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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2011
    • (edited Feb 6th 2011)

    A manifold has

    • a set of orientations;

    • an xyz of topological spin structures

    • a 3-groupoid of topological string structures;

    • a 7-groupoid of topological fivebrane stuctures, etc.

    and for some reason it is common in the literature (which of course is small in the last cases) to speak of these nn-groupoids, but not so common to speak of the xyz here:

    • A manifold has a groupoid of spin structures.

    Namely the homotopy fiber of the second Stiefel-Whitney class

    Spin(X)Top(X,BSO)(w 2) *Top(X,B 2 2). Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.

    I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.

    Do you have further references?

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 7th 2011

    @Urs This looks worth pursuing. It may relate to HQFTs. We can discuss this when you arrive (in Lisbon).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2011

    How is it going in Lisbon? I hope it’s sunny. I need some sun. Though today it seems winter is taking a break even here.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 7th 2011

    Urs, you skipped Lisboa ?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2011

    Urs, you skipped Lisboa ?

    No, I’ll go. My flight is day after tomorrow, Feb 9.

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