Round handle problem (Q2025309)

The round handle problem (RHP) asks whether for a certain collection of links, which contains the generalized Borromean rings, each such link is slice in the four ball with some round handles added. Here the precise way to add the round handles depends on the given link. The RHP was formulated in [\textit{M. Freedman} and \textit{V. Krushkal}, Forum Math. Sigma 4, Paper No. e22, 57 p. (2016; Zbl 1366.57010)], where it was shown that the 4-dimensional topological surgery conjecture together with the 4-dimensional topological \(s\)-cobordism conjecture yield a positive answer to the RHP. Thus the RHP might provide a way to disprove the surgery and \(s\)-cobordism conjecture. In the paper under review, the authors provide a simplified proof that the above mentioned conjectures imply a positive answer to the RHP. The proof in particular implies that every knot is round handle slice. The authors also show that the surgery and the \(s\)-cobordism conjecture together are equivalent to the disc embedding conjecture. This is not a new statement, but the authors provide a very nice guide to the relevant literature.