nLab Spin(5)/SU(2) is the 7-sphere



Group Theory




The coset space of Spin(5) by its subgroup SU(2) is diffeomorphic to the standard 7-sphere:

(1)Spin(5)/SU(2) diffS 7. Spin(5)/SU(2) \;\simeq_{diff}\; S^7 \,.

This is however not an isometry to the standard Riemannian manifold-structure (“round n-sphere”), whence one speaks of a squashed n-sphere.

The identification (1) follows via the exceptional isomorphisms

Spin(5)\simeq Sp(2)


SU(2)\simeq Sp(1)

as a special case of the general statement

S 4n1 diffSp(n)/Sp(n1) S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)

(see this Prop.).

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)


  • 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)

Discussion of this squashed 7-sphere coset space as a fiber for KK-compactification of 11-dimensional supergravity:

Last revised on April 29, 2019 at 08:36:47. See the history of this page for a list of all contributions to it.