# What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.

The study of characteristic classes tells us that these Schubert cells generate the cohomology of the Grassmannian and so the differentials in the cochain complex with $\mathbb{Z}/2$ coefficients are all zero.

I am studying the $RO(\mathbb{Z}/2)$-graded equivariant cohomology of Grassmann manifolds and there can be non-trivial differentials (in the appropriate spectral sequence) arising from a non-trivial attaching of a high dimensional cell to lower dimensional cells which will affect the cohomology. (This differs from the non-equivariant setting in that attaching an $n$-dimensional cell influences only the adjacent cohomology groups and no others. It turns out that attaching an $n$-dimensional cell can affect the equivariant cohomology in a larger range of dimensions.)

Hence, I am interested in knowing which of these Schubert cells are non-trivially attached to which other cells and how to detect this behavior.

-