Editing Differential structure (section)

==Differential structures on spheres of dimension 1 to 20==

{{Main|Exotic sphere}}

The following table lists the number of smooth types of the topological ''m''−sphere ''S''<sup>''m''</sup> for the values of the dimension ''m'' from 1 up to 20. Spheres with a smooth, i.e. ''C''<sup>∞</sup>−differential structure not smoothly diffeomorphic to the usual one are known as [[exotic sphere]]s.

{| class="wikitable"
|-
! Dimension
! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|-
! Smooth types
| 1 || 1 || 1 || ≥1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
|}

It is not currently known how many smooth types the topological 4-sphere ''S''<sup>4</sup> 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 ''S''<sup>4</sup> 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).