A loop theorem for duality spaces and fibred ribbon knots
type: Item
instance of: scholarly article
title: A loop theorem for duality spaces and fibred ribbon knots
zbMATH Open document ID: 0538.57003
DOI: 10.1007/BF01388533
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143064
review text: In this paper it is proved that a fibered knot in a homology 3-sphere is homotopically ribbon if and only if its (closed) monodromy extends over a handlebody. The proof of this result is based on an extension of the loop theorem as established by the authors. The authors show how their characterization of homotopically ribbon knots can be utilized to construct explicit examples of knots in the boundary of a contractible 4- manifold V, which either do not bound PL-discs in V, or which are slice but not homotopically ribbon. Finally, a question of Scharlemann concerning concordances of torus knots is answered affirmatively for the larger class of \({\mathbb{Q}}\)-anisotropic [the second author, Math. Ann. 257, 157-170 (1981; Zbl 0451.57001)] fibered knots in homology 3-spheres.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 57M35
zbMATH DE Number: 3856013
zbMATH Keywords: ribbon knots
zbMATH Keywords: slice knots
zbMATH Keywords: loop theorem in 3-dimensional Poincaré duality spaces
zbMATH Keywords: fibered knot in a homology 3-sphere
zbMATH Keywords: homotopically ribbon knots
zbMATH Keywords: concordances of torus knots
author: Andrew J. Casson
author: Cameron McA. Gordon
reviewed by: Klaus Johannson
MaRDI profile type: Publication
cites work: Mazur manifolds
cites work: Cobordism of automorphisms of surfaces
cites work: Algorithmic compression of surface automorphisms
cites work: The topology of four-dimensional manifolds
cites work: Knots, homology spheres, and contractible 4-manifolds
cites work: Homotopy equivalences of 3-manifolds with boundaries
cites work: A theorem on planar covering surfaces with applications to 3-manifolds
cites work: Q5570652
cites work: On Solid Tori
cites work: A reduction of the Poincaré conjecture to group theoretic conjectures
cites work: Group rings and their augmentation ideals
cites work: Essential tori in 4-manifold boundaries
cites work: On the loop theorem
cites work: Homology and central series of groups
cites work: On irreducible 3-manifolds which are sufficiently large
cites work: Q4172657
cites work: Surgery of non-simply-connected manifolds
cites work: On the dunce hat
description: scientific article; zbMATH DE number 3856013