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

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

2011-01-26

arxiv.org/abs/1101.5147v3

Mathematics

Geometric Topology

27 pages, applications emphasized, title extended, 2 typos corrected

Scientific paper

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).

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