Stably diffeomorphic manifolds and the realisation of modified surgery obstructions

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