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nLab > Latest Changes: spinᶜ

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2011
    • (edited Jul 17th 2013)

    I have split off spin^c from spin^c structure

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2013
    • (edited Jul 17th 2013)

    in the section As the homotopy fiber of smooth W3 I have added some more comments and a proposition to make more explicit why the “determinant line” map is given on the canonical U(1)U(1)-components by multiplication by 2.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2020

    cross-linked the definition with central product group.

    diff, v23, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2020

    added the example Spin c(3)≃U(2) Spin^c(3) \;\simeq\; U(2)

    diff, v23, current

  5. typo

    Anonymous

    diff, v26, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 16th 2023

    Moved exceptional example to under the definition, and hence before all the abstract higher-topos style description. This isomorphism is not mentioned on Wikipedia, and for some reason, when looking for it more widely, it’s not easy to find; I even missed seeing it here in my scrambling around.

    diff, v27, current

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 16th 2023

    Added mention the exceptional isomorphism covers another one, namely SO(3)≃PU(2)SO(3) \simeq PU(2).

    diff, v27, current

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023

    Added the exceptional isomorphism Spin c(4)≃U(2)× U(1)U(2)Spin^c(4)\simeq U(2)\times_{U(1)}U(2), where the fibre product is via the determinant homomorphism on both sides. A bit more minor clarifying details as well in the definition section.

    Here’s a question: what should Spin c(2)Spin^c(2) be? The underlying manifold is a 2-torus, but what’s the group structure?

    The page currently claims Spin c(n)Spin^c(n) is defined for all natural numbers, but I’m struggling to imagine Spin c(1)Spin^c(1) being useful :-) Is it just isomorphic to U(1)U(1) itself?

    diff, v28, current

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023

    Reminder to self: add the citation

    • Ozbagci, B., Stipsicz, A.I. (2004). Spin cSpin^c Structures on 3- and 4-Manifolds. In: Surgery on Contact 3-Manifolds and Stein Surfaces. Bolyai Society Mathematical Studies, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10167-4_6

    for the exceptional examples.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023

    Added the citation as noted, and also the proof of the exceptional isomorphism Spin c(4)≃U(2)× U(1)U(2)Spin^c(4) \simeq U(2)\times_{U(1)} U(2).

    diff, v29, current

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023

    Typos

    diff, v29, current

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023

    Added another exceptional case: Spin c(6)Spin^c(6) is the unique nontrivial double cover of U(4)U(4).

    [I’ll leave the identification of Spin c(5)Spin^c(5) to another adventurous soul, or perhaps future me.]

    diff, v30, current

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2023
    • (edited Sep 17th 2023)

    Hmm, isn’t Spin c(5)Spin^c(5) “just” Sp(2)⋅U(1)Sp(2)\cdot U(1) inside Sp(2).Sp(1)? Probably there’s not much more to say past that ….

    EDIT: but maybe? What is the image of Sp(2)×U(1)→U(4)Sp(2)\times U(1) \to U(4), where we take the embedding of Sp(2)Sp(2) inside U(4)U(4) and then multiply? Can we see it as the intersection of U(4)U(4) and a subgroup of GL(4,ℂ)GL(4,\mathbb{C}) larger than Sp(4,ℂ)Sp(4,\mathbb{C})? [Edit of edit: fixed the dimension]

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 18th 2023

    In discussion with John Baez on Mathstodon, I thought of the following:

    @johncarlosbaez OK, how’s this. Consider the 4x4 matrix Omega as at https://en.wikipedia.org/wiki/Symplectic_group#Sp(2n,_F), Then define a subgroup G of 4x4 invertible complex matrices M satisfying M^t Omega M = z Omega for some unit complex number z. Then I think U(4) \cap G is double covered by Spin^c(5), as a factorisation of the isomorphism Sp(2)xU(1) = [Sp(4,C) \cap U(4)]xU(1) followed by the multiplication map [Sp(4,C) \cap U(4)]xU(1) -> G \cap U(4). Because M appears twice in the LHS of the equation defining G, I think G\cap U(4) is a little too small, since the condition squares the extra factor coming from U(1)…

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2024
    • (edited Jul 5th 2024)

    changing page name (from “spin^c” to “spinᶜ”) for better looks, following discussion here

    diff, v32, current

    • CommentRowNumber16.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeJul 5th 2024
    • (edited Jul 5th 2024)

    Changed “spin^c” to “spinᶜ” and “spin^h” to “spinʰ” in related concepts. (Links will be checked.)

    Edit: Added !redirects spinʰ group and !redirects spinʰ structure on spinʰ structure to make the link work.

    diff, v33, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2024

    Great. Thanks for all this.

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