Showing non-realizability of spheres by distilling a tree

Abstract

In [Zhe20a], Hailun Zheng constructs a combinatorial 3-sphere on 16 vertices whose graph is the complete 4-partite graph K4;4;4;4. Such a sphere seems unlikely to be realizable as the boundary complex of a 4-dimensional polytope, but all known techniques for proving this fail because there are just too many possibilities for the 16 4 = 64 coordinates of its vertices. Known results [PPS12] on polytopal realizability of graphs also do not cover multipartite graphs. In this paper, we level up the old idea of Grassmann{Pl ucker relations, and assemble them using integer programming into a new and more powerful structure, called positive Grassmann{Pl ucker trees, that proves the non-realizability of this example and many other previously inaccessible families of simplicial spheres. See [Pfe20] for the full version