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

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

2006-10-27

arxiv.org/abs/math/0610825v1

Mathematics

Geometric Topology

11 pages

Scientific paper

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 $K^{3}_{9}$ triangulates the twisted $S^{2}$-bundle over $S^{1}$. 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.

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