Rigidity of self-maps of $V_{n,2}$ and classification of 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 $7 \leq k \leq n-3$, $k \neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases -- notably $V_{12,2} \times S^3$, $V_{16,2} \times S^3$, $V_{12,2} \times S^5$, $V_{10,2} \times S^5$ -- the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2} \mathbin{\#} \Sigma \times S^k$ for some exotic sphere $\Sigma$. In the general case, we identify inverses for a large subgroup of $\operatorname{Im}(\eta)$ and provide a possible way forward to the remainder.