Smoothings of sphere bundles over spheres in the stable range

Abstract

The general problem of classifying up to orientation-preserving diffeomorphism those smooth manifolds homeomorphic to a given manifold is probably too complicated to be treated in any effective uniform manner. The first case to be treated was of course the sphere, where everything has been established modulo computation of the Adams spectral sequence. The next case considered was a product of two spheres, where the author and DeSapio independently reduced the classification to homotopy theory with the exception of determining the action of ~P4k + 2 on some (4 k + 1)-dimensional products. It is completely straightforward to extend this classification to the k-sphere bundles associated to (k+l)-plane bundles over S" which have nowhere zero cross sections and satisfy k n which apparently diverge from [1] in some respects. We make two remarks for completeness. A classification of smoothings of bundles with k < n (not necessarily having cross sections) may be derived from [3, w 5]. Finally, throughout this paper the phrases "combinatorially equivalent" and "homeomorphic" are interchangeable by the Hauptvermutung for simply connected closed manifolds with torsion free homology and dimension at least 5.