New York Journal of Mathematics
Volume 27 (2021), 1258-1273


Bora Ferlengez, Gustavo Granja, and Aleksandar Milivojevic

On the topology of the space of almost complex structures on the six sphere

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Published: August 18, 2021.
Keywords: almost complex structures, six sphere.
Subject: Primary 57R22, Secondary 53C15.

The space of orientation-compatible almost complex structures on the six-dimensional sphere naturally contains a copy of seven-dimensional real projective space. We show that the inclusion induces an isomorphism on fundamental groups and rational homotopy groups. We also compute the homotopy fiber of the inclusion and the homotopy groups of the space of almost complex structures in terms of the homotopy groups of the seven-dimensional sphere. Our approach lends itself to generalization to components of almost complex structures with vanishing first Chern class on six-manifolds.


We thank Luis Fernandez and Scott Wilson for useful comments, and the referee for helping improve the exposition; the third author likewise thanks Claude LeBrun and Dennis Sullivan. The second author was partially supported by FCT/Portugal through project UIDB/MAT/04459/2020.

Author information

Bora Ferlengez:
Deparment of Mathematics and Statistics
Hunter College of CUNY
695 Park Avenue, 01220 New York, USA


Gustavo Granja:
Center for Mathematical Analysis, Geometry and Dynamical Systems
Instituto Superior Técnico
Universidade de Lisboa
Av. Rovisco Pais, 1049-001 Lisboa, Portugal


Aleksandar Milivojevic:
Mathematics Department
Stony Brook University
100 Nicolls Road
Stony Brook, 11794 New York, USA