Ideal classes and Cappell-Shaneson homotopy 4-spheres

TL;DR

This paper investigates Gompf's conjecture on Cappell-Shaneson matrices, revealing a symmetry that links different trace cases and confirming the conjecture for a broad range of traces, thereby showing many such spheres are standard.

Contribution

The paper introduces a symmetry between trace n and trace 5-n Cappell-Shaneson matrices and proves the conjecture for traces between -64 and 69, also constructing a new infinite family of standard spheres.

Findings

Confirmed Gompf conjecture for traces between -64 and 69.

Discovered a symmetry linking trace n and trace 5-n matrices.

Constructed a new infinite family of standard Cappell-Shaneson spheres.

Abstract

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace $n$ Cappell-Shaneson matrices and trace $5-n$ Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace $n$ case is equivalent to the trace $5-n$ case. We confirm Gompf conjecture for the special cases that $-64\leq \textrm{trace}\leq 69$ and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.