Showing papers in "Journal of The Mathematical Society of Japan in 1961"
••
325 citations
••
85 citations
••
62 citations
••
49 citations
••
48 citations
••
43 citations
••
32 citations
••
[...]
31 citations
••
26 citations
••
17 citations
••
••
••
••
••
••
TL;DR: In this paper, it was shown that the invariant 2' characterizes differentiable structures on S4k-1 which bound 7r-manifolds for k > 1.
Abstract: J. Milnor [2] 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 [3] 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 -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 [4]) 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 ([4]). 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 [4]. We shall use them without a special reference.
••
••
••
••
••
••
••
••
••
••
••
••