On the commutator of unit quaternions and the numbers 12 and 24

TL;DR

This paper explores the algebraic and geometric properties of unit quaternions, constructing explicit null-homotopies for their commutators, and investigates related topological actions and homotopy groups, revealing new geometric insights.

Contribution

It constructs a concrete null-homotopy of the 12th power of the quaternion commutator and develops geometric models for exotic spheres and homotopy groups.

Findings

Constructed explicit null-homotopy for the 12th power of the commutator.

Developed free S^3-actions on S^7 x S^3 with exotic quotient spheres.

Provided geometric explanations for the order of certain stable homotopy groups.

Abstract

The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction of the octonions emphasizing the inclusion SU(3) \subset G_2, a detailed study of Duran's geodesic boundary map construction, and explicit formulas for the characteristic maps of the bundles G_2 \to S^6 and Spin(7) \to S^7.