## Illinois Journal of Mathematics

### Proposed Property 2R counterexamples examined

Martin Scharlemann

#### 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
First available in Project Euclid: 21 June 2017

https://projecteuclid.org/euclid.ijm/1498032031

Digital Object Identifier
doi:10.1215/ijm/1498032031

Mathematical Reviews number (MathSciNet)
MR3665179

Zentralblatt MATH identifier
1376.57012

#### Citation

Scharlemann, Martin. Proposed Property 2R counterexamples examined. Illinois J. Math. 60 (2016), no. 1, 207--250. doi:10.1215/ijm/1498032031. https://projecteuclid.org/euclid.ijm/1498032031