**
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 S^{6} as the
unit sphere in the imaginary octonions, one detects a real projective
seven-space RP^{7} in the space of all almost complex
structures on S^{6}. 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 S^{6} agrees with that of
RP^{7}. Sullivan asked whether the inclusion of the octonionic
RP^{7} 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.