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nLab > Latest Changes: Sullivan model of loop space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2016
    • (edited Aug 15th 2016)

    I have created Sullivan model of free loop space with the formula and pointers to the literature.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2016
    • (edited Aug 16th 2016)

    There is a canonical S 1S^1-action on the free loop space. For Sullivan models (∧ •(V⊕sV,d ℒX)(\wedge^\bullet (V \oplus s V, d_{\mathcal{L}X} ) such that d Xd_X is simple enough, then it is easy to guess a Sullivan model for the S 1S^1-homotopy quotient. Namely add a generator ω 2\omega_2 of degre 2 and then modify the differential on the unshifted generators by adding a term proportional to the corresponding shifted generator wedge ω 2\omega_2.

    Is there any published statement about Sullivan models for the homotopy quotients ℒX/S 1\mathcal{L}X/S^1?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2016

    I have found a source for the proof of the Sullivan model for ℒX/S 1\mathcal{L}X/S^1. It is theorem A in Vigué-Burghelea 85. I have added the statement here.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2016

    I have added an Examples-section The 4-sphere and twisted de Rham cohomology which spells out the Sullivan model for ℒS 4//S 1\mathcal{L}S^4 // S^1 and makes an observation of how this relates to a kind of caloron correspondence.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2019

    added as a corollary the Sullivan model for based loop spaces (here) and, as an example, the Sullivan models for the iterated based loop spaces of spheres Ω kS n\Omega^k S^{n} for k<nk \lt n (here)

    diff, v27, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2019

    changed title, making it just “loop space” instead of “free loop space”

    diff, v27, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021
    • (edited Jul 2nd 2021)

    I have harmonized notation by replacing all “/^h” and “/ /” by “\sslash”

    (On the other hand, I guess there was a time when \sslash didn’t render properly on some browsers. I forget what the status is, hope it works now.)

    diff, v31, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    added pointer to:

    • Bitjong Ndombol & M. El Haouari, The free loop space equivariant cohomology algebra of some formal spaces, Mathematische Zeitschrift 266 (2010) 863–875 (doi:10.1007/s00209-009-0602-z)

    diff, v32, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    also this one:

    • Kentaro Matsuo, The Borel cohomology of the loop space of a homogeneous space, Topology and its Applications 160 12 (2013) 1313-1332 (doi:10.1016/j.topol.2013.05.001)

    diff, v32, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2021

    There used to be mentioning of “twisted de Rham cohomology” in a sub-section headline, without this ever being mentioned in the text. Now I have instead removed these words from the section headline, but added an actual remark (here) on this point.

    diff, v34, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2022

    have spelled out (here) the example of the Sullivan model for the cyclic loop space of B n+1ℚB^{n+1} \mathbb{Q}.

    diff, v35, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2023

    Below the statement of the Sullivan model of a based loop space (here) I added a remark (here) pointing to the Pontrjagin ring structure on the loop space’s homology.

    diff, v38, current

1 to 12 of 12

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