Aleksandar Milivojevic, Max Planck Insitute for Mathematics

Title: The space of almost complex structures on the six sphere

Abstract: By thinking of the six sphere S⁶ as the unit sphere in the imaginary octonions, one detects a real projective seven-space RP⁷ in the space of all almost complex structures on S⁶. On the other hand, using the Haefliger-Sullivan rational homotopy theoretic model for the space of sections of a fiber bundle applied to the twistor space construction, one can abstractly calculate that the rational homology of the space of almost complex structures on S⁶ agrees with that of RP⁷. Sullivan asked whether the inclusion of the octonionic RP⁷ into the space of all almost complex structures is a homotopy equivalence. We show that it is not, though it is a rational homology equivalence that induces an isomorphism on fundamental groups. We can further describe the homotopy fiber of this inclusion. This is joint work with Bora Ferlengez and Gustavo Granja.