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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 30th 2012

I have added to principal bundle

• a remark on their definition As quotients;

• statements about (classes of) (counter-)examples of quotients

Thanks for pointers to the literature from this MO thread!

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 30th 2012
• (edited Apr 30th 2012)

I find a bit strange the instistence on the distinction between torsors and principal bundles put in this article (in sentence 1 in the idea, as if it were central, while it is just a matter of local culture). To me, torsor and principal bundle is the same thing, except that torsor is usually used in algebraic context, with respect to Grothendieck topologies like flat or etale. My principal references are Husemoller, Eells, Postnikov etc. The equivalence of the two words is also accepted for the noncommutative generalization by most practitioners in noncommutative geometry. The article says that principal bundle includes local triviality while torsor does not. In my experience, both words can be used in locally trivial and not locally trivial case. By convention, one may drop the locally trivial prefix, if it is assumed throughout a text and mentioned at the beginning of the article.

• CommentRowNumber3.
• CommentAuthorTim_Porter
• CommentTimeApr 30th 2012

I agree with Zoran.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 30th 2012

True, that was an old remnant. Good that you spotted it, I didn’t read the Idea-section anymore!

I have now replaced it with a different discussion. I hope to find time now to bring the entire entry a bit more up to speed.

• CommentRowNumber5.
• CommentAuthorziggurism
• CommentTimeMar 3rd 2020

seems we want right G-spaces…

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 9th 2020

added more publication data to some of the references

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 18th 2020

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeJan 16th 2021

Questions related to the existence slices of G-spaces, of sections of $G$-bundles and conditions for properness of some related maps are treated in

• Richard S. Palais, On the existence of slices of actions of non-compact Lie groups, Ann. Math. 73:2 (1961) pdf
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 11th 2021

I made the remark on “Cartan principal bundles” (Palais’ terminology for the notion without the local triviality condition) a numbered Remark environment (here) and added more precise pointer to where in Palais 61 it says so (namely Def. 1.1.2)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 11th 2021

but I notice that this is still not a very useful pointer, since there are 11 exposés behind this link, none of which has a title that would suggest it introduces principal bundles.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 16th 2021

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 16th 2021

(this maths-phys text trumps every pure math textbook account that I have seen regarding exposition of the theory, both in coherent conceptual breadth and systematic account of the details)

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 17th 2021

• Loring Tu, Parts I-II in: Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)
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