Three-dimensional pseudomanifolds on eight vertices. (English) Zbl 1157.57014
This paper gives a classification of all combinatorial 3-pseudomanifolds (in the strong sense) with 8 vertices. There are exactly 35 items with at least one singular vertex, up to combinatorial automorphisms. It was previously known that there are exactly 39 distinct combinatorial types of 3-manifolds with 8 vertices and that all of them are spheres (37 polytopal and 2 non-polytopal ones). The work of the authors is based on a classification of the possible vertex links which must be triangulated surfaces with at most 7 vertices. The topological types of these 3-pseudomanifolds are almost completely classified. Some of them are branched quotients of triangulated 3-manifolds. Most of them are (topologically) suspensions of \(\mathbb{R} P^2\). One of the cases contains all \(8\choose 3\) triangles, each vertex link is the unique 7-vertex torus. In combinatorics this object was observed by A. Emch in 1929 as a 2-fold quadruple system or block design \(S_2(3,4;8)\).
Reviewer: Wolfgang Kühnel (Stuttgart)
Software:
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