The following question is particularly interesting for me:


Does the natural map $Gr(3,5)\to Gr(3,6)$ induce a surjection
$$H^4(Gr(3,6),\mathbb{Z})\to H^4(Gr(3,5),\mathbb{Z})?$$


Here $Gr(k,n)$ means the real grassmannian of rank $k$.



Cohomological behavior of the embedding $Gr(3,5)\to Gr(3,6)$
I think $H^i(Gr(k,n),\mathbb{Z})\to H^i(Gr(k,n-1),\mathbb{Z})$ is surjective if $n-k$ is odd and $H^i(Gr(k,n),\mathbb{Z})\to H^i(Gr(k-1,n-1),\mathbb{Z})$ is surjective if $n-k$ is even.


