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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 $P$ 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 “$\langle -\rangle$”.
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turns out that Weil’s unpublished note is available in his collected works, have added the pointers: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)
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