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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.

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This seems to be a difficult problem; in this link, it is mentioned that Bill Richter computed the attaching maps in his University of Washington PhD thesis using Morse theory (I haven't been able to find it online, but if you contact the university, they should be able to help). Also, they mention a "1934 Annals paper by Ehresmann" discussing this topic, and point out that this is before spectral sequences. There is an unanswered Mathoverflow question discussing all of this as well:

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