Combinatorial Spheres inside Spheres on the Same Vertex Set: Partial Results on a Question of Santos

F. Santos asked in an Oberwolfach problem session (2019) whether every combinatorial d-sphere S with n ≥ d+3 vertices is a subcomplex of a combinatorial (d+1)-sphere with the same n vertices. We give partial results. If S has an edge vw with lk(v) ∩ lk(w) = lk(vw), equivalently an edge that lies in no missing face with at least three vertices, then the union of the cones v*ast(v) and w*ast(w) over the two antistars is such a sphere. This extends Datta's construction for flag spheres and recovers his results for joins and for spheres with a vertex of degree d+1. If S has no missing face of dimension d and some vertex link is a stacked sphere, the answer is also positive. We add reformulations in terms of completable antistars, one-point suspensions and balls, closure under connected sums and stellar subdivisions, and restrictions on a smallest counterexample in dimension 3. With computer certificates that are checked by an independent program, every combinatorial 3-sphere with 6 ≤ n ≤ 10 vertices lies in a 4-sphere on the same vertex set: 1320 of the 247,882 ten-vertex 3-spheres have no edge of the above kind, and each of them has an explicit extension. All 337 combinatorial 4-spheres with 9 vertices that we found have such an edge; the completeness of this list is a solver result without proof certificates, and we know no published count to compare it with. A Z_11-invariant neighborly 3-sphere without such an edge and two Z_13-invariant ones have extensions, but none in which every facet meets a fixed pair of vertices. Of nine Z_14-invariant neighborly 3-spheres without such an edge, six extend and three are undecided. These spheres are known from the enumerations of Kühnel and Lassmann and of Köhler and Lutz. The general question remains open. This is an unrefereed note.