More Cappell-Shaneson spheres are standard

TL;DR

This paper demonstrates that a larger family of Cappell-Shaneson homotopy 4-spheres are standard, using a simpler method that exploits hidden symmetries rather than Kirby calculus, extending previous results.

Contribution

It introduces a new, simpler approach to prove more Cappell-Shaneson spheres are standard by leveraging symmetries, avoiding complex Kirby calculus.

Findings

A larger family of Cappell-Shaneson spheres are shown to be standard.

Hidden symmetries can be used to simplify the proof of standardness.

Gluck twists can sometimes be undone using symmetries of fishtail neighborhoods.

Abstract

Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akbulut and Kirby showing that the simplest example with untwisted framing is standard. Instead, hidden symmetries of the original Cappell-Shaneson construction are exploited. In the course of the proof, we give an example showing that Gluck twists can sometimes be undone using symmetries of fishtail neighborhoods.