  In 1962, Wall showed that smooth, closed, oriented, $(n-1)$-connected
$2n$-manifolds of dimension at least $6$ are classified up to connected sum
with an exotic sphere by an algebraic refinement of the intersection form which
he called an $n$-space. In this paper, we complete the determination of which
$n$-spaces are realizable by smooth, closed, oriented, $(n-1)$-connected
$2n$-manifolds for all $n \neq 63$. In dimension $126$ the Kervaire invariant
one problem remains open. Along the way, we completely resolve conjectures of
Galatius-Randal-Williams and Bowden-Crowley-Stipsicz, showing that they are
true outside of the exceptional dimension $23$, where we provide a
counterexample. This counterexample is related to the Witten genus and its
refinement to a map of $\mathbb{E}_\infty$-ring spectra by Ando-Hopkins-Rezk.
By previous work of many authors, including Wall, Schultz, Stolz and
Hill-Hopkins-Ravenel, as well as recent joint work of Hahn with the authors,
these questions have been resolved for all but finitely many dimensions, and
the contribution of this paper is to fill in these gaps.
