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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2021

    am finally giving this its own entry. Nothing much here yet, though, still busy fixing some legacy cross-linking…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    Note about how the error arose in Atiyah–Segal, and that the norm topology is still distinct (and finer) on U(H).

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021

    Thanks. BTW, it’s spelled [[norm topology]], with square brackets around it.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    Sorry :-)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021
    • (edited Sep 4th 2021)

    Interesting to look at the time stamps:

    • In Feb 2012 Uribe et al. pick up the topic and amplify (Sec. 1) the wrong statement from Atiyah-Segal.

    • In Sept 2013 Schottenloher points out the issue.

    • In Nov 2013 Uribe et al.’s article gets published.

    • In July 2014 Uribe at al.-prime notice the issue, apparently still unaware of Schottenloher’s preprint (?).

    • In Aug 2014 Uribe et al.-prime is already published, too.

    • In 2015 nothing happens.

    • In 2016 nothing happens.

    • In 2017 nothing happens.

    • In Aug 2018 (a rewrite of) Schottenloher’s preprint is finally published.

    What gives?

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    One wonders why Atiyah-Segal made the error. An early sign of what was to come?

    I tried to really in-depth read the twisted K-theory paper as a PhD student, and it was so brief on details in certain places I made no headway for a long time, and eventually gave up. A number of reasons for this, but certainly having mistakes like the one under discussion doesn’t help!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021
    • (edited Sep 19th 2021)

    One wonders why Atiyah-Segal made the error.

    People make mistakes all the time. Authorities do, too.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021

    added the statement that U()U(\mathcal{H}) in the strong topology is completely metrizable, with pointer to:

    diff, v6, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added pointer to

    for a proof that the weak and strong topology on U()U(\mathcal{H}) agree and make it a topological group

    diff, v6, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021

    added the statement that the norm topology makes U()U(\mathcal{H}) a Banach Lie group.

    Schottenloher talks about this somewhat informally, while Espinoza & Uribe point to Neeb 1997. However, I don’t see that Neeb says this explicitly.

    diff, v6, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added the statement that U() strongU(\mathcal{H})_{strong} is not locally compact, with reference to section 5 in:

    • Rostislav Grigorchuk, Pierre de la Harpe, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, in: Groups, Graphs and Random Walks, Cambridge University Press 2017 (arXiv:1404.7030, doi:10.1017/9781316576571.011)

    diff, v6, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    Is U()\mathrm{U}(\mathcal{H}) well-pointed?

    I know that

    1. S 1S^1 is well-pointed;

    2. PU()\mathrm{PU}(\mathcal{H}) is well-pointed;

    3. there is an open neighbourhood V eV_{\mathrm{e}} of e\mathrm{e} in PU()PU(\mathcal{H}) such that

      U() |V eV e×S 1\mathrm{U}(\mathcal{H})_{\vert V_{\mathrm{e}}} \simeq V_{\mathrm{e}} \times S^1

    4. products of h-cofibrations remain h-cofibrations.

    This seems like it might be getting close. Or maybe not.

    [ edit: on the other hand, it dawns on me that I don’t actually need to know the answer to do what I want to do… ]

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2021
    • (edited Sep 20th 2021)

    I have spelled out the various topologies, following the list as given by Espinoza & Uribe. Then I have further refined the list of propositions about these topologies, with references.

    It seems that all these results, except maybe concerning the compact-open topology, are already due to K-H Neeb in the 1990s.

    diff, v9, current

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