We consider here the problem
introduced by Conner and Floyd of determining necessary and sufficient conditions
for a manifold M to be cobordant to a bundle over a given sphere Sk. Two recent
studies by D. F. X. O’Reilly [7] and A. Didierjean [4] presented obstructions to
fibering manifolds over spheres in terms of the “top” Stiefel-Whitney classes of M.
While these conditions were shown by Conner and Floyd [3] and R. L. W. Brown [2]
to be sufficient when restricted to the cases of fiberings over S1 and S2, they are not
at all sufficient for guaranteeing the fibering of a cobordism class over a sphere of any
higher dimension. This is shown in O’Reilly’s study of fiberings over the
4-sphere.
In this paper we exhibit an obstruction to fibering a manifold over a sphere that
extends the obstructions mentioned above. We then essentially answer all open
questions but one regarding the problem of which cobordism classes can be
represented by a bundle over S4.
