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nLab > Latest Changes: Pontrjagin dual

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 8th 2020

    Corrected a statement about Lie groups.

    diff, v15, current

  2. Added answer to: when is the Pontryagin dual connected?

    Jens Hemelaer

    diff, v16, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021

    touched the formatting of this old entry

    diff, v18, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021

    merged little left-over material from what is now Pontryagin duality > history into here (just one reference and a list of “Related concepts”)

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021
    • (edited Aug 21st 2021)

    added pointer to the original:

    • Lev Pontrjagin, Theory of topological commutative groups, Uspekhi Mat. Nauk, 1936, no. 2, 177–195 (mathnet:umn8882)

      English translation: Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 361-388 (doi:10.2307/1968438)

    [edit: Hm, the dates suggest that the Russian version is actually translated from an English original? ]

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021
    • (edited Aug 21st 2021)

    added mentioning of the following example (here):

    For GG a finite group, the fundamental group π 1(−)\pi_1(-) of the GG-fixed locus (−) G(-)^G of the base space ℬPU(ℋ)\mathcal{B} PU(\mathcal{H}) of the universal equivariant PU(ℍ)PU(\mathbb{H})-bundle (classifying 3-twists in twisted equivariant K-theory) is

    π 1((ℬPU(ℋ)) G)≃Grps(G,S 1)=G^ \pi_1 \Big( \big( \mathcal{B} PU(\mathcal{H}) \big)^G \Big) \;\simeq\; Grps(G, S^1) \,=\, \widehat G

    (in any connected component of a “stable map” G→PU(ℋ)G \to PU(\mathcal{H}), that is) and hence is the Pontrjagin dual group when GG is abelian.

    diff, v20, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 21st 2021

    Re #5:

    [edit: Hm, the dates suggest that the Russian version is actually translated from an English original? ]

    Yes, the Russian version explicitly says so in the first footnote. It also says that Chapter II was completely rewritten in the Russian translation, though.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021

    Thanks!

    So I have adjusted the item accordingly (here)

    diff, v22, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2021
    • (edited Aug 21st 2021)

    I have added statement and proof (here) that for finite groups the Pontrjagin dual is equivalently the second integral group cohomology group.

    diff, v23, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2021
    • (edited Sep 3rd 2021)

    [ obsolete ]

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2022

    added Example cross-linking back to Brillouin torus

    diff, v25, current

1 to 11 of 11

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