Intereting Posts

What is the difference between strategies and actions?
There is no isometry between a sphere and a plane.
Permutations with restrictions on item positions
Solve the equation $(2^m-1) = (2^n-1)k^2$
$\mathbb{Q}/\mathbb{Z}$ is divisible
Monic polynomial reducible in rationals
Principal ideal of an integrally closed domain
To characterize uncountable sets on which there exists a metric which makes the space connected
Metric is continuous, on the right track?
Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?
What is the cardinality of the set of all topologies on $\mathbb{R}$?
Undecidable conjectures
Coproducts in $\text{Ab}$
Truth of $x^2-2=0$

I’m reading a paper called *An Additive Basis for the Cohomology of Real Grassmannians*, which begins by making the following claim (paraphrasing):

Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over $G_m(\mathbb{R}^{m+n})$ and let $\bar{w}=1+\bar{w_1}+\ldots+ \bar{w_n}$ be its dual. Then $H^\ast G_m (\mathbb{R}^{m+n})$ is the quotient of the polynomial algebra $\mathbb{Z}_2[1,w_1,\ldots,w_m]$ by the ideal $(\bar w_{m+1},\cdots,\bar {w}_{m+n})$ generated by the relation $w\bar{w}=1$.

- Vanishing of the first Chern class of a complex vector bundle
- Quick question: Chern classes of Sym, Wedge, Hom, and Tensor
- Chern Classes and Stiefel-Whitney Classes
- First Pontryagin class on real Grassmannian manifold?
- On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.
- Elementary proof of the fact that any orientable 3-manifold is parallelizable

The reference provided is to Borel’s *La cohomolgie mod 2 de espaces homogenes*. As the title suggests, this paper is in French, a language with which I am not familiar.

I’m familiar with the fact that the cohomology ring of the infinite Grassmannian $G_m(\mathbb{R}^\infty)$ is freely generated by $w_1,\ldots,w_m$ over $\mathbb{Z}_2$ (as proved in Hatcher’s *Vector Bundles*), but I can’t see how to prove this variant. Any help would be much appreciated. Perhaps someone can even translate the proof given in Borel’s paper.

- Second Stiefel-Whitney Class of a 3 Manifold
- Calculate the Wu class from the Stiefel-Whitney class
- Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$
- Chern Classes and Stiefel-Whitney Classes
- The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?
- Relation between Stiefel-Whitney class and Chern class
- Elementary proof of the fact that any orientable 3-manifold is parallelizable
- How to interpret the Euler class?

- bound of number of cycles in composing two permutations
- Proof/Disproof of property of perpendicular lines in the Fibonacci grid
- How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$
- Why do people interchange between $\int$ and $\sum$ so easily?
- Axiom of Choice: Can someone explain the fallacy in this reasoning?
- Non-Euclidean Geometrical Algebra for Real times Real?
- How is a set subset of its power set?
- How to integrate a vector function in spherical coordinates?
- Equivalence of Quadratic Forms that represent the same values
- cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence
- Every separable Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$
- Is there a formula for $\sum_{n=1}^{k} \frac1{n^3}$?
- Better Notation for Partial Derivatives
- limit of a sequence with $\pi $ and $ e $
- What is $\sum\limits_{i=1}^n \sqrt i\ $?