Simply Connected Manifolds With Large Homotopy Stable Classes

For every k ≥ 2 and n ≥ 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 four, we exhibit an analogous phenomenon for spin ᶜ structures on S² × S² . For m≥ 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 π ₄ₘ₋₁(SO) → π ˢ₄ₘ₋₁ .