What is a topological invariant?

This note provides a general picture of a classical problem in differential topology and geometry, namely the classification of smooth manifolds. A natural question is: Where do these objects come from? As in many branches of mathematics, physics provides many examples of interesting mathematical objects: the dynamics of mechanical systems involve "spaces" with many degrees of freedom, for example the dynamic of a particle moving under the action of a field of forces. How do we describe this dynamic? The first step is to find a model for all the possible configurations of the particle. This leads to the notion of a manifold. A smooth manifold of dimension d is an abstract topological space which is locally diffeomorphic to R^d . Roughly, a smooth manifold consist of small euclidean pieces glued together by diffeomorphisms. Examples of manifolds are the torus, the sphere and the Klein bottle. This is the image to have, but we should not think of a manifold as always sitting inside a euclidean space, but rather as an abstract object. Indeed, the space of all k-dimensional linear subspaces of a given vector space V has a natural manifold structure, but it is clear that this manifold is not contained in a euclidean space (it is quite a nice exercise to find the local model of this space!). It follows from the definition that two d-dimensional manifolds are locally the same space, but of course they are not necessarily the same global space: the 2-sphere and the 2-torus are not the same global object. How do we know if two manifolds are globally the same space or not? This motivates the problem of finding topological invariants, this means an algebraic data naturally attached to a manifold, where "naturally" means that such a data has to be preserved by homeomorphisms. The first example of a topological invariant is given by the following classification result.

Theorem 1. If M is a compact oriented 2-dimensional manifold then there exists an integer number g such that M is homeomorphic to a 2-sphere with g handles.

We refer to a 2-manifold as a surface. The integer number g is called the genus of a surface. The result above is known as the classification of compact surfaces and it says that every surface is completely determined by its genus. Hence, a 2-sphere is not globally a 2-torus, since the sphere has genus 0 and the torus has genus 1!.

What about higher dimensions?

The philosophy is simple, we need to find invariants. The idea of finding invariants goes back to Poincare and his algebraic study of manifolds. Why algebra? the answer is pretty simple, algebra allows you to make computations, so if we were able to attach an algebraic data to a manifold, then we are in algebra territory, so let us make computations and then translate our computations to a topological language. For example, consider a manifold and let us define its Poincare group: the elements of the group are loops in M identified up to homotopy and the group structure is given by concatenation of loops. This is the fundamental group of a manifold. The fundamental group is an algebraic object with the following property: two manifolds which are globally the same object, then their fundamental groups are necessarily isomorphic. The fundamental group does not classify manifolds since there exist examples of non homeomorphic manifolds with the same fundamental group (the cylinder and the circle). So if our interest in classifying manifolds remains, we need to look for more invariants, more subtle invariants. Classical invariants associated to a manifold are homotopy groups or homology and cohomology groups, but these invariants are not powerful enough to classify smooth manifolds...so...

That's all folks?

Now we bring geometry into the picture! In spite of the fact that we shall reduce even more the type of manifolds to be classified, we consider smooth manifold with extra geometrical structures as: Riemannian manifolds, complex manifolds, algebraic manifolds, symplectic manifolds, or manifolds acted on by a group of symmetries (ie; by a Lie group). The geometrical point of view has shown to be in the right direction.

Theorem 2. (Thurston) There exist just eight different types of 3-dimensional compact smooth manifolds.

The theorem above is known as Thurston geometrization conjecture. It was proved by G. Perelman (Fields medal 2006) using hard geometric-analytical techniques as the theory of geometric flows (Ricci flow and PDEs). In particular the geometrization of 3-manifolds gives a solution to a very famous problem in topology:

Conjecture 1. (Poincare) A compact 3-manifold "without holes" has the topology of the sphere.

In summary, in dimension 2, 3 we have satisfactory answers to the problem of classifying smooth manifolds, but we have not solved our problem at all, for example:

What about dimension 4?

Now we need to improve our techniques and the main idea to solving this question is to bring physics into the picture (notice that 4 is exactly the dimension of the space-time). This leads to the work of S. Donaldson (Fields Medal 1986) and the theory of connections on principal bundles, but we postpone the answer to a future note.