Before revision:  Finally, the \mathbb Z _2-reduction of the obstruction shows how to obtain a non-trivial upper bound for the K\"{u}hnel problem: determine the smallest n so that \Delta_n^{(k)} does not embed into M.
After revision:  Finally, the \mathbb{Z _2-reduction of the obstruction shows how to obtain a non-trivial upper bound for the K\"{u}hnel problem: determine the smallest n so that \Delta_n^{(k)} does not embed into M.