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

Thomas Puettmann

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arXiv · 1101.5147

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

Thomas Puettmann

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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.

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Thomas Puettmann. 2011-01-26. On the commutator of unit quaternions and the numbers 12 and 24. https://arxiv.org/abs/1101.5147

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