Combinatorial $3$-manifolds with $10$ vertices

Frank H. Lutz

Abstract: We give a complete enumeration of all combinatorial $3$-manifolds with $10$ vertices: There are precisely $247882$ triangulated $3$-spheres with $10$ vertices as well as $518$ vertex-minimal triangulations of the sphere product $S^2\times S^1$ and $615$ triangulations of the twisted sphere product $S^2\hbox{$\times \hspace{-1.62ex}_\hspace{-.4ex}_\hspace{.7ex}$}S^1$. All the $3$-spheres with up to $10$ vertices are shellable, but there are $29$ vertex-minimal non-shellable $3$-balls with $9$ vertices.

Editorial remark: Due to a mixup, the Table 3 in the first published electronical version of the paper in this volume is not the version the author wanted to submit to the journal. The obsoleted version is kept at 5v1.html. The current version represents now the same version as the printed article in this journal.}

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