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1. fix wrong definition of free group action

Alexey Muranov

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJan 9th 2019
• (edited Jan 9th 2019)

Interpreted the freeness and transitivity in terms of the map $\langle \rho, \pi_2 \rangle: G \times P \to P \times P$.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 6th 2021
• (edited Jan 6th 2021)

This entry is lacking a good textbook reference. It used to list, on the one extreme, an expository webpage, and on the other end a bunch of references which are citeable but way too over-specialized for any reader who just needs to look up what a torsor is.

I have now added pointer to

which comes closer, but still buries the simple idea under faithfully flat verbiage.

Do we maybe have a discrete group theory textbook that states the definition in a way useful for those readers who don’t already know it?

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJan 6th 2021

I don’t know about a textbook reference, but this paper by Breen cites the definition on page 2 here, and it doesn’t look too over-specialized for a quick look-up.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 6th 2021

But what I have in mind here is discussion of less than that: torsors over the point, preferably in discrete sets. Such as the default meaning when people speak of $\mathbb{Z}$-torsors. Such as in John Baez’s expository note but inside a citable textbook. Probably John wrote that note because such textbooks are not common.

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 7th 2021

Do such textbooks exist? I’ve just checked Artin, Lang, Dummit–Foote, Aluffi, and none of them mention torsors.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 7th 2021

Thanks for checking! Maybe not.

• CommentRowNumber8.
• CommentAuthorUlrik
• CommentTimeJan 8th 2021

Our Symmetry book mentions torsors (see current Sec. 4.6+4.7), and is meant to be an undergraduate textbook. But of course, the book is still far from finished, and we take as the main definition of a torsor that it’s any G-set merely equal to the principal G-torsor. (I don’t remember if we prove the equivalence with the standard definition yet, but it’ll be there in time—Ch. 4 is not very polished yet.)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

Thanks, that sounds promising! But I don’t see a book behind that link. Do you have a pointer to a pdf for me?

• CommentRowNumber10.
• CommentAuthorUlrik
• CommentTimeJan 8th 2021

Here you go — but as I said, it’s still quite preliminary.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

Okay, thanks. I have added the pointer to the list of references (here)

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeJan 22nd 2021

In a model theoretic context of definable sets, principal homogeneous spaces are studied in

• Anand Pillay, Remarks on Galois cohomology and definability, The Journal of Symbolic Logic 62:2 (1997) 487-492 doi