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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeMar 7th 2020

Corrected a statement about Lie groups.

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

Jens Hemelaer

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 21st 2021

touched the formatting of this old entry

• 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”)

• 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? ]

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

added mentioning of the following example (here):

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

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

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

• 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)

• 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.

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

[ obsolete ]

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMay 4th 2022

added Example cross-linking back to Brillouin torus

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