An Exact Sequence in Differential Topology

We now describe these groups briefly. To obtain T, divide the group of diffeomorphisms of the n — 1 sphere S" by the normal subgroup of those diffeomorphisms that are extendable to the w-ball. See [2] for details. The set 0 is the set of /-equivalence classes of closed, oriented, differentiable w-manifolds that are homotopy spheres. If M is an oriented manifold, let —If be the oppositely oriented manifold. Two closed oriented w-manifolds M and N are J-equivalent if there is an oriented w + 1-manifold X whose boundary is the disjoint union of M and — N9 and which admits both M and N as deformation retracts. We denote the /-equivalence class of M by [M]. If [M] and [N] are elements of 0, their sum is defined to be [M f N], where [ M # N] is obtained by removing the interior of an w-ball from M and N and identifying the boundaries in a suitable way. Details may be found in [1]. The group A is defined analogously using combinatorial manifolds. Instead of the interior of an w-ball, the interior of an ^-simplex is removed. If M is a combinatorial manifold, we write (M) for its / equivalence class.