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at Bockstein homomorphism in the examples-section where it says
$\mathbf{B}^n U(1) \simeq \mathbf{B}^{n+1}\mathbb{Z}$I have added the parenthetical remark
(which is true in ambient contexts such as $ETop\infty Grpd$ or $Smooth \infty Grpd$)
Just to safe the reader from a common trap. Because it is not true in $Top \simeq \infty Grpd$. The problem is that in all traditional literature the crucial distinction between $Top$ and $ETop \infty Grpd$ (or similar) is often appealed to implicitly, but rarely explicitly. In $Top \simeq \infty Grpd$ we have instead $\mathbf{B}^n U(1) \simeq K(U(1), n)$.
After geometric realization, that is! Maybe I should add more discussion, even.
NOT for abelian topological groups
That was exactly the point of my remark above!
‘Bockstein homomorphism’ usually referred to as Bockstein operation
Is it true that it is called “operation” outside of the special case of its application to Steenrod operatins?
I’ve now added a redirect from “Bockstein operation”
In the examples-section here I added a few words on the mod-2 integral Bockstein map, the first Steenrod square, and the exponential sequence Bockstein, and their relations.
Then, I added the statement, originally observed by Gomi, about the Beilinson-Deligne cup square on odd-degree ordinary differential cohomology being a differential refinement of the “first” Adem relation: here.
I’ll expand/polish this a bit more, and maybe then this example should (also) go to another entry.
I have also made explicit the integral Steenrod squares: here
(also gave them their own little entry)
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