# On the history of the Hopf problem

@article{Agricola2017OnTH,
title={On the history of the Hopf problem},
author={Ilka Agricola and Giovanni Bazzoni and Oliver Goertsches and Panagiotis Konstantis and Sonke Rollenske},
journal={arXiv: History and Overview},
year={2017}
}
This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on $S^6$?" This unsolved mathematical question was the subject of the Conference "MAM 1 $-$ (Non-)Existence of Complex Structures on $S^6$", which took place at Philipps-Universit\"at Marburg, Germany, between March 27th and March 30th, 2017.
3 Citations
On the minimal sum of Betti numbers of an almost complex manifold.
• Mathematics
• 2018
We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rationalExpand
Almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres
In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra Ca′.Expand
The almost complex (para-complex) structures on 6-pseudo-Riemannian spheres and related Schrödinger flows
In this paper, by using the $$G_{2(2)}$$ G 2 ( 2 ) -structure on $$\hbox {Im}(\mathbf{Ca'})\cong {\mathbb {R}}^{3,4}$$ Im ( Ca ′ ) ≅ R 3 , 4 of the purely imaginary Cayley’s split-octavesExpand

#### References

SHOWING 1-10 OF 55 REFERENCES
$S^6$ and the geometry of nearly K\"ahler $6$-manifolds
• Mathematics
• 2017
We review results on and around the almost complex structure on $S^6$, both from a classical and a modern point of view. These notes have been prepared for the Workshop "(Non)-existence of complexExpand
Chern's contribution to the Hopf problem: An exposition based on Bryant's paper
• Mathematics
• 2017
Abstract We give a comprehensive account of Chern's Theorem that S 6 admits no ω -compatible almost complex structures. No claim to originality is being made, as the paper is mostly an expandedExpand
Almost complex structures on spheres
• Mathematics
• 2017
In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S^2 and S^6. The proof described here uses characteristic classes and the Bott periodicityExpand
The Non-Existent Complex 6-Sphere
The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many "solutions" have been put forward, in both directions. MistakesExpand
Hodge Numbers of a Hypothetical Complex Structure on the Six Sphere
We prove that the terms Erp,q(S6) in the Frölicher spectral sequence associated to any hypothetical complex structure on S6 would satisfy Serre duality. It is also shown that the vanishing of theExpand
The complex geometry of a hypothetical complex structure on $S^6$.
• Mathematics
• 2017