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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 4th 2021

    Created:

    Idea

    A generalization of homotopy groups.

    Definition

    Given a finitely generated abelian group AA and n2n\ge2, we set

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

    where P n(A)P^n(A) is the nnth Peterson space of AA.

    Related concepts

    References

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2021

    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

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 5th 2021
    • (edited Apr 5th 2021)

    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.

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