## Attaching maps for Grassmann manifolds

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.

What is known about the attaching maps in this CW-complex structure?

I understand that a lot of work has been done to try to understand the answer to this question using things like Schubert calculus, Young diagrams, Steenrod operations, etc. I'd like to see some kind of collection of known results about the attaching maps and the specific techniques used to obtain those results.

I'm also interested in the case of the complex Grassmannians.

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Hi Bill, so do you want something more than the integral chain complex? – Ryan Budney Mar 31 2010 at 17:56
Ryan, I suppose I do want something more, since in the complex case all the differentials are zero per force. I'm really interested in knowing which cells are attached non-trivially to which other cells and how to detect this. – Bill Kronholm Mar 31 2010 at 18:13
There is an old paper by Charles Ehresmann in the 1934 Annals that studies in some detail how the cells are attached in Grassmann manifolds. – Allen Hatcher Mar 31 2010 at 20:08
Thanks Allen. Looks like it's time to dust off my French-English dictionary! :) – Bill Kronholm Mar 31 2010 at 21:13