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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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    $\begingroup$ 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. $\endgroup$ – ARupinski May 21 '11 at 18:15
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    $\begingroup$ 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"? $\endgroup$ – Qiaochu Yuan May 21 '11 at 18:34
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    $\begingroup$ You may end up getting a lot of different but equally valid answers. Should this be a Community Wiki? $\endgroup$ – David White May 21 '11 at 18:45
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    $\begingroup$ Perhaps this question would be a better fit for math.stackexchange? $\endgroup$ – Ryan Budney May 21 '11 at 20:36
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    $\begingroup$ As Clifford Taubes would say, it gives a list of all possible worlds. $\endgroup$ – mephisto May 23 '11 at 2:16
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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.

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    $\begingroup$ To amplify the 3rd point, manifolds can be viewed as the simplest examples of varieties (namely, as singularity-free varieties). Thus any classification of manifolds constrains any possible classification of varieties. Similarly, geometric methods for composing manifolds (e.g., tensor products) are isomorphic to algebraic methods for composing varieties (e.g., joins). $\endgroup$ – John Sidles May 22 '11 at 1:21
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    $\begingroup$ Ryan, your second point leads me back to the dangerous precipice question I've struggled with occasionally: I wonder how often as mathematicians we conclude something is important mostly because it is hard and required a lot of work or we haven's solved it yet. $\endgroup$ – Greg Friedman May 22 '11 at 4:30
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    $\begingroup$ Manifolds forced major ideas onto mathematics (homology, fundamental group, eventually category theory, etc) which changed the landscape of mathematics. $\endgroup$ – Ryan Budney May 22 '11 at 5:01
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    $\begingroup$ It is striking that Ryan's list of major ideas (homology, fundamental group, eventually category theory, etc) is not a list of theorems, but rather is a list of definitions. Michael Spivak has plainly stated the same point: "A fully evolved major theorem has three important attributes: (1) It is trivial. (2) It is trivial because the terms appearing in it have been properly defined. (3) It has significant consequences." Thus, classification theorems are important (according to Spivak) mainly because their proofs require us to create and/or embrace proper definitions. $\endgroup$ – John Sidles May 22 '11 at 13:07
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    $\begingroup$ The quotation of Spivak is shortly after the proof of Stokes's theorem on page 104 in his Calculus on Manifolds, and he only says that the three features are shared by many major theorems. I think classification theorems form a class of strong counterexamples to any claims of triviality. In particular, when the family of objects in question (e.g., finite simple groups, compact Lie groups) has exceptional objects or exceptional families, one is typically forced to chase down cases. Clearly some people find this aesthetically unappealing, but I think it is a fact of life. $\endgroup$ – S. Carnahan May 23 '11 at 2:16
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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.

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