On finite groups acting on acyclic low-dimensional manifolds

Alessandra Guazzi, Mattia Mecchia, and Bruno Zimmermann. "On finite groups acting on acyclic low-dimensional manifolds." Fundamenta Mathematicae 215.3 (2011): 203-217. <http://eudml.org/doc/282627>.

@article{AlessandraGuazzi2011,
abstract = {We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. Euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogous statement remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups 𝔸₅ and 𝔸₆, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.},
author = {Alessandra Guazzi, Mattia Mecchia, Bruno Zimmermann},
journal = {Fundamenta Mathematicae},
keywords = {acyclic manifold; finite group action; finite simple group},
language = {eng},
number = {3},
pages = {203-217},
title = {On finite groups acting on acyclic low-dimensional manifolds},
url = {http://eudml.org/doc/282627},
volume = {215},
year = {2011},
}

TY - JOUR
AU - Alessandra Guazzi
AU - Mattia Mecchia
AU - Bruno Zimmermann
TI - On finite groups acting on acyclic low-dimensional manifolds
JO - Fundamenta Mathematicae
PY - 2011
VL - 215
IS - 3
SP - 203
EP - 217
AB - We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. Euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogous statement remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups 𝔸₅ and 𝔸₆, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.
LA - eng
KW - acyclic manifold; finite group action; finite simple group
UR - http://eudml.org/doc/282627
ER -