Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 8th 2020

    Corrected a statement about Lie groups.

    diff, v15, current

  1. 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 PU()\mathcal{B} PU(\mathcal{H}) of the universal equivariant PU()PU(\mathbb{H})-bundle (classifying 3-twists in twisted equivariant K-theory) is

    π 1((PU()) 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” GPU()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