Embeddings of the base and bundle isomorphisms (Q1399161)

The purpose of this work is to give two generalizations of the results of \textit{R. De Sapio} and \textit{G. Walschap} [Topology 39, 921-929 (2000; Zbl 0955.55008)] on two oriented, smooth vector bundles, \(\pi_i: E_i\rightarrow M\), \(i=1, 2\), both of rank \(k\), over a smooth, closed, oriented \(n\)-dimensional manifold \(M\), and to give new counterexamples in certain situations. The author states the main theorems as follows. Theorem 1. Suppose that \(k>[(n+1)/2]+1\) and let \(F:U\rightarrow E_2\) be a smooth, orientation preserving embedding of an open neighborhood \(U\) of the zero section \(s_1(M)\) in \(E_1\) so that the composition \(\pi_2\circ F\circ s_1: M\rightarrow M\) is homotopic to a diffeomorphism \(f\). If \(f\) has degree \(+1\), then \(E_1\) and the induced bundle \(f^\ast E_2\) are isomorphic as oriented bundles. If \(f\) has degree \(-1\), then there is a smooth bundle map from \(E_1\) onto \(f^\ast E_2\) covering the identity and reversing the orientation of each fiber, hence if either \(k\) is odd or one of the bundles has a nowhere-zero cross-section, then \(E_1\) and \(f^\ast E_2\) are isomorphic oriented bundles. Theorem 2. Suppose that \(k=[(n+1)/2]+1\), that \(M\) is a homotopy \(n\)-sphere, and that one of the bundles has a nowhere-zero cross-section. If \(F: U\rightarrow E_2\) is a smooth, orientation-preserving embedding of an open neighborhood \(U\) of the zero section \(s_1(M)\) in \(E_1\) such that \(\pi_2\circ F\circ s_1\) is homotopic to a diffeomorphism of \(M\), then \(E_1\) and \((\pi_2\circ F\circ s_1)^\ast E_2\) are isomorphic oriented bundles. The author remarks that the results of this paper provide a correction to the statement of the corollary of Theorem 2 in his joint paper with Walschap mentioned above.