---
tag: V4M2HPJQPR
kind: problem
name: Complex structure on the six-sphere
msc: [32Q60, 53C15]
uses: [JSGWAN2RJV, 9SYWRGVTRS, 1HRMT3DSX2]
about: [JSGWAN2RJV, 9SYWRGVTRS]
sources:
  - url: https://en.wikipedia.org/wiki/Almost_complex_manifold
  - url: https://arxiv.org/abs/1708.01068
history:
  posed_by: "Heinz Hopf"
  first_stated: "1948"
---

## Problem

Decide whether the sphere $S^6$ admits a complex structure.
That is, decide whether there is a [[complex manifold|JSGWAN2RJV]] of complex dimension $3$ whose underlying smooth manifold is diffeomorphic to the standard $S^6$.
Equivalently, decide whether $S^6$ carries an integrable [[almost complex structure|9SYWRGVTRS]].

## Remarks

The problem is open; several announced solutions have not been accepted.
It is often called the Hopf problem, after Heinz Hopf, who raised it in a paper published in 1948.

What is known:

1. $S^6$ has almost complex structures; one comes from identifying $S^6$ with the unit imaginary [[octonions|1HRMT3DSX2]] and using octonion multiplication.
2. Borel and Serre showed that $S^2$ and $S^6$ are the only spheres with an almost complex structure; $S^2$ is the Riemann sphere, so $S^6$ is the only open case.
3. The almost complex structure coming from the octonions is not integrable.
4. LeBrun showed that no complex structure on $S^6$ is orthogonal with respect to the round metric.

By the [[Newlander–Nirenberg theorem|VCT848WV1T]], the problem asks for an almost complex structure whose Nijenhuis tensor vanishes.

## Explanations

### L1 human

The two-dimensional sphere is the Riemann sphere, the simplest compact complex manifold.
The question is whether the six-dimensional sphere can also be made into a complex manifold.
It is known to have the weaker structure of an almost complex manifold, which looks complex at each point separately.
Whether this weaker structure can be made to fit together into true complex coordinates is still unknown.