The Transition Function of $G_2$ over $S^6$

Ádám Gyenge

Mathematical Institute, University of Oxford, UK

Abstract: We obtain explicit formulas for the trivialization functions of the ${\rm SU}(3)$ principal bundle $G_2 \to S^6$ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six-sphere. In this way we obtain a new proof of the known fact that this fibration corresponds to a generator of $\pi_{5}({\rm SU}(3))$.

Keywords: $G_2$, six-sphere, octonions, fibration, transition function.

MSC: 57S15, 55R10, 55R25

Received: May 23, 2019; in final form September 26, 2019; Published online October 9, 2019

Language: English

DOI: 10.3842/SIGMA.2019.078

Bibliographic databases:

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ArXiv: 1811.03613