Topological Prismatoids and Small Non-Hirsch Spheres

We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids used in the recent counter-examples to the Hirsch Conjecture. We show that the strong d-step Theorem that allows to construct non-Hirsch polytopes from prismatoids of large width still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different non-d-step 4-dimensional topological prismatoids with 14 vertices. This implies the existence of 8-dimensional spheres with 18 vertices which do not satisfy the Hirsch bound, which is smaller that the previously known examples by Mani and Walkup (24 vertices, dimension 11). Our non-Hirsch spheres are shellable but we do not know whether they are polytopal.