Fibered spherical 3-orbifolds

Abstract

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.

Constraining mapping class group homomorphisms using finite subgroups

“…Alternatively, we may see this because all finite subgroups of SO(4) of odd order are abelian. (The classification of finite subgroups of SO(4) goes back to work of Seifert-Threlfall [40,41]; see the paper of Mecchia-Seppia for a contemporary treatment [33].) We have therefore found the required G g for all but these ten values of g. For g = 4, 11, 16, 25, 34, and 41, we may take G g = C 5 C 4 = SmallGroup (20,3); for these values of g this finite group is a subgroup of Mod(S g ), but it is not a subgroup of Homeo + (S 3 ) by Theorem 14.…”

Isometry groups and mapping class groups of spherical 3-orbifolds

“…In [MS15] the base orbifold and the Seifert invariants of the fibrations induced by the Hopf fibration on S 3 /G, for the groups G in the Du Val's list, were computed. Those formulae hold unchanged also for all groups in Table 5, allowing that the indices vary with no restrictions on m ≥ 1.…”

“…To understand the induced action on the fibration, one has first to find the base orbifold B of the Seifert fibration (this had already been done in [MS15]), and then determine the action of Isom p (S 3 /G)/Isom f (S 3 /G) on such base orbifold. This is done case-by-case, and we will treat in detail the cases of Families 1, 3 and 4, which are quite illustrating.…”

Classification of non-free Kleinian groups generated by two parabolic transformations

“…The isometry groups of the dihedral spherical orbifolds obtained as the π-orbifolds associated with 2-bridge links are calculated by [57,28]. Moreover, in the recent papers [40,41], Mecchia and Seppi classified the Seifert fibered spherical 3-orbifolds and calculated the isometry groups of such orbifolds. Since every spherical dihedral orbifold is Seifert fibered, the results in this section are implicitly contained in [40,41].…”

“…Moreover, in the recent papers [40,41], Mecchia and Seppi classified the Seifert fibered spherical 3-orbifolds and calculated the isometry groups of such orbifolds. Since every spherical dihedral orbifold is Seifert fibered, the results in this section are implicitly contained in [40,41]. However, we give a self-contained proof, because it is not a simple task to translate their results into the form we need.…”