A 3-Sphere Counterexample to the W_v-Path Conjecture

A triangulation D of the 3-sphere with 16 vertices and 90 3-simplices is exhibited. The cell complex D* dual to D has the property that each edge-path between two specified vertices visits some one of the 16 3-cells of D* at least twice. Thus D* is a counterexample to the W_v-path conjecture for S₃ and consequently implies a counterexample to the Hirsch conjecture for S₁₁ Previously known examples are of much larger size or dimension.