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

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. Posbej, 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.

Keywords

minimal triangulation;PL-manifold;homology sphere;good cover;Lusternik-Schnirelmann category;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.164
COBISS:	18671705
ISSN:	0308-2105
Views:	480
Downloads:	257
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Majhne triangulacije mnogoterosti s fundamentalno grupo, ki ni prosta
Secondary 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 (▫$d\ge 3$▫) 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.
Pages:	str. 1453-1463
Volume:	ǂVol. ǂ149
Issue:	ǂiss. ǂ6
Chronology:	Dec. 2019
DOI:	10.1017/prm.2018.136
ID:	11551954