Manifolds Homeomorphic to Sphere Bundles over Spheres
نویسنده
چکیده
1. Statement of results. Let E be the total space of a fe-sphere bundle over the w-sphere with characteristic class a 6 i „ i (SOk+i). We consider the problem of classifying, under the relation of orientation preserving diffeomorphism, all differential structures on E. It is assumed that E is simply connected, of dimension greater than five, and its characteristic class a may be pulled back to lie in 7rn-i (SO*) (that is, the bundle has a cross-section). In [l] and [2] we gave a complete classification in the special case where a = 0. The more general classification Theorems 1 and 2 below include this special case. The proofs of these theorems are sketched in §2 below; detailed proofs will appear elsewhere. J. Munkres [ô] has announced a classification up to concordance of differential structures in the case where the bundle has at least two cross-sections. (It is well known that concordance and diffeomorphism are not equivalent, concordance of differential structures being strictly stronger than diffeomorphism.)
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