Detecting exotic spheres in low dimensions using coker J

TL;DR

This paper investigates the differentiable structures of spheres in low dimensions, identifying specific dimensions where spheres have unique smooth structures, extending previous mathematical classifications.

Contribution

It extends the classification of spheres with unique differentiable structures to certain even dimensions below 140, building on prior work by Kervaire, Milnor, Hill, Hopkins, and Ravenel.

Findings

Unique differentiable structures in spheres at dimensions 2, 6, 12, 56, and possibly 4.

Confirmation of known results for odd dimensions 1, 3, 5, and 61.

New results on even dimensions below 140.

Abstract

Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which S^n has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which S^n has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.