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nLab > Latest Changes: rational parametrized homotopy theory

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2020

    starting something…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2020
    • (edited Aug 30th 2020)

    added this statement:

    Let

    F ⟶fib A ↓ p B \array{ F &\overset{fib}{\longrightarrow}& A \\ && \big\downarrow{}^{\mathrlap{p}} \\ && B }

    be a Serre fibration of connected topological spaces, with fiber FF (over any base point) also connected.

    If in addition

    1. π 1(B)\pi_1(B) acts nilpotently on H •(F,k)H_\bullet(F,k)

      (e.g. if BB is simply connected);

    2. at least one of AA, FF have rational finite type

    then the cofiber of any relative Sullivan model for pp is a Sullivan model for FF.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 29th 2020

    Did you mean π 1(Y)\pi_1(Y) acts nilpotently?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2020

    Yes, I had fixed it in the entry, didn’t fix it here.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2020
    • (edited Aug 30th 2020)

    added the addendum:

    Moreover, if CE(𝔩B)CE(\mathfrak{l}B) is a minimal Sullivan model for BB, then the cofiber of the corresponding minimal relative Sullivan model for pp is the minimal Sullivan model CE(𝔩F)CE(\mathfrak{l}F) for FF:

    CE(𝔩F) ⟵cofib(CE(𝔩p)) CE(𝔩 BA) ↓ CE(𝔩p) CE(𝔩B) \array{ CE(\mathfrak{l}F) &\overset{ cofib \big( CE(\mathfrak{l}p) \big) }{\longleftarrow}& CE(\mathfrak{l}_{{}_B}A) \\ && \big\downarrow{}^{\mathrlap{ CE(\mathfrak{l}p) }} \\ && CE(\mathfrak{l}B) }

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2020

    added pointer to:

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2020

    added the following further addendum to the proposition:

    But this cofiber, being the cofiber of a relative Sullivan model and hence of a cofibration in the projective model structure on dgc-algebras, is in fact the homotopy cofiber, and hence is a model for the homotopy fiber of the rationalized fibration.

    Therefore (eq:SullivanModelForFiber) implies that on fibrations of connected finite-type spaces where π 1\pi_1 of the base acts nilpotently on the homology of the fiber, rationalization preserves homotopy fibers.

    (This was originally proven in Bousfield-Kan 72, Chapter II.)

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2020

    added pointer also to the followup book

    where the above statement appears as Theorem 5.1

    diff, v5, current

1 to 8 of 8

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