Exotic Sphere - Applications

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 Γ_k = Θ_k. More precisely, the obstructions to the existence of any smooth structure lie in the groups H_k+1(M, Γ_k) for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups H_k(M, Γ_k). In particular the groups Γ_k vanish if k<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_n of h-cobordism classes of oriented homotopy n-spheres.
• The group of h-cobordism classes of oriented n-spheres.
• The group Γ_n of twisted oriented n spheres.
• The homotopy group π_n(PL/DIFF)
• If n ≠ 3, the homotopy π_n(TOP/DIFF) (if n=3 this group has order 2; see Kirby–Siebenmann invariant).
• The group of smooth structures of an oriented PL n-sphere.
• If n≠4, the group of smooth structures of an oriented topological n-sphere.
• If n≠5, the group of components of the group of all orientation-preserving diffeomorphisms of Sn−1.