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G2/SU(3) is the 6-sphere

Contents

Context

Group Theory

Geometry

Contents

Statement

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:

standard:
S n diffSO(n+1)/SO(n)S^n \simeq_{diff} SO(n+1)/SO(n)this Prop.
S 2k+1 diffSU(k+1)/SU(k)S^{2k+1} \simeq_{diff} SU(k+1)/SU(k)this Prop.
exceptional:
S 7 topSpin(7)/G 2S^7 \simeq_{top} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 topSpin(6)/SU(3)S^7 \simeq_{top} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 topSpin(5)/SU(2)S^7 \simeq_{top} Spin(5)/SU(2)Spin(5)/SU(2) is the 7-sphere
S 6 topG 2/SU(3)S^6 \simeq_{top} G_2/SU(3)G2/SU(3) is the 6-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

References

According to ABF 17 the statement is originally due to to

Review:

  • 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 19, 2019 at 16:19:46. See the history of this page for a list of all contributions to it.