The cup product is supercommutative, i.e the supercommutator $[-,-]$ is trivial at the cohomology level — but not at the cochain level, which allows one to produce various cohomology operations. The simplest (in some sense) of such (integral) operations is the following «Massey cube». Suppose $a$ is an integral $k$-cocycle, $k$ is odd; $[a,a]=0\in H^{2k}$, […]

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: \begin{align} w=Sq(u),\ \ \ Sq=1+Sq^1+Sq^2 +\cdots . \end{align} The Wu classes can be defined through the Steenrod square (is this right? see nLab). $$ Sq^k(x) = \begin{cases} u_k […]

By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times … \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of topological spaces such that integral cohomology $H^n(-, \mathbb{Z})$ is represented by the Eilenberg-MacLane spaces $K(\mathbb{Z}, n)$. The Yoneda lemma shows that such operations are in natural […]

Intereting Posts

Can a strictly increasing convex function $F$ meet a line segment in 3 places, without being linear?
On continuously uniquely geodesic space
Prove that $n$ divides $\phi(a^n-1)$, where $\phi$ is Euler's $\phi$-function.
Showing the square of a Markov process is or isn't Markov
$q$-norm $\leq$ $p$-norm
Analytical solution to a Quadratic form Matrix Integral
In how many ways can we arrange 4 letters of the word “ENGINE”?
Probability of rolling a 1 before you roll two 2's, three 3's, etc
Riemann rearrangement theorem
What tactics could help with this probability questions
The shape of Pringles potato chip
How to understand marginal distribution
If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$?
Combinatorial interpretation of a sum identity: $\sum_{k=1}^n(k-1)(n-k)=\binom{n}{3}$
Proof that the predictable sigma algebra is also generated by continuous and adapted processes