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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 4th 2021

    Created with the following content:

    Definition

    Given a finitely generated abelian group AA and n3n\ge 3, the nnth Peterson space P n(A)P^n(A) of AA is the simply connected space whose reduced cohomology groups vanish in dimension knk\ne n and the nnth cohomology group is isomorphic to AA.

    Existence and uniqueness

    The Peterson space exists and is unique up to a weak homotopy equivalence given the indicated conditions on AA and nn.

    There are counterexamples both to existence and uniqueness without these conditions.

    For example, the Peterson space does not exist if AA is the abelian group of rationals.

    Corepresentation of homotopy groups with coefficients

    For all n2n\ge2, we have a canonical isomorphism

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

    where the left side denotes homotopy groups with coefficients and the right side denotes morphisms in the pointed homotopy category.

    Related concepts

    References

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 4th 2021

    Added:

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 4th 2021

    Added:

    Relation to Moore spaces

    Moore spaces M n(A)M_n(A) are defined similarly to Peterson spaces, using homology instead of cohomology.

    We have natural weak equivalences

    P n(A)M n(Hom(A,Z))P^n(A) \simeq M_n(Hom(A,\mathbf{Z}))

    if AA is a finitely generated free abelian group and

    P n(A)M n1(Hom(A,Q/Z))P^n(A) \simeq M_{n-1}(Hom(A,\mathbf{Q}/\mathbf{Z}))

    if AA is a finite abelian group.

    v1, current

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 4th 2021

    Functoriality and examples.

    v1, current

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