Editing Exotic sphere (section)

==Applications==
If ''M'' is a [[piecewise linear manifold]] then the problem of finding the compatible smooth structures on ''M'' depends on knowledge of the groups {{nowrap|1=Γ<sub>''k''</sub> = Θ<sub>''k''</sub>}}. More precisely, the obstructions to the existence of any smooth structure lie in the groups {{nowrap|H<sub>''k+1''</sub>(''M'', Γ<sub>''k''</sub>)}} for various values of ''k'', while if such a smooth structure exists then all such smooth structures can be  classified using the groups {{nowrap|H<sub>''k''</sub>(''M'', Γ<sub>''k''</sub>)}}.
In particular the groups Γ<sub>''k''</sub> vanish if {{nowrap|''k'' &lt; 7}}, so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6.

The following finite abelian groups are essentially the same:
*The group Θ<sub>''n''</sub> of h-cobordism classes of oriented homotopy ''n''-spheres.
*The group  of h-cobordism classes of oriented ''n''-spheres.
*The group  Γ<sub>''n''</sub> of twisted oriented ''n''-spheres.
*The homotopy group {{pi}}<sub>''n''</sub>(PL/DIFF)
*If {{nowrap|''n'' ≠ 3}}, the homotopy group {{pi}}<sub>''n''</sub>(TOP/DIFF) (if {{nowrap|1=''n'' = 3}} this group has order 2; see [[Kirby–Siebenmann invariant]]).
*The group of smooth structures of an oriented PL ''n''-sphere.
*If {{nowrap|''n'' ≠ 4}}, the group of smooth structures of an oriented topological ''n''-sphere.
*If {{nowrap|''n'' &ne; 5}}, the group of components of the group of all orientation-preserving diffeomorphisms of ''S''<sup>''n''&minus;1</sup>.