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For every $k \geq 2$ we construct infinitely many $4k$-dimensional manifoldsthat are all stably diffeomorphic but pairwise not homotopy equivalent. Each ofthese manifolds has hyperbolic intersection form and is stably parallelisable.In fact we construct infinitely many such infinite sets. To achieve this weprove a realisation result for appropriate subsets of Kreck's modified surgerymonoid $\ell_{2q+1}(\mathbb{Z}[\pi])$, analogous to Wall's realisation of theodd-dimensional surgery obstruction $L$-group $L_{2q+1}^s(\mathbb{Z}[\pi])$.

Stably diffeomorphic manifolds and modified surgery obstructions