nLab 6-sphere

Idea

The n-sphere of dimension $n = 6$.

Properties

Coset structure

The 6-sphere, as a smooth manifold is diffeomorphic to the coset space

$$S^6 \simeq G_2/ SU(3)$$

of G₂ (automorphism group of the octonions) by SU(3) (Fukami-Ishihara 55).

For more see at G₂/SU(3) is the 6-sphere.

The induced action of G₂ on $S^6$ induces an almost Hermitian structure which makes it a nearly Kaehler manifold?.

Review in is in Agrikola-Borowka-Friedrich 17

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

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

Complex structure — The Hopf problem

A famous open problem — known as the Hopf problem, after Heinz Hopf 1947 — was the question whether the 6-sphere admits an actual complex structure.

For review see Bryant 2014.

An AI-generated proof proposal of the existence of such complex structure was presented by Alpöge 2026 and was then claimed by Alexeev 2026 to have been verified via the Lean proof assistant.

Related entries

• fuzzy 6-sphere

• 2-sphere

• 3-sphere

• 4-sphere

• 7-sphere

• n-sphere

References

General

• T. Fukami, S. Ishihara, Almost Hermitian structure on $S^6$ , Tohoku Math J. 7 (1955), 151–156.

• Ilka Agricola, Aleksandra Borówka, Thomas Friedrich: $S^6$ and the geometry of nearly Kähler 6-manifolds, Differential Geometry and its Applications 57 (2018) 75-86 [arXiv:1707.08591, doi:10.1016/j.difgeo.2017.10.007]

Complex structure

On the Hopf problem of existence of a complex structure on the 6-sphere:

• Robert Bryant, S.-S. Chern’s study of almost-complex structures on the six-sphere (arXiv:1405.3405)

• Robert Bryant: Remarks on the geometry of almost complex 6-manifolds (arXiv:math/0508428)

Claim of a proof of existence:

• Levent Alpöge (24 Aug 2026) [x.com:2091639597193368014]

  (AI-generated proof proposal)

• Boris Alexeev: HopfProblem (2026) [github:plby/HopfProblem]

  (claims proof check via Lean)