# Classifying manifolds

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

• I suppose a general way to look at it is that classifications of objects are natural things to consider in mathematics; by being able to completely classify all objects with some given property, one has developed a deep understanding of the inner workings of the objects in question. Alternatively, a classification shows how 'extreme' objects can get within a set of parameters, and also shows how these extremes are achieved. – ARupinski May 21 '11 at 18:15
• What kind of answer are you looking for beyond "manifolds appear a lot in mathematics, so if we can classify them we can use that classification in a lot of mathematics"? – Qiaochu Yuan May 21 '11 at 18:34
• You may end up getting a lot of different but equally valid answers. Should this be a Community Wiki? – David White May 21 '11 at 18:45
• Perhaps this question would be a better fit for math.stackexchange? – Ryan Budney May 21 '11 at 20:36
• As Clifford Taubes would say, it gives a list of all possible worlds. – mephisto May 23 '11 at 2:16

On many occasions, one finds that when solving some problem (not necessarily directly related to geometry) one needs to understand the structure of a certain set, which might happen to be a differentiable manifold. Knowing, for example, that a set is diffeomorphic to $[0,1]$ or $\mathbb{S}^1$ let's you attack the problem in a concrete way(at least knowing a priori that what you're studying is similar to these concrete manifolds), instead of trying to prove theorems for general manifolds which might or might not be true.