  For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically
inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which
are stably diffeomorphic to one another. Each of these manifolds has hyperbolic
intersection form and is stably parallelisable. In dimension $4$, we exhibit an
analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$.
  For $m\geq 1$, we also provide similar $(4m{-}1)$-connected $8m$-dimensional
examples, where the number of homotopy types in a stable diffeomorphism class
is related to the order of the image of the stable $J$-homomorphism
$\pi_{4m-1}(SO) \to \pi^s_{4m-1}$.
