Math 2.0

Several complex variables and analytic spaces Compact complex manifolds Problem

Complex structure on the six-sphere

Builds on Complex manifoldAlmost complex manifoldOctonions

Problem

Decide whether the sphere S6S^6 admits a complex structure. That is, decide whether there is a complex manifold of complex dimension 33 whose underlying smooth manifold is diffeomorphic to the standard S6S^6. Equivalently, decide whether S6S^6 carries an integrable almost complex structure.

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. S6S^6 has almost complex structures; one comes from identifying S6S^6 with the unit imaginary octonions and using octonion multiplication.
  2. Borel and Serre showed that S2S^2 and S6S^6 are the only spheres with an almost complex structure; S2S^2 is the Riemann sphere, so S6S^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 S6S^6 is orthogonal with respect to the round metric.

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

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Almost complex manifold, concept, no evidence levelAlmost complex manifo…conceptComplex manifold, concept, no evidence levelComplex manifoldconceptGroup, concept, no evidence levelGroupconceptNonassociative ring, concept, no evidence levelNonassociative ringconceptRing, definition, no evidence levelRingdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleOctonions, object, no evidence levelOctonionsobjectComplex structure on the six-sphere, problem, no evidence levelComplex structure on …problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost complex manifold concept
  2. Complex manifold concept
  3. Group concept
  4. Nonassociative ring concept
  5. Ring definition
  6. Unit of a ring definition
  7. Division ring definition
  8. The quaternions example
  9. Octonions object
  10. Complex structure on the six-sphere problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Almost_complex_manifold
  2. Web https://arxiv.org/abs/1708.01068
About this page
TagV4M2HPJQPR
URIm2://V4M2HPJQPR
Kindproblem
WritingW0 stub
MSC32Q60, 53C15
StatusThis copy of the site has no build data. Run m2 status V4M2HPJQPR.
AboutComplex manifold, Almost complex manifold
UsesComplex manifold, Almost complex manifold, Octonions

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