Geodesic
The Transition Function of $G_2$ over $S^6$
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2019_15_a77,
     author = {\'Ad\'am Gyenge},
     title = {The {Transition} {Function} of $G_2$ over $S^6$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a77/}
}
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Ádám Gyenge. The Transition Function of $G_2$ over $S^6$. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a77/

[1] Baez J. C., “The octonions”, Bull. Amer. Math. Soc., 39 (2002), 145–205, arXiv: math.RA/0105155 | DOI | MR | Zbl

[2] Bott R., “The stable homotopy of the classical groups”, Ann. of Math., 70 (1959), 313–337 | DOI | MR | Zbl

[3] Chaves L. M., Rigas A., “Complex reflections and polynomial generators of homotopy groups”, J. Lie Theory, 6 (1996), 19–22 | MR | Zbl

[4] Ehresmann C., “Sur les variétés presque complexes”, Proceedings of the International Congress of Mathematicians (Cambridge, Mass., 1950), v. 2, Amer. Math. Soc., Providence, R.I., 1952, 412–419 | MR

[5] Lamont P. J. C., “Arithmetics in Cayley's algebra”, Proc. Glasgow Math. Assoc., 6 (1963), 99–106 | DOI | MR | Zbl

[6] Lee J. M., Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, Springer-Verlag, New York, 2003 | DOI | MR

[7] 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 | DOI | MR | Zbl

[8] Postnikov M., Lectures in geometry. Semester V: Lie groups and Lie algebras, Mir, M., 1986 | MR

[9] 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 | DOI | MR | Zbl