On the high-dimensional geography problem

In 1962, Wall showed that smooth, closed, oriented, .n1/-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.We complete the determination of which n-spaces are realizable by smooth, closed, oriented, .n1/connected 2n-manifolds for all n 63.In dimension 126, the Kervaire invariant one problem remains open.Along the way, we completely resolve conjectures of Galatius and Randal-Williams and Bowden, Crowley and 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 E 1 -ring spectra by Ando, Hopkins and Rezk.By previous work of many authors, including Wall, Schultz, Stolz, and Hill, Hopkins and 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.