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

TL;DR

This paper investigates the topological properties of the space of almost complex structures on the six-sphere, revealing its fundamental group and rational homotopy group relations with projective space and spheres.

Contribution

It establishes an isomorphism between the fundamental groups of the space of almost complex structures and real projective space, and computes the homotopy fiber and groups in relation to the sphere.

Findings

Inclusion induces an isomorphism on fundamental groups.

Computed homotopy fiber and homotopy groups of the space.

Results applicable to six-manifolds with vanishing first Chern class.

Abstract

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.