Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle

Via a computer search, Altshuler and Steinberg found that there are 1296 + 1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold K 9 triangulates the twisted S -bundle over S . It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by proper bistellar moves to a 9-vertex neighbourly 3-manifold. 2000 Mathematics Subject Classification. 57Q15, 57R05.