Embeddings of the base and bundle isomorphisms

Let πi :Ei→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections si :M→Ei. Suppose that U is an open neighborhood of s1(M) in E1 and F :U→E2 a smooth embedding so that π2∘F∘s1 :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E1 and the induced bundle f∗E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<k⩽n we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f∗E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.