nLab G2/SU(3) is the 6-sphere



Group Theory




The coset space of the exceptional Lie group G2 by its special unitary subgroup SU(3) is diffeomorphic to the 6-sphere:

G 2/SU(3)S 6. G_2/SU(3) \;\simeq\; S^6 \,.

coset space-structures on n-spheres:

S n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
S 7 diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 diffSpin(5)/SU(2)S^7 \simeq_{diff} Spin(5)/SU(2)since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
S 6 diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15 diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7)Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)


According to Gray-Green 70, p. 2 the statement is originally due to

  • Deane Montgomery, Hans Samelson, Transformation Groups of Spheres, Annals of Mathematics Second Series, Vol. 44, No. 3 (Jul., 1943), pp. 454-470 (jstor:1968975)

According to ABF 17 the statement is originally due to to


  • Alfred Gray, Paul S. Green, p. 2 of Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)

  • Robert Bryant, Section 1.3 of Manifolds with G 2G_2-Holonomy, lecture notes, 1998 (web)

  • Robert Bryant, Section 2.3 of Some remarks on G 2G_2-structures, Proceedings of 12th Gokova Geometry-Topology Conference, 2005 (pdf)

  • Simon Salamon, p. 6 of A tour of exceptional geometry, Milanj. math.72 (2002)1-0 (web)

  • A. J. MacFarlane, The sphere S 6S^6 viewed as a G 2/SU(3)G_2/SU(3)-coset space, International Journal of Modern Physics A Vol. 17, No. 19, pp. 2595-2613 (2002) (doi:10.1142/S0217751X02010650)

  • A. Gyenge, Section 4.2 of On the topology of the exceptional Lie group G 2G_2, 2011 (pdf)

  • Ilka Agricola, Aleksandra Borówka, Thomas Friedrich, S 6S^6 and the geometry of nearly Kähler 6-manifolds (arXiv:1707.08591)

Last revised on April 29, 2019 at 08:40:31. See the history of this page for a list of all contributions to it.