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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.

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    $\begingroup$ I don't know if this is what you're looking for, but for me the Wu manifold is important because it generates the group of closed 5-dimensional manifolds under cobordism. Its class in the cobordism ring is the lowest-degree class not generated by projective spaces. It also represents the lowest-degree torsion in the oriented cobordism ring. $\endgroup$ – Arun Debray Apr 11 '18 at 15:12
  • $\begingroup$ google says it has a metric with non-negative sectional curvature, (which is apparently quite rare) $\endgroup$ – Nick L Apr 11 '18 at 16:27
  • $\begingroup$ Would "Geometric properties of the Wu manifold" be a better title for this question? $\endgroup$ – j.c. Apr 11 '18 at 16:56
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    $\begingroup$ @j.c. I changed the title, but I still want rational homology spheres to be there, because it is a primary reason why I am interested in the Wu manifold. $\endgroup$ – Piotr Hajlasz Apr 11 '18 at 17:49
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    $\begingroup$ Could you say more on what you mean by "geometric properties"? And what mappings you wish to construct, i..e., from where to where? In any case I woud start by searching in mathscinet for SU(n)/SO(n) in the title field. In my mirror it yields 66 items many of which seem relevant. $\endgroup$ – Igor Belegradek Apr 11 '18 at 19:21
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Another feature of the Wu manifold which makes it an important example is that it is not spin${}^c$. In contrast, every closed orientable manifold of dimension at most four is spin${}^c$.

One geometric property of the Wu manifold is that it admits a metric of positive scalar curvature. Gromov and Lawson showed that a simply connected non-spin manifold of dimension at least five always admits a metric of positive scalar curvature.

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