Aleksandar Milivojevic, Max Planck Institute for Mathematics, Bonn
"Topological aspects of almost complex structures on the six sphere"
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 (orientation-compatible) 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.
On a related note, over six-manifolds, almost complex structures correspond to embedded half-dimensional submanifolds of the twistor space, and hence one obtains numerical invariants via their homological intersection. Time permitting, we compute these numbers concretely over the six-sphere and other six-manifolds, and comment on their relation to integrability.
This is joint work with Bora Ferlengez and Gustavo Granja.