Triangulations with few vertices of manifolds with non-free fundamental group
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by
Petar Pavešić
2019 p1-11
Abstract
<jats:title>Abstract</jats:title>We study lower bounds for the number of vertices in a PL-triangulation of a given manifold <jats:italic>M</jats:italic>. While most of the previous estimates are based on the dimension and the connectivity of <jats:italic>M</jats:italic>, we show that further information can be extracted by studying the structure of the fundamental group of <jats:italic>M</jats:italic> and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a <jats:italic>d</jats:italic>-dimensional manifold (<jats:italic>d</jats:italic> ⩾ 3) whose fundamental group is not free has at least 3<jats:italic>d</jats:italic> + 1 vertices. As a corollary, every <jats:italic>d</jats:italic>-dimensional homology sphere that admits a combinatorial triangulation with less than 3<jats:italic>d</jats:italic> vertices is PL-homeomorphic to <jats:italic>S</jats:italic><jats:sup><jats:italic>d</jats:italic></jats:sup>. Another important consequence is that every triangulation with small links of <jats:italic>M</jats:italic> is combinatorial.
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