On finite groups acting on acyclic low-dimensional manifolds

• Autorzy: Alessandra Guazzi , Mattia Mecchia , Bruno Zimmermann
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 57S25: Groups acting on specific manifolds
• 57S17: Finite transformation groups
• 57M60: Group actions in low dimensions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 215
• Numer: 3
• Strony: 203-217

Daty

wydano

2011

Twórcy

autor

Alessandra Guazzi

• SISSA, Via Bonomea 256, 34136 Trieste, Italy

autor

Mattia Mecchia

• Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, 34100 Trieste, Italy

autor

Bruno Zimmermann

• Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, 34100 Trieste, Italy

Identyfikatory

DOI

10.4064/fm215-3-1