What are the attaching maps for the real Grassmannian?

Solution 1:

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:https://mathoverflow.net/questions/19980/attaching-maps-for-grassmann-manifolds