Diffeomorphism of total spaces and equivalence of bundles
It has been known, due to the first named author [Math. Z. 107, 232-236 (1968; Zbl 0162.55302)], that any stably trivial oriented real smooth \(n\)-plane bundle over a homotopy \(n\)-sphere \((n\neq 3,4)\) and any bundle with the same properties over the standard \(n\)-sphere \(S^n\) are equivalent as oriented \(n\)-plane bundles over \(S^n\) if their total spaces are diffeomorphic, with orientation preserved. In the paper under review, the authors prove that if the total spaces of two smooth oriented \(k\)-plane bundles over \(S^n\) are diffeomorphic, with orientation preserved, then the bundles are smoothly equivalent up to orientation of the base whenever \(k>[(n+1)/2]+1\) (here \([x]\) denotes the integer part of \(x\in \mathbb R\)). In a more general context, the following are the main results of the paper. Theorem 1. Let \(\pi_i: E_i\rightarrow M\), \(i=1,2\), be smooth, oriented, real \(k\)-plane bundles over a closed, oriented, smooth manifold \(M\), with zero cross-sections \(s_i: M\rightarrow E_i\), and let \(F:E_1 \rightarrow E_2\) be an orientation-preserving diffeomorphism such that \(\pi_2\circ F\circ s_1\) is homotopic to the identity map of \(M\). If \(k>[(n+1)/2]+1\), then the oriented bundles \(\pi_1\) and \(\pi_2\) are smoothly equivalent. Theorem 2. Let \(\pi_i: E_i\rightarrow M\), \(i=1,2\), be as in Theorem 1, but now let \(M\) be a homotopy \(n\)-sphere. If \(k>[(n+1)/2]+1\) and \(F:E_1 \rightarrow E_2\) is an orientation-preserving diffeomorphism such that \(\pi_2\circ F\circ s_1\) has degree \(1\), then the oriented bundles \(\pi_1\) and \(\pi_2\) are smoothly equivalent. The same is true for \(k=[(n+1)/2]+1\), provided that \(\pi_1\) or \(\pi_2\) has a nowhere-zero cross-section. Among other aspects of their results, the authors discuss an application to Riemannian geometry.
scientific article; zbMATH DE number 1472307