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September 2022 Generalized square knots and homotopy $4$-spheres
Jeffrey Meier, Alexander Zupan
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J. Differential Geom. 122(1): 69-129 (September 2022). DOI: 10.4310/jdg/1668186788


The purpose of this paper is to study geometrically simply-connected homotopy $4$-spheres by analyzing $n$-component links in $S^3$ with a Dehn surgery realizing $\#^n (S^1 \times S^2)$. We call such links $n$R-links. Our main result is that a homotopy $4$-sphere that can be built without $1$-handles and with only two $2$-handles is diffeomorphic to the standard $4$-sphere in the special case that one of the $2$-handles is attached along a knot of the form $Q_{p,q} = T_{p,q} \# T_{-p,q}$, which we call a generalized square knot. This theorem subsumes prior results of Akbulut and Gompf.

Along the way, we use thin position techniques from Heegaard theory to give a characterization of $2$R-links in which one component is a fibered knot, showing that the second component can be converted via trivial handle additions and handleslides to a derivative link contained in the fiber surface. We invoke a theorem of Casson and Gordon and the Equivariant Loop Theorem to classify handlebody-extensions for the closed monodromy of a generalized square knot $Q_{p,q}$. As a consequence, we produce large families, for all even $n$, of $n$R-links that are potential counter-examples to the Generalized Property R Conjecture. We also obtain related classification statements for fibered, homotopy-ribbon disks bounded by generalized square knots.


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Jeffrey Meier. Alexander Zupan. "Generalized square knots and homotopy $4$-spheres." J. Differential Geom. 122 (1) 69 - 129, September 2022.


Received: 12 June 2019; Accepted: 26 February 2020; Published: September 2022
First available in Project Euclid: 11 November 2022

Digital Object Identifier: 10.4310/jdg/1668186788

Rights: Copyright © 2022 Lehigh University


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Vol.122 • No. 1 • September 2022
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