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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    $\begingroup$ Hi Bill, so do you want something more than the integral chain complex? $\endgroup$ – Ryan Budney Mar 31 '10 at 17:56
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    $\begingroup$ 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. $\endgroup$ – Bill Kronholm Mar 31 '10 at 18:13
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    $\begingroup$ 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. $\endgroup$ – Allen Hatcher Mar 31 '10 at 20:08
  • $\begingroup$ Thanks Allen. Looks like it's time to dust off my French-English dictionary! :) $\endgroup$ – Bill Kronholm Mar 31 '10 at 21:13

The degrees of the attaching maps (and hence the integral chain complex) for the real Grassmannians have been determined in

  • L. Casian and Y. Kodama. On the cohomology of real Grassmann manifolds. arXiv:1309.5520 (Link to arXiv)

The integral chain complex is determined by an explicit combinatorial procedure involving the Young diagram representations of Schubert cells. There are other papers by the authors in which they determine the incidence graphs of Grassmannians. Maybe the techniques used there can also help to obtain more precise information on the attaching maps.

Concerning the attaching maps for the complex Grassmannians, there is some discussion in the following paper:

  • C. Lenart. The combinatorics of Steenrod operations on the cohomology of Grassmannians. Adv. Math. 136 (1998), no. 2, 251–283. (link to paper on ScienceDirect)

In the final section, you find the specific example $Gr_2(\mathbb{C}^5)$ and remarks on how to detect non-trivial cell attachment using the action of Steenrod operations.

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