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Triangulations with few vertices of manifolds with non-free fundamental group

Part of: Polytopes and polyhedra PL-topology

Published online by Cambridge University Press:  15 January 2019

Petar Pavešić
Petar Pavešić*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenija (petar.pavesic@fmf.uni-lj.si)
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Abstract

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

Keywords

minimal triangulationPL-manifoldhomology spheregood coverLusternik-Schnirelmann category

MSC classification

Primary: 57Q15: Triangulating manifolds
Secondary: 52B70: Polyhedral manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1453 - 1463
Copyright
Copyright © Royal Society of Edinburgh 2019 

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