Classifying Manifolds: Why Celebrated?

[jem05]

What yould you answer if a professor asks you,

Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?

 

[Tinyboss]

Mathematicians spend a lot of time and effort classifying all kinds of mathematical objects. It let's us prove results like, "for any manifold (module, group, etc.) X, statement Y is true". Without a classification of manifolds (modules, groups, etc.), how would you go about proving such a claim?

Often classification of all objects of some type is too difficult, and we choose to restrict our attention to some subclass in order to get useful results. For instance, instead of investigating all manifolds, we might look at closed 2-manifolds. Instead of all groups, we can look at finite simple groups. Instead of all modules, we can examine finitely-generated modules over a PID.

In general, the stronger the restrictions we place on the objects of study, the stronger the results we can prove. The really good theorems are the ones that give us very useful results while placing only mild restrictions on the objects they apply to.

 

[lavinia]

What yould you answer if a professor asks you,

Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?

I would say that they are important because they lead to new techniques that apply more generally.