- Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented [stable special orthogonal $SO$] manifold)?
Dold manifold $(\mathbf{CP}^2 \times S^1)/\mathbf{Z}_2$, where we identify the $(z,x) \in (\mathbf{CP}^2 \times S^1)/\mathbf{Z}_2$ with $(\bar{z},-x) \in (\mathbf{CP}^2 \times S^1)/\mathbf{Z}_2$.
Wu manifold: $SU(3)/SO(3)$.
- What are examples of the 6-dimensional oriented manifold, which have Dold manifold and Wu manifold as two boundaries?
p.s. Of course, there is a sophisticated way of checking the $\Omega_5^{SO}=\mathbf{Z}_2$ has only a cobordism invariant $w_2(M)w_3(M)$. It can be shown that both Dold manifold and Wu manifold take nontrivial values under $w_2(M)w_3(M)$. But this is not intuitive enough for me to fully understand the manifold structure. Any simple reasonings, derivations, intuitions and arguments are welcome!
