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nLab > Latest Changes: stable splitting of mapping spaces

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2018

    starting something, not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2018

    added pointer to the original article Snaith 74 and to the interpretation of the stable splitting as the Goodwillie-Taylor tower of the mapping space by Arone 99

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeNov 2nd 2018
    Should that be $\Sigma^{+}_{\infty}Maps(X,A)$ or is that in based CW complexes?
    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeNov 2nd 2018

    Sorry, I meant Σ + ∞Maps(X,A)\Sigma^{\infty}_{+}Maps(X,A) of course

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2018
    • (edited Nov 2nd 2018)

    I think the mapping space is/must be regarded as being pointed already, so that Σ ∞Maps(X,A)\Sigma^\infty Maps(X,A) is correct.

    Abstractly, the stable splitting is the Goodwillie derivative of the functor Maps(X,−)Maps(X,-), and that is a functor from and to pointed topological spaces.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018
    • (edited Nov 3rd 2018)

    added pointer to

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    added now detailed definitions and statements up to Snaith 74 theorem 1.1 and Boedigheimer 87, Examples 2 and 5.

    So far, this material is duplicated also at configuration space of points. I think that’s okay. Once the discussion here enters the generalization as in Arone 99, the two entries will diverge.

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 3rd 2018

    There are diffeomorphic-invariant results on stable splitting here if needed:

    • Richard Manthorpe, Ulrike Tillmann, Tubular configurations: equivariant scanning and splitting, (arXiv:1307.5669)
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    Thanks! I’ll have a look.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2018
    • (edited Nov 6th 2018)

    I have now added a section (here) indicating how the stable splitting of mapping spaces is the limiting Goodwillie-Taylor tower of the mapping space functor.

    I am just following the basic observations about excisiveness of the stages in the splitting formula, from the second page of Arone 99, but regarding assumptions on the space that we are homming out, and the notation and conventions on the mapping space, I am sticking with what I already had in the entry. (See the tables there for matching notation to the literature.)

    diff, v7, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2018
    • (edited Nov 16th 2018)

    Fixed the notation from the point on where the decomposition Conf n(X,Y)≃Conf n ord(X)∧ Σ n(Y/∂Y) ∧ nConf_n(X,Y) \simeq Conf_n^{ord}(X) \wedge_{\Sigma_n} (Y/\partial Y)^{\wedge_n} is used (the superscriped for the ordered configuration space had been missing)

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 16th 2018

    Comp=ConfComp=Conf in #11?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2018

    Yes, thanks, fixed.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2019

    added pointer to

    • Lauren Bandklayder, Stable splitting of mapping spaces via nonabelian Poincaré duality (arxiv:1705.03090)

    and cross-linked with nonabelian Poincaré duality

    diff, v13, current

  15. The reference does not deal with general n-connective spaces.

    Anonymous

    diff, v14, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    added pointer to:

    • Douglas Ravenel, What we still don’t understand about loop spaces of spheres, Contemporary Mathematics 1998 (pdf)

    diff, v15, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2023

    My notation for this equivalence was odd, I have now adjusted it.

    diff, v17, current

1 to 17 of 17

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