Calegari's homotopy 4-spheres from fibered knots are standard

TL;DR

This paper proves that all smooth homotopy 4-spheres constructed from fibered knots by Calegari are actually standard 4-spheres, using handlebody techniques and mapping class group results, with implications for the Schoenflies conjecture.

Contribution

It demonstrates that Calegari's homotopy 4-spheres are diffeomorphic to the standard 4-sphere, providing new insights into 4-manifold topology and potential counterexamples to the Schoenflies conjecture.

Findings

Calegari's homotopy 4-spheres are standard

Handlebody techniques are effective in 4-manifold classification

Potential counterexamples to the smooth 4D Schoenflies conjecture are identified

Abstract

In 2009, Calegari constructed smooth homotopy 4-spheres from monodromies of fibered knots. We prove that all these are diffeomorphic to the standard 4-sphere. Our method uses 5-dimensional handlebody techniques and results on mapping class groups of 3-dimensional handlebodies. As an application, we present potential counterexamples to the smooth 4-dimensional Schoenflies conjecture which are related to the work of Casson and Gordon on fibered ribbon knots.