A homological generalized Property R conjecture is false

TL;DR

This paper disproves a generalized form of the Property R conjecture by providing counterexamples of 2-component links in S^3 that produce certain 3-manifolds through surgery but are not handleslide equivalent to split links.

Contribution

It introduces counterexamples to a broader generalization of the Property R conjecture, showing that certain links do not behave as previously conjectured under surgery.

Findings

Counterexamples of 2-component links surgering to connected sums of homology S^1×S^2's

These links are not handleslide equivalent to split links

Disproves a generalized Property R conjecture in homological settings

Abstract

The generalized Property R conjecture (GPRC) predicts that if framed surgery on an $n$-component link $L$ in $S^3$ produces $\#^{n} (S^1\times S^2)$, then $L$ is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an $n$-component link in $S^3$ surgers to the connected sum of $n$ three-manifolds with the homology of $S^1 \times S^2$, then this link should be handleslide equivalent to an $n$-component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in $S^3$ that surger to a connected sum of homology $S^1\times S^2$'s but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.