Generalized square knots and homotopy $4$-spheres

Abstract

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.