manifold. A smooth manifold is a manifold equipped with a globally defined differential structure. It can be thought of as a geometric object that is a higher-dimensional analogue of a curve or surface. For instance, a 2-dimensional sphere or a torus can be considered smooth manifolds. The study of these manifolds includes examining their local properties (like tangent spaces and vector fields) and their global properties, such as their topology and the existence of particular types of differentiable functions.
In differential topology, much attention is given to the concept of smooth maps and diffeomorphisms. A smooth map is a function between manifolds that is infinitely differentiable, and a diffeomorphism is a bijective smooth map whose inverse is also smooth. Diffeomorphisms are of particular interest because they allow the exploration of manifold structures that are "essentially the same" in the smooth sense, helping to classify manifolds up to smooth structure. This classification can involve intricate techniques and theorems, such as the famous Morse theory, which relates the topology of manifolds to the critical points of smooth functions defined on them.
Moreover, differential topology utilizes tools like vector bundles and fiber bundles to further analyze and understand manifolds. A vector bundle is a topological construction that associates a vector space with each point of a manifold, smoothly varying from point to point. This concept is crucial for understanding phenomena such as tangential and normal spaces, which are fundamental in fields like physics and engineering. Fiber bundles generalize this idea by allowing the fiber to be any type of space, not just a vector space, leading to richer structures and more generalized applications. Through these and other constructs, differential topology offers profound insights into the mathematical framework underlying many aspects of both theoretical and applied science.