Cohomology ring of Grassmanian

Question

I am having trouble proving $H^*(G_n(\mathbb{R}^{n+k});\mathbb{Z}_2)=\mathbb{Z}_2[w_1,...,w_n]/<\bar{w}_{k+1},...,\bar{w}_n>$, where $w_i$ are the Stiefel-Whittney classes of the tautological bundle $\gamma^n$, and $\bar{w}_i$ are the dual classes, considered as polynomials in $w_i$.

A search through this site led me to: Cohomology ring of Grassmannians, where a comment reads

One approach might be to note that the relations hold on the infinite level, so via inclusion, you have a surjection from the algebra mod the relation onto the cohomology of the m-Grassmannian. Now, use the cell structure and make a dimension counting argument to prove it must be an isomorphism.

which seems to be accepted as a valid answer to the question.

Now I think I understand the approach outlined but I am having difficulty implementing it. From what I understand I have to show

(1) $H^*(G_n(R^{\infty})) \to H^*(G_n(R^{n+k}))$ induced by inclusion is surjective,

(2) the polynomials $\bar{w}_{k+1},...,\bar{w}_n$ are zero $H^*(G_n(R^{n+k}))$, thus $H^*(G_n(R^{\infty}))/<\bar{w}_{k+1},...,\bar{w}_n> = \mathbb{Z}_2[w_1,...,w_n]/<\bar{w}_{k+1},...,\bar{w}_n>\to H^*(G_n(R^{n+k}))$ is well-defined and surjective,

(3) verify that the dimension of both sides match thus conclude the map is an isomorphism.

I don't really understand why (1) is true, and worse still, I doubt how (2) can be true. Consider the following argument: Let $\xi$ be the n-fold Cartesian product of the tautological bundle $\gamma^1$ on $P^{n+k}=G_1(\mathbb{R}^{n+k})$, ie. $\xi=\gamma^1 \times...\times \gamma^1$ over $P^{n+k}\times...\times P^{n+k}$. There exists a canonical bundle map $\xi \to \gamma^n$. Meanwhile a calculation shows $w_j(\xi)$ is the $j^{th}$elementary symmetric polynomial of $\pi_i^*(\gamma^1)$, where $\pi_i:P^{n+k}\times...\times P^{n+k} \to P^{n+k}$ is the projection to $i^{th}$ summand. If $\bar{w}_{k+1}$, as polynomials in $w_j(\gamma^n)$, were zero in $H^*(G_n(\mathbb{R}^{n+k}))$, pulling back to $H^*(P^{n+k} \times ...\times P^{n+k})$ implies that same polynomial in $w_j(\xi)$, which is a degree $k+1$ polynomial in $\pi_j^*({\gamma^1})$ will be zero. But polynomials in $H^*(P^{n+k} \times ...\times P^{n+k})=\mathbb{Z}_2[\pi_1^*(\gamma^1),...,\pi_n^*(\gamma^1)]/<\pi_1^*(\gamma^1)^{n+k+1},...,\pi_n^*(\gamma^1)^{n+k+1}>$ won't vanish until we reach terms of degree $n+k+1$!

It will be very much appreciated if someone can point out what went wrong in my argument, and also indicate how steps (1) and (2) can be done. Reference to other proofs of this result are also welcomed. Thank you in advance.

Answer 1

The cohomology of $\mathrm{Gr}(k, n)$ the space of $k$-planes in $\mathbb{R}^n$ is usually described by the Schubert calculus. Here is an element of the Grassmanian:

$$\left\{ \begin{array}{ccc} x_1 &=& 0 \\ x_1 &=& 0 \\ &\vdots& \\ x_k &=& 0 \end{array} \right.$$ so that in away, the intersection theory of hyperplanes in $\mathbb{R}^n$ generalizes the study of simultaneous equations. Indeed certain basis of cohomology elements are indexed by their row-echelon form.

I am hesitant to say much about your torsion group $\mathbb{Z}_2$ - nor can I say a whole lot about your infinite bundle approach. However, I do encourage you to read Fulton's book Young Tableaux which discusses these ideas throughly.

---

An alternative approach by Allen Knuton and Terence Tao

• The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture
• The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone
• Honeycombs and sums of Hermitian matrices
• Puzzles and (equivariant) cohomology of Grassmannians

The 2nd and 4th references are certainly the most relevant to you, but do not take the Stiefel-Whitney class approach. Instead they motivate Schubert calculus from some rather boring problems on Hermitian matrices and Linear Algebra

---

These notes on Vector Bundles and K-Theory state this much:

Thm 3.9 $H^\ast(G_n, \mathbb{Z}_2)$ is the polynomial ring on the Stiefel-Whitney classes $w_i = W_i(E_n)$ of the universal bundle $E_n \to G_n$.

Stiefel-Whitney classes are for real vector bundles and Chern classes are for Complex vector bundles. The structure of this ring is pretty complicated, but also involves partitions of integers (and potentially, tableaux).