What would you answer if someone asks you,
Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?
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Why are the classification theorems of manifolds so important? Why was the classification of surfaces celebrated?
I suppose there's several closely-related reasons. One would be that manifolds are homogeneous objects -- they look the same near any point in them. So there's no natural thing to start counting if you're trying to classify them. In that regard they're fairly "slippery" objects. Secondly, I suppose this is hind-sight but perhaps you could come to this conclusion from the 1st point, classification of manifolds is a hard problem in that quite a lot of ideas go into their classification. And I suppose a 3rd point is that manifolds are fairly generic geometric objects (according to things like Sard's Theorem) so it's nice to know something about objects that appear frequently in many branches of mathematics and in applied subjects.
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.