For most n, self-maps of V_{n,2} homotopic to almost diffeomorphisms are determined, and manifolds tangentially homotopy equivalent to V_{n,2}×S^k are classified up to almost diffeomorphism in many dimensions.
Crowley,The classification of highly connected manifolds in dimensions7and15, Ph.D
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Rigidity of self-maps of $V_{n,2}$ and manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$
For most n, self-maps of V_{n,2} homotopic to almost diffeomorphisms are determined, and manifolds tangentially homotopy equivalent to V_{n,2}×S^k are classified up to almost diffeomorphism in many dimensions.