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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 5th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea A generalization of [[homotopy groups]]. ## Definition Given a finitely generated [[abelian group]] $A$ and $n\ge2$, we set $$\pi_n(X,A)=[P^n(A),X],$$ where $P^n(A)$ is the $n$th [[Peterson space]] of $A$. ## Related concepts * [[Peterson space]] ## References * [[Franklin P. Peterson]], _Generalized Cohomotopy Groups_. American Journal of Mathematics 78:2 (1956), 259–281. [doi:10.2307/2372515](http://dx.doi.org/10.2307/2372515) * [[Joseph A. Neisendorfer]], _Homotopy groups with coefficients_, Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. [doi:10.1007/s11784-010-0020-1](http://dx.doi.org/10.1007/s11784-010-0020-1). <a href="https://ncatlab.org/nlab/revision/homotopy+groups+with+coefficients/1">v1</a>, <a href="https://ncatlab.org/nlab/show/homotopy+groups+with+coefficients">current</a>

   Created:

   Idea

   A generalization of homotopy groups.

   Definition

   Given a finitely generated abelian group $A$ and $n\ge2$, we set

   $$\pi_n(X,A)=[P^n(A),X],$$

   where $P^n(A)$ is the $n$th Peterson space of $A$.

   Related concepts

   • Peterson space

   References

   • Franklin P. Peterson, Generalized Cohomotopy Groups. American Journal of Mathematics 78:2 (1956), 259–281. doi:10.2307/2372515

   • Joseph A. Neisendorfer, Homotopy groups with coefficients, Journal of Fixed Point Theory and Applications 8:2 (2010), 247–338. doi:10.1007/s11784-010-0020-1.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for these pointers, hadn't seen that before. For a moment I thought Peterson's "Generalized Cohomotopy" might subsume [[schreiber:Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation|twistorial Cohomotopy]], but it doesn't: He considers replacing spheres by homology spheres etc. Regarding Neisendorfer's terminology "with coefficients": This seems a little unfortunate to me, as (co)homology "with local coefficient" is commonly understood to refer to *twisted* (co)homology, which doesn't seem what this connects to? Regarding the typesetting: I have made some more of the technical terms hyperlinked (e.g. [[finitely generated object|finitely generated]]) and added table of contents and floating context menu. Also made the page name singular, to comply with running convention. Last not least, I added pointer back to this entry here from "Related concepts" at _[[homotopy group]]_. <a href="https://ncatlab.org/nlab/revision/diff/homotopy+group+with+coefficients/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/homotopy+group+with+coefficients/2">v2</a>, <a href="https://ncatlab.org/nlab/show/homotopy+group+with+coefficients">current</a>

   Thanks for these pointers, hadn’t seen that before. For a moment I thought Peterson’s “Generalized Cohomotopy” might subsume twistorial Cohomotopy, but it doesn’t: He considers replacing spheres by homology spheres etc.

   Regarding Neisendorfer’s terminology “with coefficients”: This seems a little unfortunate to me, as (co)homology “with local coefficient” is commonly understood to refer to twisted (co)homology, which doesn’t seem what this connects to?

   Regarding the typesetting:

   I have made some more of the technical terms hyperlinked (e.g. finitely generated) and added table of contents and floating context menu. Also made the page name singular, to comply with running convention. Last not least, I added pointer back to this entry here from “Related concepts” at homotopy group.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 5th 2021
   • (edited Apr 5th 2021)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItex> Regarding Neisendorfer’s terminology “with coefficients”: This seems a little unfortunate to me, as (co)homology “with local coefficient” is commonly understood to refer to twisted (co)homology, which doesn’t seem what this connects to? But where do you see “local” in this article? “Homology groups with coefficients in an abelian group” and “cohomology groups with coefficients in an abelian group” are perfectly standard terms. Why should “homotopy groups with coefficients in an abelian group” be called any different? I can also imagine a twisted version with coefficients in a local system, just like for homology and cohomology, we could have “homotopy groups with coefficients in a local system of abelian groups”. But this would be a different concept.

   Regarding Neisendorfer’s terminology “with coefficients”: This seems a little unfortunate to me, as (co)homology “with local coefficient” is commonly understood to refer to twisted (co)homology, which doesn’t seem what this connects to?

   But where do you see “local” in this article?

   “Homology groups with coefficients in an abelian group” and “cohomology groups with coefficients in an abelian group” are perfectly standard terms.

   Why should “homotopy groups with coefficients in an abelian group” be called any different?

   I can also imagine a twisted version with coefficients in a local system, just like for homology and cohomology, we could have “homotopy groups with coefficients in a local system of abelian groups”. But this would be a different concept.