Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-n9lxd Total loading time: 0.211 Render date: 2022-10-04T05:08:30.865Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Representing elements of π1M3 by fibred knots

Published online by Cambridge University Press:  24 October 2008

John Harer
Affiliation:
Columbia University, New York

Extract

Every smooth, closed, orientable manifold of dimension 3 contains a fibred knot, i.e. an imbedded circle K such that MK is the total space of a fibre bundle over S1 whose fibre F is standard near K. This means the boundary of the closure of F is K so that K is null-homologous in M. A natural problem suggested by Rolfsen (see (2), problem 3·13) is to determine which elements of π1M3 may be represented by fibred knots. We deal with this by proving the

Theorem. The collection of all elements of π1 M3 which can be represented by fibred knots is exactly the commutator subgroup [π1 M3, π1 M3].

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Haker, J. How to construct all fibered knots and links. Preprint (1979).Google Scholar
(2)Kirby, R.Problems in low dimensional manifold theory. A.M.S. Proceedings of Symposia in Pure Mathematics 32 (1978), 273312.CrossRefGoogle Scholar
(3)Meyers, R.Notices A.M.S. 22 (1975), A 651.Google Scholar
(4)Stallings, J.Constructions of fibered knots and links. A.M.S. Proceedings of Symposia in Pure Mathematics 32 (1978), 5560.CrossRefGoogle Scholar
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Representing elements of π1M3 by fibred knots
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Representing elements of π1M3 by fibred knots
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Representing elements of π1M3 by fibred knots
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *