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.