Simply-connected manifolds with large homotopy stable classes

Abstract

For every $k\ge 2$ and $n\ge 2$ 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 $m\ge 1$, 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 $\ddot{I}{€}_{4m-1}(SO)\to \ddot{I}{€}_{4m-1}^s$.

Minor revision: corrected a mistake in the proof of Theorem 4.11 and improved exposition in Section 4. 27 pages