Differential Structure
It is not currently known how many smooth types the topological 4-sphere S4 has, except that there is at least one. There may be one, a finite number, or an infinite number. The claim that there is just one is known as the smooth Poincaré conjecture (see generalized Poincaré conjecture). Most mathematicians believe that this conjecture is false, i.e. that S4 has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).
Read more about Differential Structure: Differential Structures On Topological Manifolds
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