Remarks on differentiable structures on spheres
Journal of the Mathematical Society of Japan
J. Milnor  defined the invariant 2' for compact unbounded oriented differentiable (4k-1)-manifolds which are homotopy spheres and boundaries of irmanifolds at the same time, and proved that the invariant 2' characterizes the J -equivalence classes of these (4k-1)-manifolds for k > 1. Recently S. Smale  has shown that a compact unbounded (oriented) differentiable n-manifold (n > 5) having the homotopy type of Sn is homeomorphic to Sn and that two such manifolds belonging to the same f
... to the same f -equivalence class are diffeomorphic to each other if n * 6. Hence it turns out that the invariant 2' characterizes differentiable structures on S4k-1 which bound 7r-manifolds for k > 1. In this note we shall compute the invariant 2' of B",1 (S3 bundles over S4, see  ) and show that every differentiable structure on S7 can be expressed as a connected sum of Bm,1. We shall obtain also a similar result on S15. Furthermore we shall show that Bm" U1 D8 such that m(m+l) = 0 mod 56 are 3connected compact unbounded differentiable 8-manifolds with the 4 th Betti number 1 and differentiable 8-manifolds of this type are exhausted by them, where Bm,1 are 4-cell bundles over S4 (). This will reveal that Pontrjagin numbers are not homotopy type invariants. Notations and terminologies of this note are the same as in the previous :paper  . We shall use them without a special reference.