关于单位四元数的换位子与数12和24
On the commutator of unit quaternions and the numbers 12 and 24
摘要：
四元数是不可交换的。对交换性的偏离被封装在单位四元数的换位子中。已知交换子的k次幂是零同伦的当且仅当k能被12整除。本文的主要目的是构造一个具体的交换子的12次幂零同伦。随后，我们构造了S^7 × S^3上的自由S^3-作用，它的子空间是奇异的7-球面，并给出了稳定同伦群π_{n+3}（S^n）的阶的几何解释.中间结果可能是独立的兴趣是强调包含SU（3）\subset G_2的八元数的构造，杜兰测地边界映射构造的详细研究，以及丛G_2 \to S^6和Spin（7）\to S^7的特征映射的显式公式。
英文摘要：
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 π_{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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