Intereting Posts

How find this integral limt
Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.
What is bigger, $p(\mathbb{N})$ or $\mathbb{R}$?
Simple proof of L'Hôpital's Rule
At least one prime between $\sqrt{n}$ and $n$?
Real analyticity of a function
Why is $\tan((1/2)\pi)$ undefined?
Visualizing quotient groups: $\mathbb{R/Q}$
Is the statement $A \in A$ true or false?
How to calculate the sumation of a function in one step?
Is it possible that all subseries converge to irrationals?
In how many ways can five letters be posted in 4 boxes?
How many Turing degrees are there?
“Semidirect product” of graphs?
Prove that $f$ has a minimum

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 $w\bar{w}=1$.

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.

- Hatcher chapter 0 exercise.
- covering map with finite fibres and preimage of a compact set
- If $\|\left(f'(x)\right)^{-1}\|\le 1 \Longrightarrow$ $f$ is an diffeomorphism
- Higher homology group of Eilenberg-Maclane space is trivial
- Long exact sequence for cohomology with compact supports
- Quotient space and Retractions

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.

- Homology groups of the Klein bottle
- Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions
- Formula relating Euler characteristics $\chi(A)$, $\chi(X)$, $\chi(Y)$, $\chi(Y \cup_f X)$ when $X$ and $Y$ are finite.
- Is composition of covering maps covering map?
- Homology of the Klein Bottle
- Embeddability of the cone of Klein bottle to $\mathbb R^4$
- $H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$ are not isomorphic as rings
- Homotopy groups of $S^2$
- Contractible vs. Deformation retract to a point.
- Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

- maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$
- Is the pointwise maximum of two Riemann integrable functions Riemann integrable?
- How to prove that a simple graph having 11 or more vertices or its complement is not planar?
- Creating a sequence that does not have an increasing or a decreasing sequence of length 3 from a set with 5 elements
- (Fast way to) Get a combination given its position in (reverse-)lexicographic order
- simplify $\sqrt{5+2\sqrt{13}}+\sqrt{5-2\sqrt{13}}$
- Highest power of a prime $p$ dividing $N!$
- Estimate total song ('coupon') number by number of repeats
- What is the fastest way to multiply two digit numbers?
- Is $$ the union of $2^{\aleph_0}$ perfect sets which are pairwise disjoint?
- Does an absolutely integrable function tend to $0$ as its argument tends to infinity?
- Evaluating the sums $\sum\limits_{n=1}^\infty\frac{1}{n \binom{kn}{n}}$ with $k$ a positive integer
- Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
- Proof that $\displaystyle\sum_{n=0}^{\infty}{\frac{x^{n}}{n!}}={\left(\sum_{n=0}^{\infty}{\frac{1}{n!}}\right)}^{x}$?
- Dropping the “positive” and “decreasing” conditions in the integral test