We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of Z 2 -homology 3-spheres (i.e., with the Z 2 -homology of the 3-sphere where Z 2 denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on Z 2 -homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group A 5 ∼ = PSL(2, 5) or the binary dodecahedral group A * 5 ∼ = SL(2, 5); most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S 3 . Roughly, in the case of Z 2 -homology 3-spheres the groups PSL(2, 5) and SL(2, 5) get replaced by the groups PSL(2, q) and SL(2, q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2, q) and SL(2, q) on Z 2 -homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on Z 2 -homology 3-spheres remains still open). 676 M. Mecchia, B. Zimmermann
The only finite nonabelian simple group acting on a homology 3-spherenecessarily non-freely -is the dodecahedral group A 5 ∼ = PSL(2, 5) (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group A * 5 ∼ = SL(2, 5)). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups A 5 ∼ = PSL(2, 5) and A 6 ∼ = PSL (2, 9). From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-spheres.1991 Mathematics Subject Classification. 57M60, 57S17, 57S25
Abstract. In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of SO(4), which gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we deal with the fibered case and in particular we give explicit formulae relating the finite subgroups of SO (4) with the invariants of the corresponding fibered 3-orbifolds. This allows us to deduce directly from the algebraic classification topological properties of spherical 3-orbifolds. IntroductionGeometric 3-manifolds and 3-orbifolds play an important role in the geometrization program of Thurston (completed at the beginning of this century by Perelman).Roughly speaking, a n-orbifold is a Hausdorff topological space locally modelled by quotients of R n by finite groups of isometries. To each point x of the orbifold is associated the minimal group Γ x such that x has a neighborhood modelled on R n /Γ x . If Γ x is non-trivial the point is called singular. Complete geometric orbifolds are orbifolds diffeomorphic to the quotient of a geometric space (e.g. spherical, Euclidean and hyperbolic space) by a discrete groups of isometries. In particular an orientable spherical 3-orbifold is a quotient of S 3 by a finite subgroup of SO(4). For basic definitions about orbifolds see for example [BMP].In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of SO (4) (4) gives immediately an algebraic classification of spherical 3-orbifolds, but from a topological point of view this classification is not completely satisfactory because it does not give any direct information about the topological structure of the orbifold (underlying topological space and singular set).W.D. Dunbar wrote two classical papers about geometric 3-orbifolds. In [Dun2] he classified the Seifert fibered geometric 3-orbifolds with underlying topological space S 3 in terms of the invariants of the fibration and the singular set of these orbifolds was explicitly drawn. In [Dun3], he described the topology of a non-fibered spherical 3-orbifold starting from the corresponding finite subgroup of SO(4). Up to conjugacy the groups giving a non-fibered 2010 Mathematics Subject Classification. 57M60, 57M12, 57M50.
We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z 2 -homology 3-sphere are the linear fractional groups PSL(2, q), for an odd prime power q (and the dodecahedral group A 5 ∼ = PSL(2, 5) in the case of an integer homology 3-sphere), by showing that G has dihedral Sylow 2-subgroups and applying the Gorenstein-Walter classification of such groups. We also discuss the minimal dimension of a homology sphere on which a linear fractional group PSL(2, q) acts.
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 analogue 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 A 5 and 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.
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