On The Acyclic MacPhersonian

The acyclic MacPhersonian ${\text{AM}}_n^d$ is the poset of acyclic oriented matroids of rank $d+1$ on $n$ labelled elements, the combinatorial model of the space of configurations of $n$ points spanning ${\mathbb{R}}^d$ up to affine equivalence, which is the Grassmannian $G(d,n-1)$. We study it inside the larger poset ${\text{Sep}}_n^d$ of \emph{spherical separoids}, those acyclic separoids whose Radon complex is a sphere of the dimension of the space of affine dependences. Our main unconditional results are as follows. Over every poset of spherical separoids that contains the realizable oriented matroids there is a tautological sphere fibration, and its Stiefel–Whitney classes show that the mod~$2$ cohomology ring of $G(d,n-1)$ is a ring retract of that of the poset, for all $n$ and $d$. For $d=n-2$ the poset of spherical separoids is homotopy equivalent to $G(n-2,n-1)$, and contains ${\text{AM}}_n^{n-2}$ as a homotopy equivalent subposet. A fat model of ${\text{AM}}_n^d$ made of piecewise geodesic antipodal spheres in the round sphere is weakly homotopy equivalent to its order complex. The chirotope of every such sphere is recovered from the degrees of its coordinate projections. We then analyse a strategy based on the mean curvature flow. Any argument that follows the combinatorial type of a sphere generically along the flow would connect every uniform oriented matroid to a realizable one by mutations. By a theorem of Gu and Knauer this is impossible in every rank $r\ge 7$ on $n\ge r+17$ elements, so the flow must leave the oriented matroids. We show that the basin hypothesis of the flow, together with the property that $\mathrm{\parallel }{\text{AM}}_n^d\mathrm{\parallel }$ is a homology manifold, implies $\mathrm{\parallel }{\text{AM}}_n^d\mathrm{\parallel }\simeq G(d,n-1)$, without following types along the flow. We conclude with the open problems and conjectures that remain.