Introduction to Differential Topology by Prof. Justin Sawon
Differential topology is a subject in which geometry and analysis are used to obtain topological invariants of spaces, often numerical. Some examples are the degree of a map, the Euler number of a vector bundle, the genus of a surface, the cobordism class of a manifold (the last example is not numerical).
A manifold is a topological space which is locally homeomorphic to R^n. It is made of open subsets of R^n glued together by homeomorphisms. If these gluing maps are diffeomorphisms, then we obtain a smooth (or differentiable) manifold. These are the basic objects of study of differential topology. They are also the basic objects of study of differential geometry, but whereas differential geometry is concerned mainly with local invariants (for example, curvature), differential topology is concerned with global issues.
Manifold occur in many branches of mathematics, for example as Lie groups in algebra, as space-time in relativity, as phase-space in mechanics, as state-space in dynamics and differential equations. An important idea in differential topology is the passage from local to global information. As an illustration of the distinction consider differential equations : two solutions which may look similar locally can turn out to have very different global properties (one may be periodic, the other convergent).
In the above examples there is typically additional structure, such as a group structure (multiplication and inverses), a metric, or a symplectic structure. In differential topology we are interested in the manifold itself and the additional structure is just a tool. This is similar to algebraic topology, which employs triangulations and uses combinatorial methods to obtain topological information.
As an example, let's look at the Euler characteristic of the sphere S^2. In algebraic topology we have a combinatorial definition. We take a triangulation, and then count the number of vertices, minus the number of edges, plus the number of faces, to get two. In the case of a Platonic solid this is just Euler's theorem. In differential topology we count the number of zeros of a smooth vector field, weighted by their indices, and once again get two. There can be no non-vanishing smooth vector field, a fact known as the hairy ball theorem. Finally, a more geometric definition requires us to choose a metric on the sphere then take the Gaussian curvature K of the corresponding connection. Integrating over the sphere we one again get two (up to a factor of 2Pi). This is the Gauss-Bonnet formula. In all of these definitions we employ some additional structure (a triangulation, a vector field, a metric) but ultimately define a topological invariant which is independent of the choices made.
From a topological point of view, manifolds have no local invariants. They look the same at every point, and there are symmetries taking a given point to any other in the same connected component. The kinds of questions that one asks in differential topology are therefore global. For example, can we embed one manifold M in another N? If M is homeomorphic to N, is it diffeomorphic to N? Given M, does there exist N such that M is the boundary of N?
Note: The above text is an excerpt from the course synopsis of MAT 566, Fall 2002, taught at State University of New York/Stony Brook by Prof. Justin Sawon (now teaching in the Mathematics Dept at Colorado State). Permission to use the above excerpt was duly granted by author on 15 January 2007 via E-mail correspondance.No copyright assigned.