Start a new discussion

Not signed in

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

Site Tag Cloud

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

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 3rd 2021

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

• 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).

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

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

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

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

added the statement that the norm topology makes $U(\mathcal{H})$ 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.

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

added the statement that $U(\mathcal{H})_{strong}$ 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)
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 19th 2021
• (edited Sep 19th 2021)

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

I know that

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

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

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

$\mathrm{U}(\mathcal{H})_{\vert V_{\mathrm{e}}} \simeq V_{\mathrm{e}} \times S^1$

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.