Contents

group theory

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Statement

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

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

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)

and

SU(2)$\simeq$ Sp(1)

as a special case of the general statement

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

(see this Prop.).

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 