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nLab > Latest Changes: Boardman homomorphism

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2016

    created some minimum at Boardman homomorphism (the thing generalising the Hurewicz homomorphism)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2018

    added statement (here) of the (“tight”) bound on kernel/cokernel of the Boardman homomorphism from stable cohomotopy to ordinary cohomology from Arlettaz 04, theorem 1.2

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2018
    • (edited Oct 26th 2018)

    spelled out an example (here) for Arlettaz’ estimates of the torsion of the cokernel of the cohomological Boardman homomorphism to ordinary cohomology, just so as to make explicit the numbers that appear in low degrees.

    I am slightly worried that I am misreading theorem 1.2 in Arlettaz 04, since the stated conditon that the space be (m−1)(m-1)-connected makes sens in the conclusion of the theorm only if he really means that mm is the maximal number such that the space is (m−1)(m-1)-connected.

    For the example that I am interested in this doesn’t affect the desired conclusion either way, but it feels disconcerting nevertheless

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2018
    • (edited Oct 27th 2018)

    In the Examples-section I have added statement and detailed proof (here) that for 2I2I the binary icosahedral group we have a Boardman iso-morphism

    𝕊 4(S 7/2I)⟶≃βH 4(S 7/2I,ℤ) \mathbb{S}^4\left( S^7/2I \right) \underoverset{\simeq}{\beta}{\longrightarrow} H^4\left( S^7/2I , \mathbb{Z} \right)

    from stable cohomotopy in degree 4 to integral cohomology in degree 4.

    (First I thought that I’d need this fact for the last slides of Equivariant Stable Cohomotopy and Branes (schreiber). Now I don’t think I actually need it, but it’s maybe still a fun example of Boardman homomorphisms.)

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2020

    added pointer to this recent article on the Boardman homomorphism (generalized Hurewicz homomorphism) to tmf:

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2021

    added these pointers:

    • John Hunton, The Boardman homomorphism, Contemporary Mathematics 181, 251-251, 1995

    • Hadi Zare, On the image of the unstable Boardman map (arXiv:1806.07079)

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2021
    • (edited Jan 18th 2021)

    I have added pointer to:

    • Michael Boardman, The eightfold way to BP-operations, in: Current trends in algebraic topology, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1. Providence 1982 (ISBN:978-0-8218-6003-8)

    I haven’t actually seen that article, but Hunton 95 claims that this is where the concept (the Boardman homomorphism) first appeared in print

    diff, v14, current

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