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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexat [[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 $\mathbf{B}^n U(1) \simeq K(U(1), n)$.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexAfter geometric realization, that is! Maybe I should add more discussion, even.

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

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 22nd 2011
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   Author: jim_stasheff
   Format: TextJust 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

   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
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorjim_stasheff
   • CommentTimeSep 23rd 2011
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   Author: jim_stasheff
   Format: TextYes and no - but you are right that, at least according to google and wiki, Bockstein homomorphism is the more common term.

   Yes and no - but you are right that, at least according to google and wiki, Bockstein homomorphism is the more common term.
6. • CommentRowNumber6.
   • CommentAuthordomenico_fiorenza
   • CommentTimeSep 23rd 2011
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   Author: domenico_fiorenza
   Format: MarkdownItexI've now added a redirect from "Bockstein operation"

   I’ve now added a redirect from “Bockstein operation”

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 19th 2018
   • (edited Feb 19th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn the examples-section [here](https://ncatlab.org/nlab/show/Bockstein+homomorphism#Examples) 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](https://ncatlab.org/nlab/show/Bockstein+homomorphism#SteenrodSquaresAndDBCupProductOnOddClasses). I'll expand/polish this a bit more, and maybe then this example should (also) go to another entry.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 19th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have also made explicit the [[integral Steenrod squares]]: [here](https://ncatlab.org/nlab/show/Bockstein+homomorphism#IntegralSteenrodSquares) (also gave them their own little entry)

   I have also made explicit the integral Steenrod squares: here

   (also gave them their own little entry)