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

Pavešić, Petar (author)

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 ($d\ge 3$) whose fundamental group is not free has at least $3d+1$ vertices. As a corollary, every $d$-dimensional ($\mathbb{Z}_p$-)homology sphere that admits a PL-triangulation with less than $3d$ vertices is homeomorphic to $S^d$. Another important consequence is that every triangulation with small links of $M$ is combinatorial.

• Language: English
• Keywords: minimal triangulation, PL-manifold, homology sphere, good cover, Lusternik-Schnirelmann category
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 1453-1463
• Numbering: Vol. 149, iss. 6
• PID: 20.500.12556/RUL-115222
• UDC: 515.164
• ISSN on article: 0308-2105
• DOI: 10.1017/prm.2018.136
• COBISS.SI-ID: 18671705
• Publication date in RUL: 18.04.2020

Record is a part of a journal

• Title: Proceedings of the Royal Society of Edinburgh
• Shortened title: Proc. R. Soc. Edinb., Sect. A, Math.
• Publisher: Royal Society of Edinburgh
• ISSN: 0308-2105
• COBISS.SI-ID: 26180608

Secondary language

• Language: Slovenian
• Title: Majhne triangulacije mnogoterosti s fundamentalno grupo, ki ni prosta

Abstract

V članku obravnavamo spodnje meje za število oglišč v PL-triangulacijah dane mnogoterosti $M$. Prej znane ocene uporabljajo dimenzijo in povezanost $M$, mi pa pokažemo, da je mogoče dobiti boljše ocene z uporabo fundamentalne grupe in tehnik Lusternik-Schnirelmannove kategorije. Glavni rezultat je, da je za PL-triangulacijo $d$-razsežne mnogoterosti ($d\ge 3$) katere fundamentalna grupa ni prosta potrebnih vsaj $3d + 1$ oglišč. Posebej, vsaka $d$-razsežna homološka sfera, ki dopušča kombinatorno triangulacijo z manj kot $3d$ oglišč je PL-homeomorfna običajni $d$-sferi. Druga pomembna posledica je, da so vse triangulacije z majhnimi okvirji kombinatorne.