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SIGMA 15 (2019), 078, 16 pages arXiv:1811.03613
https://doi.org/10.3842/SIGMA.2019.078
The Transition Function of $G_2$ over $S^6$
Ádám Gyenge
Mathematical Institute, University of Oxford, UK
Received May 23, 2019, in final form September 26, 2019; Published online October 09, 2019
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 $\pi_{5}({\rm SU}(3))$.
Key words: $G_2$; six-sphere; octonions; fibration; transition function.
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References
- Baez J.C., The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205, arXiv:math.RA/0105155.
- Bott R., The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313-337.
- Chaves L.M., Rigas A., Complex reflections and polynomial generators of homotopy groups, J. Lie Theory 6 (1996), 19-22.
- Ehresmann C., Sur les variétés presque complexes, in Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Vol. 2, Amer. Math. Soc., Providence, R.I., 1952, 412-419.
- Lamont P.J.C., Arithmetics in Cayley's algebra, Proc. Glasgow Math. Assoc. 6 (1963), 99-106.
- Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Vol. 218, Springer-Verlag, New York, 2003.
- Pirisi R., Talpo M., On the motivic class of the classifying stack of $G_2$ and the spin groups, Int. Math. Res. Not. 2019 (2019), 3265-3298, arXiv:1702.02649.
- Postnikov M., Lectures in geometry. Semester V: Lie groups and Lie algebras, Mir, Moscow, 1986.
- Püttmann T., Rigas A., Presentations of the first homotopy groups of the unitary groups, Comment. Math. Helv. 78 (2003), 648-662, arXiv:math.AT/0301192.
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