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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-1,
author = {Alessandra Guazzi and Mattia Mecchia and Bruno Zimmermann},
title = {On finite groups acting on acyclic low-dimensional manifolds},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {203-217},
zbl = {1244.57033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-1}
}
Alessandra Guazzi; Mattia Mecchia; Bruno Zimmermann. On finite groups acting on acyclic low-dimensional manifolds. Fundamenta Mathematicae, Tome 215 (2011) pp. 203-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-3-1/