nLab Chern-Weil homomorphism -- references

Chern-Weil homomorphism

Original articles

The differential-geometric Chern-Weil homomorphism (evaluating curvature 2-forms of connections in invariant polynomials) first appears in print (Cartan's map) in:

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, May 1950 (numdam:SHC_1949-1950__2__A18_0)

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, Georges Thon 1951 (GoogleBooks, pdf)

reprinted in the appendix of:
- Victor Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, Springer 1999 (doi:10.1007/978-3-662-03992-2)

(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. The second one briefly attributes the construction to Weil, but without reference.)

and around equation (10) of:

Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

It is the independence of this construction under the choice of connection which Chern 50 attributes (below equation 10) to the unpublished

André Weil, Géométrie différentielle des espaces fibres, unpublished, item [1949e] in: André Weil Oeuvres Scientifiques / Collected Papers, vol. 1 (1926-1951), 422-436, Springer 2009 (ISBN:978-3-662-45256-1)

The proof is later recorded, in print, in: Chern 51, III.4, Kobayashi-Nomizu 63, XII, Thm 1.1.

But the main result of Chern 50 (later called the fundamental theorem in Chern 51, XII.6) is that this differential-geometric “Chern-Weil” construction is equivalent to the topological (homotopy theoretic) construction of pulling back the universal characteristic classes from the classifying space $B G$ along the classifying map of the given principal bundle.

This fundamental theorem is equation (15) in Chern 50 (equation 31 in Chern 51), using (quoting from the same page):

methods initiated by E. Cartan and recently developed with success by H. Cartan, Chevalley, Koszul, Leray, and Weil [13]

Here reference 13 is:

Jean-Louis Koszul, Homologie et cohomologie des algebres de Lie, Bull. Soc. Math. France vol. 78 (1950) pp. 65-127 (numdam:BSMF_1950__78__65_0)

More in detail, Chern’s proof of the fundamental theorem (Chern 50, (15), Chern 51, III (31)) uses:

the fact that invariant polynomials constitute the real cohomology of the classifying space, $inv(\mathfrak{g}) \simeq H^\bullet(B G)$ , which is later expanded on in:
- Chern 51, III.5
- Raoul Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)
Some authors later call this the “abstract Chern-Weil isomorphism”.
existence of universal connections for manifolds in bounded dimension (see here), which is later developed in:
- Mudumbai Narasimhan, Sundararaman Ramanan, Existence of Universal Connections, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 (jstor:2372896)
- Mudumbai Narasimhan, Sundararaman Ramanan, Existence of Universal Connections II, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 (jstor:2373211)
- Roger Schlafly, Universal connections, Invent Math 59, 59–65 (1980) (doi:10.1007/BF01390314)
- Roger Schlafly, Universal connections: the local problem, Pacific J. Math. Volume 98, Number 1 (1982), 157-171 (euclid:pjm/1102734394)

Review

Review of the Chern-Weil homomorphism:

Shiing-Shen Chern, Chapter III of: Topics in Differential Geometry, Institute for Advanced Study (1951) (pdf)
Shoshichi Kobayashi, Katsumi Nomizu, Chapter XII in: Foundations of Differential Geometry, Volume 1, Wiley 1963 (web, ISBN:9780471157335, Wikipedia)
Shiing-Shen Chern, James Simons, Section 2 of: Characteristic Forms and Geometric Invariants, Annals of Mathematics Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 (jstor:1971013)

(in the context of Chern-Simons forms)
John Milnor, Jim Stasheff, Appendix C of: Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229)
Eckhard Meinrenken, Section 5 of: Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)

(via equivariant de Rham cohomology)
Mike Hopkins, Isadore Singer, Section 3.3 of: Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. 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, Advances in Theoretical and Mathematical Physics 16 1 (2012) 149-250 [doi:10.1007/BF02104916, arXiv:1011.4735, euclid:1358950853]

(in generalization to principal ∞-bundles)
Daniel Freed, Michael Hopkins: Chern-Weil forms and abstract homotopy theory, Bull. Amer. Math. Soc. 50 (2013) 431-468 [doi:10.1090/S0273-0979-2013-01415-0, arXiv:1301.5959]

(using the stacky language of FSS 10)
Adel Rahman: Chern-Weil theory (2017 $[$ pdf]
Domenico Fiorenza, Hisham Sati, Urs Schreiber: Section 8 of: The Character Map in Non-Abelian Cohomology, World Scientific (2023) [doi:10.1142/13422]

(as a special case of the character map on generalized nonabelian cohomology)

With an eye towards applications in mathematical physics:

Mikio Nakahara, Chapter 11.1 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Gerd Rudolph, Matthias Schmidt, Chapter 4 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)