Manifold covers of 3-orbifolds with geometric pieces

McCullough, Darryl; Miller, Andy

doi:10.1016/0166-8641(89)90080-1

Cited by 17 publications

(17 citation statements)

References 17 publications

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“…By [11], & has a geometric decomposition and so also a finite covering by a manifold. That every automorphism of u x & is induced by a diffeomorphism of & follows as Theorem 1 in [15] or from [7]. That diffeomorphisms inducing the same outer automorphism are isotopic is a consequence of the main results in [16,17].…”

Section: J? N (K)\k 0^s \Kmentioning

confidence: 81%

“…Now by Thurston's orbifold geometrization theorem (see [11]; see also [9]) the orbifold ££ n (K), having non-empty singular set, either has itself a geometric structure or has a decomposition into geometric pieces (along euclidean 2-suborbifolds). Then by results of McCullough and Miller [7], it has a finite covering by a manifold. By [15], Theorem 1 or [7], closed irreducible geometrically decomposable 3-orbifolds with infinite orbifold fundamental groups (and finite coverings by manifolds) are determined, up to orbifold homeomorphism, by their fundamental groups.…”

Section: Theorem 2 the Symmetry Group Sym(s Z K): = N 0 T>iff(s 3 mentioning

confidence: 93%

“…Then by results of McCullough and Miller [7], it has a finite covering by a manifold. By [15], Theorem 1 or [7], closed irreducible geometrically decomposable 3-orbifolds with infinite orbifold fundamental groups (and finite coverings by manifolds) are determined, up to orbifold homeomorphism, by their fundamental groups.…”

Section: Theorem 2 the Symmetry Group Sym(s Z K): = N 0 T>iff(s 3 mentioning

confidence: 93%

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On groups associated to a knot

Zimmermann

1991

Math. Proc. Camb. Phil. Soc.

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LetKbe a knot in the 3-sphereS3; letmbe a meridian and1a longitude ofK. In [1] we gave a detailed study of the ‘π-orbifold group’O(K): π1(S3\K)/〈m2〉 of the knot, where 〈m2〉 is the subgroup normally generated by the square of the meridian. In particular, we found out to what extent the π-orbifold group classifies knots (and links) and determines the symmetry group of a knot. In the present paper, we consider the groupsGn(K): = π1(S3\K)/〈ln〉. Our main results are the following.

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Section: J? N (K)\k 0^s \Kmentioning

confidence: 81%

Section: Theorem 2 the Symmetry Group Sym(s Z K): = N 0 T>iff(s 3 mentioning

confidence: 93%

Section: Theorem 2 the Symmetry Group Sym(s Z K): = N 0 T>iff(s 3 mentioning

confidence: 93%

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On groups associated to a knot

Zimmermann

1991

Math. Proc. Camb. Phil. Soc.

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“…It is a consequence of the Geometrization Theorem (cf. [5,32]) that if a 3-orbifold does not contain any bad 2-suborbifolds, then it must be very good, i.e., it is the quotient of a 3-manifold by a finite group action. For simplicity, we shall call such a 3-orbifold good.…”

Section: Recollections and Preliminary Lemmasmentioning

confidence: 99%

“…Then there is an orientable 3-manifold Y ′ equipped with a finite group action of G, such that Y 0 = Y ′ /G (cf. [5,32]). Since π orb 1 (Y 0 ) = π orb 1 (Y ), z(π orb 1 (Y 0 )) is also infinite, and consequently, z(π 1 (Y ′ )), which contains π 1 (Y ′ ) ∩ z(π orb 1 (Y 0 )), is infinite.…”

Section: Injectivity Of S 1 -Actions When π 1 Has Infinite Centermentioning

confidence: 99%

Fixed-point free circle actions on 4–manifolds

Chen

2015

Algebr. Geom. Topol.

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Abstract. This paper is concerned with fixed-point free S 1 -actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) S 1 -action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free S 1 -actions under some further conditions on the fundamental group. The connection between the classification of the S 1 -manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the S 1 -manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela [39]. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free S 1 -action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold.

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Realizing automorphisms of the fundamental group of irreducible 3-manifolds containing two-sided projective planes

Kalliongis

1992

Math. Ann.

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Manifold covers of 3-orbifolds with geometric pieces

Cited by 17 publications

References 17 publications

On groups associated to a knot

On groups associated to a knot

Fixed-point free circle actions on 4–manifolds

Realizing automorphisms of the fundamental group of irreducible 3-manifolds containing two-sided projective planes

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