Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 1

Round handle problem

Pages: 237 – 347

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a6


Min Hoon Kim (Department of Mathematics, Chonnam National University, South Korea)

Mark Powell (Department of Mathematical Sciences, Durham University, Durham, United Kingdom)

Peter Teichner (Max Planck Institut für Mathematik, Bonn, Germany)


We present the Round Handle Problem (RHP), proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings (GBRs), are slice in a $4$‑manifold $R$ constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and the $s$-cobordism conjecture for $4$‑manifolds with free fundamental group.


round handle problem, topological surgery, $s$-cobordism

2010 Mathematics Subject Classification

Primary 57M25, 57M27. Secondary 57N13, 57N70.

The first author was partly supported by NRF grant 2019R1A3B2067839.

The second author was supported by an NSERC Discovery Grant.

Received 21 October 2019

Accepted 6 July 2020

Published 11 April 2021