On finite groups acting on acyclic low-dimensional manifolds

Alessandra Guazzi¹ ; Mattia Mecchia² ; Bruno Zimmermann²

¹ SISSA Via Bonomea 256 34136 Trieste, Italy
² Dipartimento di Matematica e Informatica Università degli Studi di Trieste 34100 Trieste, Italy

Fundamenta Mathematicae, Tome 215 (2011) no. 3, pp. 203-217.

Résumé

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 ${\rm O}(3)$ or ${\rm O}(4)$, 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 $\mathbb A_5$ and $\mathbb A_6$, 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.

Détail

DOI : 10.4064/fm215-3-1

Keywords: consider finite groups which admit faithful smooth action acyclic manifold dimension three five euclidean space first main result states finite group acting acyclic manifold isomorphic subgroup orthogonal group respectively analogous statement remains dimension five where arbitrary continuous actions however prove only finite nonabelian simple groups admitting smooth action acyclic manifold alternating groups mathbb mathbb deduce short list finite groups closely related finite subgroups which candidates orientation preserving actions acyclic manifolds