a classical question in differential topology is the following: classify all simply-connected, closed, smooth (2n)-manifolds whose only non-trivial homology groups are $h_0, h_n$ and $h_{2n}$. in this talk i will survey the history of the high dimensional side of this question and how its resolution requires a surprisingly deep understanding of the adams spectral sequence computing the stable homotopy groups of spheres. time permitting, i will then discuss how the situation changes as we relax our topological restrictions on the manifold (for example allowing $h_{n-e}$, $h_{n-e+1}$, ... $h_{n+e}$ to be non-trivial for a small number e). \\ \\ this talk represents joint work with jeremy hahn and andy senger.