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added to Chern-Weil homomorphism the description of the construction of the refined CW homomorphism by differential functions built using the universal connection as described by Hopkins-Singer.
The letter is used in the first paragraph both for the invariant polynomial and for the total space of the principal bundle. I did not change it as I do not know which one to change to stay in accordance with other entries on the topic. Urs ?
Thanks. I have changed the invariant polynomial notation to “”.
Thanks!
I have moved the section to before the “refined” version, renamed to “The plain Chern-Weil homomorphism” (okay?) and instead added pointer to the actual reference Kobayashi-Nomizu 63
All right.
I only see now that the threads split, I had been replying, of course, to your message here.
L Probably it’s the hyphen bug at work, which keeps haunting the nLab.
Isn’t it that the category of the other thread is ’nLab’ rather than ’Latest Changes’ here?
Yes, David C is correct. Let me know if the two threads should be merged. I am not aware of any present hyphen bug :-).
I think there is still a problem with some links on the nLab not working, because hyphens that look the same have different character encodings. I stopped reporting that long ago, but if you have the energy, I will drop a message next time I encounter it.
Ah, yes, please do. We have a similar issue registered on the Technical TODO list (nlabmeta). I think people do it unintentionally, but if anyone is deliberately making a choice, I’d suggest to keep it simple and use the usual Ascii hyphen rather a unicode em or en dash :-).
I think the problem comes not so much from people making choices, but from some non-trivial transformation happening in the process of a) typing a hyphen into the source code, b) it being rendered (and maybe differently in bulk text and headlines?) and this rendered output c) being copy-and pasted into the next source.
But I’ll check.
added pointer to:
added pointer to
turns out that Weil’s unpublished note is available in his collected works, have added the pointers:
I finally realized that Cartan’s article exists in two different versions. Have now made the citation read as follows:
Henri Cartan, Section 7 of: _Cohomologie réelle d’un espace fibré principal différentiable. I : notions d’algèbre différentielle, algèbre de Weil d’un groupe de Lie _, Séminaire Henri Cartan, Volume 2 (1949-1950), Talk no. 19, 10 p. (numdam:SHC_1949-1950__2__A18_0)
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Henri Cartan, Section 7 of: Notions d’algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, in: Centre Belge de Recherches Mathématiques, Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950, Geroges Thon 1951 (GoogleBooks)
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(These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed.)
Funny how it goes:
Cartan gives prominently placed seminars about the idea, and publishes it in an on-topic book collection.
Three months later Chern gives a talk with quick reference to an unpublished and unavailable note by Weil, and henceforth Cartan’s idea is known as “Chern-Weil theory”. :-)
added pointer to Section 2 of
Have added more references, such as to the universal connections that Chern had been appealing to. Also pointers to these further reviews:
{#HopkinsSinger} Mike Hopkins, Isadore Singer, Section 3.3 of: Quadratic Functions in Geometry, Topology,and M-Theory J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)
Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Section 2.1 in: Cech Cocycles for Differential characteristic Classes (schreiber), Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735, euclid:1358950853, doi:10.1007/BF02104916)
Daniel Freed, Michael Hopkins, Chern-Weil forms and abstract homotopy theory, Bull. Amer. Math. Soc. 50 (2013), 431-468 (arXiv:1301.5959, doi:10.1090/S0273-0979-2013-01415-0)
If the group has no faithful finite-dimensional representation, how do we know there’s a finite-dimensional “level-” classifying space?
This is the Narasimhan–Ramanan theorem, which indeed assumes compactness.
But this theorem is not necessary to define the Chern–Weil homomorphism.
Indeed, Chapter XII in Kobayashi–Nomizu, which is already referenced in the article, constructs the Chern–Weil homomorphism in full generality without any assumptions on the Lie group G. The differential refinement also uses the computation of integral and real cohomology of the classifying space, which is insensitive to compactness because the inclusion of any maximal compact subgroup into a connected Lie group is a homotopy equivalence.
And if one is looking for a proof using universal connections, then the modern proof by Freed–Hopkins is much more elegant and also does not require G to be compact.
If you know how to generalise, then you should do so :-)
I was just pointing out one fairly obvious place in the argument on the page (mention of the fin. dim. level- class. space) where it seems compactness of was used. If that step can be gotten around by something in Freed–Hopkins, then we should use that, instead of what is currently there.
Okay, I added a new section with the modern construction, best if a few people take a look at it now to make sure my description is consistent.
Also, I believe Urs figured this out somewhere in his work, but he can probably supply a reference much faster than I can.
Sorry for the slow reaction, I am operating on stolen moments this long weekend.
But I am afraid I don’t have anything substantial to offer regarding the assumptions in the classical CW theory.
Our ambition in Chapter 7 (pp. 74) of The non-abelian character map was just to show that the classical construction (specifically in its Hopkins&Singer-incarnation) is an example of the character map on non-abelian cohomology.
Looking back at it, I see that we do assume compact , in order to be able to easily quote standard results. I’d be interested in seeing this generalized. (Back then I would have been eager to join you looking into this, now I am afraid that I am too busy on other questions.)
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