nLab Cartan's map

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Contents

Idea

Let GG be a compact Lie group and let XX be a smooth manifold equipped with a smooth action of GG.

If this is a free action, hence if XX is the total space X=PX = P of a GG-principal bundle P→BP \to B, then the Cartan map (or Cartan’s map or similar) is a quasi-isomorphism from the Cartan model for the equivariant de Rham cohomology of X=PX = P to the ordinary de Rham complex model for the ordinary de Rham cohomology of the base manifold BB:

((Ω •(P)⊗ℝ[{r a} a]) G,d dR+r aι v a)Cartan model ⟶≃ qiCartan's map (Ω dR •(B),d dR)de Rham complex ω⊗⟨−⟩ ↦ ω hor⊗⟨F ∇⟩ \array{ \overset{ \color{blue} \text{Cartan model} }{ \bigg( \Big( \Omega^\bullet\big(P \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \bigg) } & \overset{ \color{blue} \text{Cartan's map} }{ \overset{ \simeq_{qi} }{\longrightarrow} } & \overset{ \color{blue} \text{de Rham complex} }{ \Big( \Omega^\bullet_{dR}(B), \, d_{dR} \Big) } \\ \omega \otimes \langle - \rangle & \mapsto & \omega_{hor} \otimes \langle F_\nabla\rangle }

given by choosing an Ehresmann connection ∇\nabla on P→BP \to B and inserting its curvature form into the invariant polynomials ⟨−⟩\langle-\rangle (essentially the Chern-Weil homomorphism).

This is the dg-algebraic reflection (a model in rational homotopy theory under the fundamental theorem of dgc-algebraic rational homotopy theory) of the weak homotopy equivalence

P⫽G⟶≃ wheP/G=B P \sslash G \overset{\simeq_{whe}}{\longrightarrow} P/G \,=\, B

from the homotopy quotient/Borel construction to the plain quotient space, which holds for free actions (not otherwise, though).

A full proof is due to Guillemin-Sternberg 99, Chapter 5 (which reproduces Henri Cartan‘s original articles in its appendix). Review includes Meinrenken 06, around (20) and (30), Albin-Melrose 09, Theorem 11.1.

References

In equivariant de Rham cohomology

Due to Henri Cartan. Proof appears in

Review in:

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:


(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 BGB 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:

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

  1. the fact that invariant polynomials constitute the real cohomology of the classifying space, inv(𝔤)≃H •(BG)inv(\mathfrak{g}) \simeq H^\bullet(B G), which is later expanded on in:

    Some authors later call this the “abstract Chern-Weil isomorphism”.

  2. existence of universal connections for manifolds in bounded dimension (see here), which is later developed in:

Review

Review of the Chern-Weil homomorphism:

With an eye towards applications in mathematical physics:

See also:

Enhancement of the Chern-Weil homomorphism from ordinary cohomology-groups to dg-categories of ∞ \infty -local systems:

Last revised on October 30, 2020 at 09:42:43. See the history of this page for a list of all contributions to it.

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