I am interested in simply connected rational homology spheres. The first such example is in dimension 5 and it is the Wu manifold $SU(3)/SO(3)$. You can find a discussion about simply connected rational spheres in: Simply-connected rational homology spheres.
I need to understand the geometric properties of this manifold in order to construct mappings with certain properties. So my questions are:
1. Why is the Wu manifold important except for being a simply connected rational homology sphere?
2. Where can I read about the geometric properties of the Wu manifold?
I am interested more in geometric properties of $SU(3)/SO(3)$ rather than its applications to algebraic topology since, it seems, I have to construct certain mappings explicitly without using algebraic topology. I will describe my problem later in another post.