Illinois Journal of Mathematics

Proposed Property 2R counterexamples examined

Martin Scharlemann

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In 1985, Akbulut and Kirby analyzed a homotopy $4$-sphere $\Sigma$ that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysi showed that $\Sigma$ was one of an infinite collection of examples, all of which were (sadly) the standard $S^{4}$, but with an unusual handle structure.

Recent work with Gompf and Thompson, showed that the construction gives rise to a family $L_{n}$ of $2$-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each $L_{n}$, one component is the simple square knot $Q$, and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the $L_{n}$. In view of the continuing interest and potential importance of the Cappell-Shaneson-Akbulut-Kirby-Gompf examples (e.g., the original $\Sigma$ is known to embed very efficiently in $S^{4}$ and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.

Article information

Source
Illinois J. Math., Volume 60, Number 1 (2016), 207-250.

Dates
Received: 2 October 2015
First available in Project Euclid: 21 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1498032031

Mathematical Reviews number (MathSciNet)
MR3665179

Zentralblatt MATH identifier
1376.57012

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Citation

Scharlemann, Martin. Proposed Property 2R counterexamples examined. Illinois J. Math. 60 (2016), no. 1, 207--250. https://projecteuclid.org/euclid.ijm/1498032031


Export citation