# Questions tagged [cohomology-operations]

Use this tag for questions about natural transformations from a functor defining a cohomology theory to itself. Common examples include Steenrod squares in mod 2 cohomology and Adams operations in K-theory.

23 questions
Filter by
Sorted by
Tagged with
1answer
53 views

### Is it possible to find a torsion free space with non-trivial $\operatorname{Sq}^{2^n}$

Torsion free means integral cohomology is torsion free. When $n \leq 3$, such spaces are projective space $\mathbb{CP}^n$, $\mathbb{HP}^n$ and the Cayley plane. $\operatorname{Sq}^{2^n}$ takes the ...
0answers
56 views

### Why is the Bockstein morphism a derivation?

I'm trying to understand the Bockstein morphism in cohomology, and one of the points is that $\delta : H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$ is a derivation that squares to $0$. I could ...
1answer
46 views

1answer
216 views

### Why does «Massey cube» of an odd element lie in 3-torsion?

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 ...
1answer
1k views

### The simplest nontrivial (unstable) integral cohomology operation

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 ...
1answer
113 views