Contents

group theory

# Contents

## Statement

The coset space of Spin(7) by the exceptional Lie group G2 is homeomorphic to the 7-sphere:

$Spin(7)/G_2 \;\simeq\; S^7 \,.$

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

(from FSS 19, 3.4)

## References

• 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)

• Veeravalli Varadarajan, Theorem 3 in Spin(7)-subgroups of SO(8) and Spin(8), Expositiones Mathematicae, 19 (2001): 163-177 (pdf)

Last revised on April 29, 2019 at 04:36:32. See the history of this page for a list of all contributions to it.