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nLab > Latest Changes: cyclic loop space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2017

    I need a word for the homotopy quotient (ℒX)/S 1(\mathcal{L}X)/S^1 of free loop spaces ℒX\mathcal{L}X by their canonical circle action. It seems that the only term in use with respect to this is “twisted loop space”, which however usually refers just to the constant loops (ℒ constX)//S 1(\mathcal{L}_{const}X)//S^1. Since under nice conditions the derived functions on the ℒSpec(A)/S 1\mathcal{L}Spec(A)/S^1 is the cyclic homology complex of AA, I suggest that a good name is “cyclic loop space”. I made a quick note at cyclic loop space, just to fix and disambiguate terminology.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2021
    • (edited Jun 21st 2021)

    I have added pointer to cylic loop orbifolds in

    Also added pointer to our

    and

    from which some of the entry material had been taken.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    added pointer to:

    diff, v11, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    also:

    diff, v11, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    also:

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021
    • (edited Jul 2nd 2021)

    added (here) statement of the stable splitting theorem by Carlsson & Cohen 1987:

    Ω ∞Σ ∞(Maps(S 1,ΣX)⫽S 1)≃(Ω ∞Σ ∞BS 1)×∏k∈ℕ ≥1Ω ∞Σ ∞((E(ℤ/k)) +∧ ℤ/kX ∧ n) \Omega^\infty \Sigma^\infty \big( Maps(S^1, \Sigma X) \sslash S^1 \big) \;\; \simeq \;\; \big( \Omega^\infty \Sigma^\infty B S^1 \big) \times \underset{ k \in \mathbb{N}_{\geq 1} }{\prod} \Omega^\infty \Sigma^\infty \big( (E (\mathbb{Z}/k))_+ \wedge_{\mathbb{Z}/k} X^{\wedge^n} \big)

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2021

    also pointer to

    where this result was announced

    diff, v12, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2021

    added pointer to:

    for the terminology “string space” (Section 4.8.1)

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2022

    I have adjusted the wording in this partagraph in the Idea-section, which previouly was somewhat mixed up.

    diff, v14, current

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