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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 22nd 2011

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 $\mathbf{B}^n U(1) \simeq K(U(1), n)$.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 22nd 2011

After geometric realization, that is! Maybe I should add more discussion, even.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeSep 22nd 2011
Just reading these posts, I find them at least ambiguous,
especially
B^nU(1)≃K(U(1),n)
since the latter notation is standard for abelian groups NOT for abelian topological groups
of course the above could be taken as a definition
ditto for referring to `Bockstein homomorphism' usually referred to as
Bockstein operation
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 22nd 2011

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?

• CommentRowNumber5.
• CommentAuthorjim_stasheff
• CommentTimeSep 23rd 2011
Yes and no - but you are right that, at least according to google and wiki, Bockstein homomorphism is the more common term.
1. I’ve now added a redirect from “Bockstein operation”

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 19th 2018
• (edited Feb 19th 2018)

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeFeb 19th 2018

I have also made explicit the integral Steenrod squares: here

(also gave them their own little entry)