The Non-Existent Complex 6-Sphere

2016/10/25 by Michael Atiyah, Atiyah, Michael
Mathematics · #53A55 #53B15 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1610.09366

openalex publication_date 2016/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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. Mistakes have always been found. In this paper I present a short proof of the non-existence, based on ideas developed, but not fully exploited, over 50 years ago. The only change in v2. is in section 3, where the notation has been clarified.

Related

• Non-existence of orthogonal complex structures on 6-sphere with a metric close to the round one
• The complex structure on the six dimensional sphere
• Hartogs phenomenon and the problem of complex spheres
• S.-S. Chern's study of almost-complex structures on the six-sphere
• The Non-existence of Complex Sphere Sⁿ (n>2)