Rigidity of self-maps of $V_{n,2}$ and manifolds tangentially homotopy equivalent to $V_{n,2}\times S^k$

We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2}\times S^k$ up to almost diffeomorphism, for $k=3,5$ or $7\le k,k\mathrm{\ne }{2}^i-2\text{ and }Dim(V_{n,2}\times S^k)\mathrm{\ne }{2}^i-2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases – notably $V_{12,2}\times S^3$, $V_{16,2}\times S^3$, $V_{10,2}\times S^5$ – the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2}\mathrm{\#}\mathrm{\Sigma }\times S^k$ for some exotic sphere $\mathrm{\Sigma }$. In the general case, we identify inverses for a large subgroup of $\mathrm{Im}(\eta )$ and provide a reasonable direction for the remainder.