Finite group actions on 3-manifolds

Meeks, William H.; Scott, Peter

doi:10.1007/bf01389073

Cited by 126 publications

(120 citation statements)

References 29 publications

Supporting

Mentioning

117

Contrasting

Order By: Relevance

“…Let T.M / be the JSJ-decomposition of M . By Meeks and Scott [20], we may assume that T.M / is h-invariant. For each component T of T.M / such that h.T / D T and h exchanges the sides of T , replace T by two parallel copies that are interchanged by h. Denote this new collection of tori by T C .M /.…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

View full text Add to dashboard Cite

First we clarify the extension of the main result of [11] to knots in solid tori that is described in the Appendix of [11]. In Section 3, we define a family of hyperbolic knots J " .`; m/ in a solid torus, " 2 f1; 2g and`; m integers, each of which admits a half-integer surgery yielding a toroidal manifold. (The knots J " .`; m/ in the solid torus are the analogs of the knots k.`; m; n; p/ in the 3-sphere.) We then use [11] to show that these are the only such:Theorem 4.2 Let J be a knot in a solid torus whose exterior is irreducible and atoroidal. Let be the meridian of J and suppose that J. / contains an essential torus for some with . ; / 2. Then . ; / D 2 and J D J " .`; m/ for some ";`; m.This theorem along with the main result of [11] then allows us to describe in Theorem 5.2 the relationship between the torus decomposition of the exterior of a knot K in S 3 and the torus decompositon of any non-integral surgery on K . In particular, Theorem 5.2 says that the canonical tori of the exterior of K and of the Dehn surgery will be the same unless K is a cable knot (in which case an essential torus of the knot exterior can become compressible), or K is a k.`; m; n; p/, or K is a satellite with pattern J " .`; m/ (in the latter cases a new essential torus is created).We apply these theorems about non-integral Dehn surgeries to address questions about unknotting number. The knots k.`;m;n;p/ (J " .`; m/) are strongly invertible. Their quotients under the involutions give rise to EM-knots, K.`;m;n;p/ (EM-tangles, A " .`; m/ resp.) which have essential Conway spheres and yet can be unknotted (trivialized, resp.) by a single crossing change. Theorem 6.2 descibes when a knot with an essential Conway sphere or 2-torus can have unknotting number 1. This is naturally stated in the context of the characteristic decomposition of a knot along

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…M restricts to hW X ! X , and we can isotop T.X / in X to be h-invariant [20]. Let T be a component of T.X /.…”

Section: Main Theoremmentioning

confidence: 99%

Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…The claim is a direct corollary of a finiteness result on the conjugacy classes of finite group actions on those N i (see [2] or [25], both using [13]). …”

Section: Claim 56 If the Family {N I I ∈ N} Is Infinite Up To Homentioning

confidence: 90%

Nonzero degree maps between closed orientable three-manifolds

Derbez

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. This paper adresses the following problem: Given a closed orientable three-manifold M , are there at most finitely many closed orientable three-manifolds 1-dominated by M ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold M 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of M .

show abstract

“…We will regard H as the quotient of Proof. If F is orientable, this is immediate from Theorem A of [5] or Theorem 8.1 of [8]. Suppose F is nonorientable.…”

Section: Involutions Of I-bundlesmentioning

confidence: 96%

“…The lifts of h to F × I generate an action of Z/2 × Z/2 on F × I which preserves F × ∂I (the lifts do not generate a cyclic group of order 4, because τ × r interchanges the components of ∂ F × I). By Theorem 8.1 of [8], the action of this group is conjugate to an action which preserves the product structure. This induces a conjugation of h to an action of the form [h 1 × 1 I ].…”

Section: Involutions Of I-bundlesmentioning

confidence: 99%

Orientation-reversing involutions on handlebodies

Kalliongis

McCullough

1996

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The observation that the quotient orbifold of an orientationreversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the I-fibers of some I-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses. IntroductionThroughout this paper h will be an orientation-reversing involution on a 3-dimensional orientable handlebody H. Moreover, h is assumed to be tame, so the action of h on an invariant ball about a fixed point must be equivalent either to the antipodal map on the 3-ball, or to a reflection across a 2-disc in the 3-ball, or (precisely when the fixed point lies in ∂H) to a reflection across a half 2-disc in a half 3-ball. The fixed point set fix(h) is thus a union of finitely many isolated fixed points and finitely many 2-manifolds properly imbedded in H. It is known (Theorem 6.1 of [2]) that the 2-dimensional components are incompressible, although this will be seen in passing when we examine the structure of h, and is immediate from the structure theory we will develop.A simple type of involution occurs when h is a product h 1 × h 2 on A × I, where A is an orientable 2-manifold with boundary, each h i is either the identity or an involution, and exactly one of the h i is orientation-reversing. Also, when H is the orientable I-bundle over a nonorientable 2-manifold with boundary, one has similar examples. By gluing such actions together along invariant discs in the boundary, one constructs more complicated examples. Our main structure results, Theorems 5.1 and 5.6, show that all orientation-reversing involutions are built up in this way, and furnish descriptions of the component actions in terms of the fixed-point data of h and the genus of H. The actions on the pieces in these decompositions are sometimes but not always unique up to equivalence; the conditions under which uniqueness holds and the extent to which it fails are fully analyzed in auxiliary theorems. As corollaries of the main structure results, one obtains the following simple characterizations of fiber-preserving involutions in terms of the fixed point data.

show abstract

Finite group actions on 3-manifolds

Cited by 126 publications

References 29 publications

Knots with unknotting number 1 and essential Conway spheres

Knots with unknotting number 1 and essential Conway spheres

Nonzero degree maps between closed orientable three-manifolds

Orientation-reversing involutions on handlebodies

Contact Info

Product

Resources

About